Act 1 of A Little Bit of Fluff by Walter W. Ellis. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Characters John Ayers, read by Thomas Peter. Bertram Tolle, his friend, read by Campbell Shelp. Nixon Trippett, inspector of claims for the motor bus company. Read by Chuck Williamson. Dr. Biglin, also for the motor bus company. Read by Todd. Pamela, Mrs. John Ayers, read by Emma Hatton. Mammy Scott, from the 500 Club. Read by T.J. Burns. Aunt Hannah, Mr. Tolle's aunt. Read by Abaii. Ursula, Mr. Tolle's maid. Read by Alicia Messiah. Chalmers, Mrs. Ayers maid. Read by Eva Davis. Set Directions. Read by Elizabeth Holland. A Little Bit of Fluff. Act 1. Scene. John Ayers flat in Bayswater. The scene represents a room in the well-to-do flat of Mr. John Ayers in the district of Bayswater. A door to the right leads into the bedroom, and another door left leads to the hall and street. There are two French windows at the back with balconies beyond. A fireplace right above door. Mirror on mantelpiece. Easy chair right. Table upright center above door with a pot of marguerites upon it. A writing desk up left center with telephone. A fancy table down left with papers on it. Telegraph boy knock and ring heard off left. Chalmers, a maid, enters at left with one telegram on solver, and crossing meets Pamela's center who is entered by door right. Pamela is a smart woman of 35, handsome and beautifully gowned. What is it, Chalmers? Seeing telegram. Oh! Telegram, madam. Thank you. Opens and reads, gives vent to a sigh of satisfaction. Ha! She thinks. Exit Chalmers left.  Chalmers enters with second telegram. What's that? Chalmers still holding solver. Another one, madam. Oh! Reads again. Yes. All right. Chalmers is going. Oh! Chalmers? I'm expecting three more. Telegrams, madam. Yes. Bring them to me directly they arrive. Very good, madam. Exits left. Pamela glances again at telegrams, and then going up places them carefully on table right center. Door slam is heard off left. Pamela surveys the room quickly and noticing her hat on table down left crosses over and conceals it with papers. Runs up to window left center and withdraws behind the window curtains. John Ayers enters door left. He looks a very smart and evening dress with coat and crush hat. He yawns and gazes sleepily around. Then crosses to armchair humming a tune and taking off coat, which he places over back of armchair, goes to mantelpiece over fireplace, and looks in mirror.  Pulls himself together, takes face from mantelpiece, places against his forehead, and exits down right. Slams the door after him. Pamela comes from hiding place and listens a door right. Then picks up John's coat, comes center, and searches the inside pocket, takes out letters but finds nothing incriminating, puts them back again. She pulls the sleeves of the coat out and sniffs twice and a long whole length of sleeve. Then pulls necklace out of side pocket. Oh, oh. She replaces necklace and puts coat on back of chair, left of table right. Coming to door right, she taps loudly on it. John, heard off irritably. What is it? Pamela repeats the knocking, then crosses to center. John is obviously changing his clothes and enters chest with morning trousers and braces showing. What is it? What the devil? Oh, it's you Pam. I didn't know you were home. Haven't you been to Folkestone? Of course I have. Mother wasn't well, so we came back yesterday. Yesterday? Oh, oh, oh. Strolls off right to get his waistcoat and jacket. Heard off. Did you sleep at, uh, at mother's last night? Pamela does not answer, but is apparently annoyed. Re-enter John, butting waistcoat.  Where did you sleep? Where did I sleep? I'm asking you. What a funny question to ask anyone. I slept at home. In there, of course. Obviously. Naturally. Whenever you adopt that innocent attitude, I always know you are telling me a willful lie. I couldn't tell you a lie if I tried. Do you remember that phrenologist we went to at Eastbourne? He told me I had an enormous bump of veracity. This is nothing to do with phrenology. I am to believe that you slept at home last night? Y-yes, of course. Why? I slept at home, too. Strange. We didn't meet. Yes, that is funny. I locked that bedroom door from half past eleven last night until nine o'clock this morning. Well, if you lock the bedroom door, how can you possibly expect me to sleep at home? Absurd. Getting into jacket. No, I'll tell you the whole facts of the case, Pam. We went to the opera last night. We? Yes, sir. My friend Tully and I. Tully had some tickets given him. And you came home together? Y-yes. And, uh, I slept at Tully's. What opera did you go to? John Braugh gestures. The, uh... The, uh... That's rather a silly question. No one ever goes to an opera and remembers anything about the performance. But the name of the opera? Oh. Oh. The name. You mean what the opera was called? Exactly. The, uh... The title? Yes. Well, uh... You know the opera where the girl comes on with a sewing machine? No. You know what I mean? A spinning wheel. Two long plates. Marguerite. Faust. That's it. Faust? Yes, Faust. With the devil in it. Business of putting fingers to forehead. And so you both went to see Faust? After that we came home. Crosses to left, laughing. Ha. I remember making a joke to Tully. Never mind the joke. Well, it was just then that I missed my latchkey. You missed your latchkey? And it was rather late to rouse Chalmers. So Tully offered me a shakedown at his place. And I stayed there. There's a good deal of Tully about it. But if you lost your latchkey, how did you get in just now? John smiling. Oh, I found the key afterwards. Well, give it to me. John hesitates. Give it to me, please. John obeys. She goes up to fireplace. While I pay the rent of the flat. Oh, don't say that. It isn't cricket, Pam, to throw the rent up in my face. After all, it was you who made me give up my office in the city. For the simplest reason you were making. I was making a profit of five pounds a week. And it was costing me another fifteen pounds to keep the office open. Coming down to John. Now look here. I have enough for both, so long as you do not work in the city. Well, I can't grab along on five pounds a week like some people. Must I remind you that I have been allowing you forty pounds a month? No. Excuse me, dear. It was agreed between us that my allowance should be fifteen pounds only. Telegraph knock and ring off left. I'm quite aware it was agreed, but you kept borrowing on account. Even now you are two years ahead with your money. As much as that. Two years. Ha! Time flies. But I shall pay it back. But let us keep to the point. Chalmers enters with three telegrams on Solver. About last night. For me? No, sir, for the mistress. Oh, er, Chalmers. Reading telegrams. Will you just knock at the flat next door and ask if Mr. Tully is at home? And if so, will he kindly look in here for a moment? Chalmers going. Very good, madam. Chalmers, Chalmers. Beckoning Chalmers to stop. To Pamela. You dare not do such a thing. Pamela to Chalmers. Do as I tell you, Chalmers. Exit Chalmers. John, as Chalmers is going off. Chalmers, cha, cha. Turns to Pamela. You are not going to show me up before my friends. There will be no showing up, John, if what you say is true. Moves up to Table Right Center. No, no, of course not. Moves to telephone. Besides, I don't suppose your friend Tully would give you away anyway. Men are such cunning brutes. John, with a burst which he checks instantly. Ha! Pamela looks round, then goes on reading telegram. At back, John, whispering into telephone. Give me Region 346. Region 346. Pamela, without turning. It's no use your phoning Mr. Tully. I should be bound to hear what you said. I was only trying to get him to come up, dear. Chalmers is quite capable of taking a message. John, rising and crossing to Pamela. Hang it all, Pam. Don't you believe what I've told you? Pamela, turning sharply to John. Not one word. Why not? This morning I sent a reply paid wire to your friends at Q. Harry Crombley? Yes.  This is his reply. Hands wire. Read it. Read it out, please. John takes wire gingerly. Reads. Yes. John stayed here last night. Aside. Silly owl. Well? Dear old Harry. I expect you thought she would be worrying about me. He is very thoughtful, is Harry. Give Swire back. Wait. I also sent a wire to your friend Blackiston at Kensington. Asking the same question. His reply. Hending second wire to John. John amazed. Read it. Read that out, please. John takes wire. Reads. Yes. John stayed here last night. Pauses. One hour. I can tell how this happened. Gives back wire. Wait.  I sent three other wires to Mr. Marshall, Gus, Dan Hope, and Drailing. They all replied that you stayed with them. Read for yourself. Hands wires to John, which he does not take. I can explain it all, dear.  They're all jolly good friends. Yes. They must all be very, very good friends or else they must have a shocking opinion of your habits. I can explain everything. I believe you could explain the Tower of London away, but you can't have slept in six different beds in one night unless you were a sleep-walker. I still maintain that I slept last night at Tully's. We shall see. Places telegrams on Table Right Centre. Enter Chalmers. Mr. Tully, madam. Holds store open until Tully is on, then exits, closing the door. Tully is rather a spare man with drooping mustache and rather sanctimonious and miserable looking. He enters and stands just above the small table down left, nervously twisting his hands. On Tully's entrance John makes a dive for Tully. I say, Tully, didn't I? Pamela catches John by right arm and pulls him down right, advances to Tully. How do you do, Mr. Tully? Shakes hands. How do you do, Mrs. Ayers? Morning, John. Morning, Tully. Good morning. Ah, you see he wasn't up when I left this morning, lazy beggar. Pamela, centre to Tully. I have to apologise, Mr. Tully, for bringing you out. Oh, not at all. But we, er, John and I are in a little difficulty, and if you could see your way to answer a few questions, it would be doing us a great favour, and it might save both of us life-long misery. My wife won't believe that I... Will you be quiet? You're breaking down the one slender thread that holds our married life together. I want Mr. Tully's version of last night without your assistance. Turning to Tully. Now, may I ask, did you have anyone staying with you last night at the flat? Tully shaking his head. No, not to my knowledge. John is pointing to himself frantically. No one stayed at your place at all? Tully seeing John. Oh, er... With a gulp. John stayed there. Pamela turns quickly, almost catches John pointing to himself. John makes a dive for book on table right centre and turns pages over quickly. Pamela turns again to Tully. But just now you said no one stayed there. We... we never count John as anyone. John rubs hands with glee. No, dear. I'm nobody. Pamela, give John a freezing look. Then again to Tully. Now, would you mind telling me how you passed the evening? Last night. Looking at John. Last night. Well, we er... let me think. We er... yes. John points to window. We went out. And where did you go, might I ask? John is gesticulating with one hand on his chest and open mouthed as an opera. Tully, felling to interpret John's signals. It's rather difficult to remember offhand. One night is so very much like another. Try to think. John still gesticulating and open mouthed. I think we must have been in a boat on the serpentine. Pamela turns quickly. John goes up to table right centre and smells marguerites. Tully very embarrassed. These are very beautiful flowers, dear. Did these come from Covent Garden? Please don't interrupt. Oh, now I remember. It's about last night you wanted to know. Last night. Oh, last night we went to Covent to... to... to the opera. John nods and smiles at Tully. John had tickets given to him. John annoyed. No, no, those tickets were given me to give to you. They were at present from Mr. Baxter. Oh, I didn't quite understand. I must write and think Mr. Baxter. Baxter! Baxter! Baxter, Baxter. Who is Mr. Baxter? Don't you know, dear? Do you? Of course. He's Mr. Baxter. Baxter. Pamela to Tully. And did you enjoy the opera? Not very much. I really prefer the singing down at our chapel. What opera was it? I don't think I noticed. Didn't notice? Of course not, dear. No decent person ever dies. It's bad form. Silence to Tully. Haven't you any idea of the name of the opera? Not for the moment, er... er. John points to Pot of Marguerite's. I er Daisy... Daisy D'Adriens. I can't say I have ever heard of an opera of the name of Daisy D'Adriens. John is now holding a plate made from his handkerchief to the back of his head. Tully watching John. Was it something to do with er... something hanging from the back of the head? And you can't remember the name of the opera. Not for the moment. John points to Marguerite's again. Are you sure it wasn't Daisy or Daisy's or Marguerite er...? John nods his head. Marguerite er... er Faust of course. Yes, dear. Faust of course. Pamela turns quickly to John. John just as quickly turns his back. Now are you satisfied? And after you left the opera? To Tully. We came home. What induced John to sleep at your place, seeing your door is next to ours? John signalling Key and Door, and then lost. Oh, he couldn't find his keyhole. Pamela turns quickly, round to John. No, no dear. We simply went to the opera and saw Daisy er... er Faust, I mean. Came out, had a drink. I told Tully I couldn't find my key. I suppose he thought I said keyhole. He offered me a shakedown, and I stayed there. And I think such a clear explanation ought to satisfy anyone. Yes, I suppose so. John going to fireplace. Then everything is quite in order. Very satisfied. Tully sighs. Pamela doubtfully. Y-yes, yes, yes, except... Picking up John's coat with left hand. Could either of you explain this? John, coming down to Pamela. That's my coat. No, taking necklace from pocket with right hand, and holding it up. This! Pause. John and Tully both amazed. Oh, that's nothing to do with me. What is it, dear? What is it? A pearl necklace. Turning to Tully. I suppose you don't wear pearl necklaces, do you, Mr. Tully? No, no. I can tell you all about that, dear. I saw that in a sharp window, and I picked it up very cheaply. I'm sure it's a bargain. And who was it intended for, may I ask? Who should I buy pearl necklaces for? For me! For me, I suppose! Boiling with rage, and throwing coat up to setee centre. Of course. Naturally. Ask Tully. Tully goes to chair by telephone left centre. Stands perfectly still, unnerved. John below table right centre stands blinking and looking into space. Pamela goes to table down left, uncovers her hat, picks it up, goes toward door right as she passes John. Goes to door right, opens door. Exits door right, bangs door after her. Tully, flopping into chair he is standing by. Oh, I'm all of a tremble. John crosses up to setee, puts coat on setee, then up to Tully.  If you had told me yesterday that you could lie like that, I should never have believed you. John, coming down right. You did your share very well. What's going to happen now? She'll probably pack up and go home to her mother's. Tully, rising and crossing to John. John, where did you go last night? I took a little friend out to dinner, and then we went out to the palace. And after that we had supper at the Five Hundred Club. We watched them dancing and had a dancer to ourselves. But it's perfectly absurd if a man can't have a little innocent enjoyment and a couple of dances with a little bit of fluff with all this absurd fuss. But the hour? At the club we kept it up a bit late, that's all. We had breakfast at Jimmy Dawson's flat and cooked bacon and eggs. Won't you promise never to do such a thing again? John, crossing to left. I promise never to poach an egg in an opera hat again. I can't possibly live without some relaxation now and then. But must you really go out and about with little bits of fluff? Most certainly, if I want to. What have you got to say to that? Oh, dear, dear, dear. Everything would have been all right only.  I'm sure Marguerite in Plates was perfectly clear. If you had only said Faust without any hesitation, everything would have been all right. But it's so risky they play a different opera every night like Heaven Garden. I know they do. I wonder what they did play. Where's the newspaper? Looking round for paper. Seeing paper on table right below door. Crosses over, gets paper. Here it is. Crosses to telly. Now, if my luck's in, they played Faust. Both look at paper together. Last night. Here we are. Theatras. Last night. Heaven Garden. Pictures. John tears the paper in two. Gives half to telly. Here, tear that up. Handing other half. And this bit too. Get rid of it somehow. Telly tears paper in pieces and puts bits in handkerchief pocket. John crosses to door right. How do you think you will get out of this? John. Crossing center. Quite easily. Tacked and diplomacy. Offering his head to telly. Feel that bump. They say I have a bigger bump of tact than Lloyd George. Telly. Feeling head. How I say. Both stand to attention as Pamela re-enters right. Wearing a hat and carrying a small suitcase. The pearl necklace is also in her hand. She crosses over to door right. Not looking at either of the men.  She stops as John speaks. Pam. Pam. Pamela. Coming down to table left and placing suitcase on table. I am going home to mothers. You'll hear from her later. And probably the solicitors. Well, Pam. I think you're awfully silly. And after I've bought you a pearl necklace too. I doubt very much if the necklace was intended for me. Oh, Tully, did you hear that? The only woman in the world I have ever loved. Sinking into chair left of table right center. Tully, who has been standing watching very nervously by table with telephone left center, crosses to Pamela. I don't think you ought to say such things, Mrs. Ayers. Pamela shrugged shoulders and turns back on him. He crosses to John. Do leave us for a few moments, John. I... But Tully, I... I'll put it all right. But Tully, I... John is persuaded to go off right by Tully. John exits muttering. Tully shuts door. Tully crossing to Pamela. One moment, Mrs. Ayers. You know I feel somehow that I am to blame for all of this. I don't want to pose as a hyper-religious man. But everyone says I'm very good and I wouldn't deceive you for the world. I'm sure that necklace was intended for you. Pamela, opening suitcase. Well, in any case, I value my feelings at something more than a... a one-and-eleven-penny pearl necklace. Drops necklace into case and shuts it. Really, I think you are doing John an injustice. I don't think you quite understand his little ways. I understand as much as is fit for me to understand. No, really. I know John doesn't behave in a conventional manner as a rule, but he is quite harmless. Pamela, raging. Up to Tully, then crossing down right. Harmless! Harmless! A man who can sleep in six different beds in one night. Harmless! Throwing arms up on last harmless. Six impossible. It would be a record. Pamela, up to table, right center. Picks up bundle of telegrams. Handsome to Tully. Read for yourself. John stayed here, cue. John stayed here, Bloomsbury. John stayed Barnes. Kensington, Bloomsbury to cue. Cue to Kensington. Kensington to Barnes. It couldn't be done in the time. Oh, I can quite understand this. It's all John's friends. All anxious to shield him from the fury of his wife. I beg your pardon. Snatches telegrams from Tully.  John has such a host of good friends. There isn't one who wouldn't lay down his life for him. Why, John's one of the best in the world. Pamela, crossing to left by table down left. I am quite a broad-minded woman, Mr. Tully. I don't expect men to be angels, but there is a limit to everything. Tully, crossing to Pamela. I quite agree with you in that, Mrs. Ayers, but as a broad-minded woman, you must see that a man like John wants a little relaxation. And there's really no harm if he does go out to dinner occasionally with... what was it he called them? Little pieces, no little bits of fluff. What? What? Tully recoils. Pamela follows him up to center. You expect me to sit at home while my husband goes out with little bits of fluff? Tully pulls out handkerchief with pieces of paper. Well, you know, it's a term, a joke. Tries to conceal pieces of paper with his feet. I'm surprised, Mr. Tully, that your mission teaching should have put such ideas into your head. Crossing to table, picking up case. As little bits of fluff. Good day. Exit Pamela down left, banging door behind her. The front door is then heard to slam. Tully stands looking into space for a time, then proceeds to pick up torn paper. John, cautiously peeping in door right. What are you doing? Sleeping up Covent Garden. John, crossing to right center. Has she gone? Yes, I'm so sorry, John. John, crosses to Tully.  Did you really sleep in six different beds? Oh, don't be silly. Is there any chance of her returning? Of course she'll come back. She does this sort of thing about every fortnight. Do you sleep out as often as that? No. She does it with the idea that I shall go and fetch her back. Well, why don't you? Because once I do that my authority will be gone.  Things have never gone so badly as this before. I think the pearl necklace did it, don't you? John, suddenly aroused. Ah, the necklace. Where is it? Where is it? Have you got it? No. No. Why should I have it? Then where is it? Where is it? The necklace. Looks about wildly for the necklace. Look for it. Don't stand like an anemic camel. Look for it. Tully, jumping about in a silly fashion. Where? Where? Everywhere. All over the place. Paras is on the floor. Look for it. Both look about for the necklace. Ah, and maybe under the table. They both dive under the table from opposite ends. Their heads collide. They both come up holding their heads in pain. Can't you see where you are going? I can only see stars. Your head's like iron. But where's the necklace? Moving armchair from right of table right center to below table about three feet. That's the question. I've got it. Where? No, not the necklace. I've got an idea. I expect Mrs. Ayers took it. You practically gave it to her, didn't you? You think she took it? Yes, I remember now while I was talking to her just now. I saw her drop it into her bag. Are you sure? Crossing to left. Doesn't matter. You can get it back from her. John, still looking about for necklace. She'll never part with it. She loves jewelry. Well, you can easily buy another. Putting hand in pocket. I'll lend you the one in eleven pence. One and eleven pence? One and eleven pence? Do you know that necklace is worth five hundred pounds? Five hundred pounds? Yes. It was lent to little Mami Scott by the Raja of Changpu. She took a fancy to the necklace, and he lent it to her to wear just for the evening. There was a big crush as we came out of the club last night, and Mami asked me to put the necklace in my pocket for safety's sake as the clasp was broken, which I did, of course. Apparently we both forgot all about it. She'll be in an awful stew. She promised faithfully to return the necklace to the Raja today. Oh, dear, dear, dear. Oh, damn, damn, damn! What can I do? What can I say? What will Mami think of me? Tully is twiddling the chair right center about. Oh, don't fiddle about with that chair. Tully stops footling. Is there no way of getting it back from Mrs. Ayers? I tell you she'll never part with it, and she may not be home for several days, possibly a week. In the meantime the Raja will be clamoring for his pearls. I shall be branded as a—well, there's no telling what it may lead to. Great heavens!  Crossing to chair left, downstage. Couldn't you tell Miss Fluffy Scott you've lost it and buy her another? Didn't you hear me say that necklace cost five hundred pounds? Tully, twirling chair round on one leg. Yes, that is awkward. Oh, do put that chair down. Advancing to Tully. Tully sits center. Couldn't you borrow the money? John, crossing sits down left.  It's easy enough I had a letter from someone only this morning offering to lend me any sum from ten pounds to a ten thousand pounds without any security. He enclosed his photograph. Such a nice, kind, honest open face. You innocent lamb. Well, I suppose if I can't give it back I shall have to find the money. You will? Oh, it is a fix. Biting his nails. John, rises, goes up to Tully. I say, Tully, I suppose you don't happen to have five hundred that you don't want? Not that I don't want. Poor old Tully. You never seem to have any money. I don't know what you live on.  You know, John, if I had the money I couldn't refuse you. You do know that, don't you, John? John, patting Tully on back. Of course I do, dear old Tully. Dear old Tully. Comes down left. Tully, rising.  I do wish you'd call me Bertram. You know when anyone calls me Bertram? I feel I could do any mortal thing in the world for them. Well, you get me out of this hole, and I'll call you Bertram till I bust. Sits down left. Will you? I think I know where you could get the money. Comes down to John. John, rising suddenly. Where? Where? Sit down. John sits. Keep calm. Dick Turner thinks the world of you. Yes, I know, but he hasn't much money. I know, I know, but he was in a bus accident last Friday, and he's claiming five hundred pounds compensation from the motor bus company. He'll never get it. Oh, I think he will. In fact, it's nearly settled. And if you approach him in the matter, I feel sure he would lend you the five hundred pounds. But I was in that bus with him coming from Q. That's right, coming from Q. And if Dick Turner could get five hundred? I'm positive I could. Well, I'm sure he's going to get it. But there was scarcely any damage done. I didn't receive a scratch, neither did Dick Turner.  Still you can't always tell at the time of the accident. Injuries sometimes develop afterwards. Business of drawing patterns on carpet with foot. John, rises and crosses slowly right. Yes, of course. Especially after you've seen your solicitor. Er, I... Don't talk to me. My brain's working. You know, John, in all cases of bus accidents the bus companies have to pay out according to what the doctors think. The question is to make the doctors think. Why should Dick Turner get five hundred, and I get nothing? I suppose he was really injured. Don't talk, don't talk. I've got the most wonderful brain. Hand to forehead. Have you? Yes. Feel that bump. Tully obeys. Oh, did you get that under the table?  It's a natural bump. Excited. It's all so simple. It's wonderful how I get myself out of every difficulty. Now, will you run down to the doctors for me? Going up to telephone. The last block of flats, you know. Doctor Green. John, looking through telephone book for number. Yes, that's it. Ask him to call at once. John, you're not going to pretend to the doctor that you are ill. Now, don't ask any questions. Oh, no, John. Working fingers along back of armchair. I couldn't do a thing like that. It's not fair. It's not honest. Protest in action against a suggestion until John says, Bertram, when a broad smile comes across his face. John, rising and crossing to Tully. Bertram, Bertram. Tully, giggles effectively. Oh, John. Crossing to door left. Oh, John. Giggles. Oh, John. Giggles to loft or left. Quick exit. John goes to telephone at phone. Give me a region to one, four, three, six quickly, Miss, please. Yes. Hello. Hello. Are you the motor omnibus company? Yes. Yes. Mrs. John Ayres speaking. Mrs. John Ayres. Yes.  My husband. And he's very ill indeed. Yes. Eh? Oh, speak up. I can't hear a damned word you're saying. Hand over phone for a second. Oh, he didn't notice it at the time. He has witnesses to prove everything. Eh, I can't hear. Oh, you'll send your inspector round to look into it. Eh, you'll send your inspector round to look into it. Oh, very good. But don't send him immediately as the patient is asleep. Eh, yes. In about half an hour's time. We're quite close to your depot. We're quite close to your depot. Number 13, St. Mark's Mansions. Yes. Very well. Thank you. Goodbye. Puts up receiver. He looks round and takes off jacket. Chalmers enters left. What is it? What is it? A lady to see you, sir. She smiles. To see me? What are you laughing at? Chalmers pulls herself together. Miss Scott, I think she said. Good Lord. Oh, I'm busy. Dressing for breakfast. Not at home. Crossing to door right. Chalmers is going. Wait. I'd better see her. Open store with right hand, holding it open. Sure in here. Chalmers in doubt. In there, sir? Pointing to door right. John pointing back into room with left hand. No, here. Here. Exit down right. Exit Chalmers door left. Chalmers shows in Mamie Scott. She is a girl about twenty-seven. Petite, but pretty. Dressed with many fur-belows and other fluffy things. She looks around as she enters with a swagger air, sees Chalmers smiling, freezes her with a look. Chalmers straightens herself and goes off door left, with nose and air. Mamie looks round room, humming or singing a tune. Places parasol and sati at back, and comes down center. Enter John undoing collar and tie. Hello, Jack.  In a playful temper. But you mustn't come here, really. Why not? I thought you said the cat was away at Folkstone. And please don't call my wife a cat. Exit John into room right. Mamie with an elaborate curtsy. Oh, I beg the cat's pardon. Sweeping round room, she sees photograph on table left center. Say, Jack, whose picture is this? John, spoken off. Which one? This one, here, by the telephone. Oh, that is my wife. Your wife? Some girl! She's not the sort of first wife I'd pick out if I was going to be her second. Why not? Looks too darned healthy. I'd have to wait too long for you. Enter John door right. She's come home unexpectedly. Mamie jumps in terror and makes a dive for vanity bag she has placed on table left center. Jack! Oh, it's all right. She's out just now. Phew! You did give me a fright. But it's true. She is home all the same. Exit into room right. Well, come out here and talk to me. I won't keep you long. John, spoken off. I can't. I'm only half dressed. Well, I'll come in there. Crossing to door right. No, no, this is a bedroom. I'm not afraid of bedrooms. Give me a minute. Just a minute. Come out as you are. I'm not particular. I won't be two ticks. Righto. Sits in armchair down right center and commences to powder her face. I say, Jack, do you know that you didn't give me back the necklace last night? John enters and creeps off again. I say, Jacko, do you know you didn't give me back that necklace last night? The words slightly smothered by using powder puff on mouth. Enter John in dressing gown. I say, Mamie, that hat does suit you. You look awfully sweet. You go on, Jack. You're the champion long distance kidder in the universe. John, crossing to left of Mamie. But I mean it. It suits you awfully. Oh, awfully. Mockingly. Do you know you didn't give me back the necklace last night? You know, the pearl necklace? John, hesitating. No, I know I didn't. We both forgot all about it, didn't we? We did. Both laugh amusedly, thinking it a great joke. I left it in my coat. And I left the coat at the club. Mamie, rising, alarmed. Jack, it isn't lost. John, pressing her gently into chair. Sit down. Sit down and don't worry. It can't be lost. If it is, I'll buy you another, that's all. Five hundred pounds. Yes, I can't forget that. But it's a mere flea-bite to me. Jack, you don't understand. The Raja looks on it as an heirloom. He wouldn't part with it for the world. That's why I wanted to wear it. It was such a cute idea. But I promise faithfully to return it to the Raja today. Can't you make some excuse? Can I? Have you been to the club? No, I can't possibly go down there for a day or two. For a particular reason. Oh, I do hope it isn't lost. Can't you phone? Oh, yes, I did phone. But the club phone seems to be out of order. Let's torn it. What will the Raja think of me? Now don't worry. If you'll only wait, everything will be all right. In any case, if it is lost, I'll buy you another exactly like it. I can't say more, can I? You really mean that? Of course I do. I never break my word. I'm even going to get some money today. Out of accidents. I mean, in case of accidents. Now listen, I'm awfully glad you've called. My wife's left me. Mamie, rising and throwing arms around John's neck. Jack, darling. John, gently but firmly disengaging her arms. Yes, but only for a little while. And I want you to do me a favor. Of course I will, Jack. I'm expecting a man here presently to examine me. To examine you? Yes, I'm very ill, you know. I was in a bus accident the other day and things have been getting worse. Poor old Jack. I-I am sorry. Pulling his face to her with hand under his chin. But you don't look ill. John, turning face again to front. No, I'm one of those who bear up to the last. Now listen, when this man calls, I want you to pretend that I'm bad. Of course I am bad, but while he is here I'm sure to be a little worse. Mrs. Ayers, that is me, I've been speaking to him on the phone and naturally when he comes, he'll expect to see me, that is Mrs. Ayers. I-I see, you don't understand. Mamie, very sympathetically. Jack, dear, you haven't injured your head, have you? No, it's quite alright. Nothing to do, but, uh, don't say you're my wife. Just pat me on the head now and then and moan, poor John, you understand? Yes, I understand. Poor John. But say, this is a spoof. You're not really ill, Jack, are you? Of course not. Oh, uh, yes I am, but don't worry, I'm going to get better. Just poor John, laid on thick. I see. Poor John. Crosses up to Mantlepiece, removes hat, and tidies her hair at glass. Tully enters hurriedly, sees Mamie, makes a bolt for door left, John catches him by coat, and pulls him back. It's alright, John. As he enters. John, to Mamie. Excuse me a moment. Tully, to John. Dr. Green was out, but they'll send him round directly he comes back. He's out on the case about a poor little woman.  Whispers in John's ear. Who? Then allowed. Both both doing well. Well, that's more than we can say. Now I must go and finish dressing, or rather undressing. Sees Mamie.  Tully alarmed. Is she, is she fast? Is she a hussy? You like her immensely. Come on. Tully in terror. No, no, I couldn't. I've never spoken to anyone like that in my life. John, taking hold of Tully.  Oh no, no, no. What would they think of me down at the mission? Besides, I wouldn't know what to say to her. Why not? I've never met a fluff. You get hold of the most extraordinary expressions. Calling to Mamie. Mamie, let me introduce you to a very old chum of mine, Mr. Bertram Tully. Miss Mamie Scott. John crosses to door right. Mamie crosses over to Tully. Mamie, taking Tully's hand.  Pulls a long face. Talked to him, Mamie. Wonderful flow of conversation. I shan't be long. Exit John door right. Mamie beckons Tully with head and eyes, and edges down to armchair right. Sits. Tully, very nervous.  Do you ever go, or go, go, no. Tries again. You like to, to, to, no. Has another try. It's, it's wonderful how the fine weather lasts. Mamie, very amused all the time. Ripping, isn't it? Yes, isn't it? Are you married? No, I regret to say. A bit of luck in store for someone. Oh, thank you. I expect you have a gay old time. Tully, twiddling his fingers down his leg. No, not so very gay. I know.  No, I assure you all my people are most respectable.  Tully, mystified. A dark horse? You know, one of those outsiders who comes up with a rush on the rails at the last minute and wins by a short head. Do you get me? I don't quite understand what you mean.  She raises her dress and picks a piece of fluff from the hem, blows it into space. No, I don't go. Cease may be exposing a deal of leg. He is very embarrassed, wipes his forehead with handkerchief. No, I don't go at all. Rising and backing away from her. What do you do to amuse yourself? I go to chapel on Wednesdays and Saturdays.  And I attend the mission on Tuesdays and Fridays. Again down to her and seeing leg, stumbles backwards and wiping forehead with handkerchief, keeps up this business, doing a sort of skating waltz. Did they teach you that rag time down at the mission? Jumping up. I like your drunken step. I must get a hold of that. Catches Tully and forces him round the room as if dancing a rag time. Mamie sings and dances as well. Da-da-da-da-da-da-da. Tully breaks away from Mamie and rushes to door right and knocking on door feverishly. John, John. John, spoken off. What is it? What is it? I'm being tempted. Well, stick it.  Tully rushes up to window right, then down again to armchair. Mamie follows him up and down on left side of table.  He succumbs, sits gingerly on front of chair. Mamie sits on arm of chair and puts right arm around his neck. Tully snatches it away nervously. Now tell me, what's this mission for? It's for the poor people. Sees Mamie's exposed ankle, turns away nervously. We give them musical evenings to keep them out of the public houses. I play the flute. You do what? I play the flute. Oh, help. Oh, they like it. Bell hurt outside door left. Mamie, starting. I wonder what that is. Goes to door right, calling to John. Jack. Jack. There's a ring at the bell. Do you think it can be the cat? Tully, rising and going up center. A cat wouldn't ring the bell, Shirley. Enter John dressed in pajamas and dressing gown from door right. She couldn't possibly be here yet a while, whatever happened. Enter Chalmers left with card on solver and crosses to John. And please don't call my wife a cat. I'm sorry. Sits in armchair. John, reading card. Good. Show him in at once. Exit Chalmers door left. It's Mr. Nixon Trippett. Mr. How much? Mr. Nixon Trippett, the inspector from the motor bus company. The man I told you about is going to examine me. Sit down and ask him to wait. Say I shan't be long. Tully, perplexed. What have we got to do? Mamie will tell you all about it. Exit John right. Now listen here, birdie bright eyes. I'll stop it. Down center. All we've got to do is keep saying poor John. Poor John. I'm to pretend I'm John's wife. Poor John. Mamie, rising. What do you mean? Nixon Trippett enters, shan't by Chalmers. He is a very ugly man of forty, dressed in frock coat and wearing spectacles. He is almost shabby Gentile. Chalmers retires. Tully nervously retreats from Mamie and seeing Trippett advances with uncertainty. Tully to Trippett. Poor John. I mean Mr. Ayers won't be a minute. Will you take a seat? Brings chair down from table left center and places it center in a line with armchair right and small chair left. Trippett places hat on table left center coming down to chair center. Thank you sir. All sit. Mamie in armchair. Trippett chair center. Tully chair left. Thank you sir. Removes gloves. John groans loudly off right. All rise slowly and simultaneously with eyes fixed on door right then sit again. John groans again very loudly. All rise. Mamie gets behind armchair. Trippett drags chair up to table left center keeping eyes on door right all the time. Tully stands by chair left gazing at door right. John enters groaning from door right. He is in pajamas with the blanket wrapped around him. Mamie assists him into armchair. John groaning all the time. Trippett advancing timidly to John. Mr. Ayers. Are you the injured person? Ah. Ah. Ah. Trippett again advancing cautiously. Might I ask if you are the injured person? Don't I look like it. Do you think I'm doing this to be funny? Mamie patting John's head. Poor John. Poor John. Trippett glares at Tully. Then over to him. Do you think it would be advisable for me to call another day? No, it's all right. I can stick it. He's awfully brave, you know. You ought to have kept in bed. Going to table right center. It would have been better if I hadn't let you know I was coming. Places gloves on table. Oh, no it wouldn't. Poor John. Poor John. Trippett gazes first at Mamie and then at Tully. Well now, may I ask a few questions that will help me make out my report? Pulls notebook and pencil out and looks round for something to write on. Yes, ask as many as you like. To Tully. Bring that table over for Mr. Sticks and Triplets. Not Sticks and Triplets, Nixon Trippett. Tully brings small table from down left and places it on left of armchair. Trippett brings chair from left of table right center and places it on left of small table. I beg your pardon. Tully moves round to back of armchair on left of Mamie. Trippett sits and preparing to write a notebook. Now then, Mr. John Ayers, isn't it? Yes, John Ayers. Trippett writes. Oh, my back. Poor John. Poor John. Trippett looks at Tully then writing again. Tell me, Mr. Ayers, are you married? Of course. Absent mindedly, taking Tully's hand and mistake for Mamie's and placing it by his face.  Yes, of course. Any family? Yes. No. Trippett writing in book. Yes and no. What is your height? Four feet four and a bit. Age? Forty-two. Chest measurement? Forty-two, two. Ever been vaccinated? Well, my godfather was Mr. Tully and my godmother was... I said, vaccinated. Oh, I beg. Oh no, he never catches anything. Trippett writing again. No, Mr. Ayers. You said you were traveling in one of the company's buses when this accident took place? Of course I was. Last Friday, coming from Q... Poor John. Poor John. John, to Mamie. I'm afraid he won't have me with you much longer, darling. Oh. Tully leaning over and looking into Trippett's face. It's hard to see him struck down like this. Trippett rises slightly annoyed. Tully retreats to window right, then out of window and in by window left. Starts back when he sees Trippett still watching and pointing at him with his pencil. Trippett, resuming. Could you tell me who was inside the bus or describe the people in any way? There was Mr. Richard Turner. Yes, we have acknowledged his claim. A check for 500 was sent him this morning. John, jumping up and leaning over to Trippett. What? Recovering himself and sitting again. Oh, it's only a spouse in that cell. Oh, I am bad. Could you describe anyone else who was in the bus?  Trippett writes, Tully sidles around to back of armchair again. Did these people make any statement or pass any remark? When the collision occurred, someone said it was like being out at the front. The stout lady said that. Mammy turns away smiling. Tully shows surprise and disgust. No, Mr. Trippett, the soldier. Poor John. Tully, who is now on left of Trippett, pats Trippett's head. Poor John. Trippett turns on Tully, very annoyed, then back to John. Now, may I ask, why didn't you report this at the time? How could I? I was too stunned, I suppose. I quite appreciate what you say, Mr. Ayers. But it's one of our rules that you should have lodged your complaint at the time the accident occurred. I suppose if a man was killed stone dead, he ought to leave his name and address. If he knew where he was going. But in this case, the situation is quite different. The Mr. Turner you mentioned just now informed us that he was the only passenger injured in the accident and the other occupants of the bus rather bear out his statement. How does he know? He couldn't see my back. But you have no witnesses. Shrugs. No witnesses indeed. Oh yes I have.  My friend Tully here was sitting next to me in the bus the whole of the time. Tully almost collapses. Oh indeed, indeed. Tully, quickly and very agitated, down to left of Trippett. But I make no claim indeed. I don't know. I make no claim. I make no claim at all. I don't think I have your name and address. Mr. Bertram Josiah Tully. Number 14, St. Mark's Mansions. Trippett, riding, then to Tully. And you yourself were not injured? Not at present. I mean not a scratch. Poor John. Poor John. Trippett looks at John. How do you account for that, if he was sitting next to you, Mr. Ayers? When the collision came I fell forward on the two soldiers. They had been in training for months and were as hard as nails and naturally I sprained my back. Well Mr. Tully here shot forward right on top of the fat old woman. And not hurt? She was enormously fat. Trippett, to Tully, who was now up again and behind armchair. And did you pass any comment at the time? No, but the woman did. I think I said, oh, dear, dear, dear. Of course I shall have to take proceedings against your company if it costs me every penny my wife's got. I mean, that I've got. I don't think that will be necessary, Mr. Ayers. Our company is a very generous one. And although we cannot acknowledge an illegal obligation, we like to treat our passengers as fairly as we can. I'm sure you do. We like to make friends. You have a friendly face, Mr. Trippett. We want to see you riding in our buses again.  They're such a pretty color. Mamie digs Tully in ribs. And if this matter could be settled at once, I'm sure you would be most satisfied. I'm sure I should. No, speaking without prejudice, what sum of money do you fancy would compensate you? John, to Mamie. What do you think, dear? Tully signalling five hundred on fingers. You see, there'll be all the doctor's expenses, a terrible loss of time and money, probably funeral expenses. Oh, and bear it. John, to Trippett. Suppose we say five hundred, without prejudice, as you say. Trippett raises his eyebrows. I'm afraid that's quite out of the question. Do you realize what five hundred means? I'm afraid I couldn't entertain anything like that. But I'll tell you what I will do. If you like to settle the matter offhand now, and give me your signature, I'll pay down at once the sum of... Taking note from pocket and presenting it to John. Five pounds. Don't be absurd. A five-pound Bank of England note, Mr. Ayers. You could go away for a nice little holiday on a five-pound note. John, rises, anger rising. Really, I think you've come here to insult me. Certainly not, Mr. Ayers. And without prejudice, I think you would be well advised to accept my offer. John, up to Trippett.  Trippett rises. Amy, comforting John. Don't upset yourself, John. Why doesn't he offer me a bag of nuts or a balloon? Trippett, getting gloves from table right center. I'm sorry you look at things in that light, Mr. Ayers. Tully, during this speech, gets Trippett's hat and holds it perched high up on his right hand. With his other hand, he holds the door left open. All I can do is hand in my report. Going left. Tony's doctor will come and examine you, and the matter will be out of my hands. Noxon to Tully, sees hat, takes it, bows to Tully, goes to door left, turns. I wish you good day, sir. To John. And I hope you'll get better soon. Exit left. Trippett, followed by Tully. I don't think I shall. Five pounds for a broken back. Tully, rushing on from door left. It's all right, John. Mrs. Ayers has come back. What? Your wife, Jack. Hide me. Mamie tries to get under table right center. John pulls her back. No, that way. Hide her, Tully. Tully wandering aimlessly about. John pushes him up to window right center. Mamie gathers up hat, et cetera, and goes window right center. Tully gets Mamie's parasol from Satie, and John pushes him out of window. Quick behind those curtains. I take those things away. Throwing Mamie's gloves after Tully. John gets into easy tear quickly, with blankets still round him. Growns. Enter Pamela, door left. Pamela, seeing John alarmed. John? John! I didn't expect to find you like this. And I didn't expect to see you back quite so soon. I've come to say I'm sorry. Mother has seen that necklace you gave me. Placing her bag on table down center. And she says it's worth five hundred pounds. Mother knows. But it is valuable. Of course it is. Instead of spending my money on riotous living, I've been spending it on you. How good of you. But do tell me, what has happened? Don't be alarmed. You know I was in a bus accident the other day. You were not hurt. Things have developed since. I think they are going to compensate me. Pamela, joyfully. Then you are not really ill? Goes down below table. John, rising. That depends. I'm going into that bedroom. Pointing right. And I'm not coming out until that bus company gives me five hundred pounds. Not if I've got to lie there for a month. Oh, don't say that John. I know what I'm doing. I'll teach them to offer me a balloon. I mean, a five pound nut. No, not nut. Note. Now please go and get the bed ready. Leading Pamela to door right. But John? Do go to oblige me. I'm expecting the doctor here at any minute. Pushes Pamela off door right. John signals to Tully, who drags Mamie out by the hand. They come down a few steps. Quick, quick as you can. Pamela re-enters. Tully and Mamie get behind curtains again quickly. But John, it may be weeks and weeks before these people pay out the money. John, holding blanket high up to obscure Pamela's view of the room. Now do please, do as I ask you. If the doctor finds me out of bed, it will ruin me. Pamela, going back into room right. Oh, very well. Exit Pamela. John signals and Tully drags Mamie across to door left. Go on, hurry up, hurry up. They are nearly across to door when Pamela re-enters. But John, it's just occurred to me. Tully and Mamie turn and Pamela faces them. John falls over blanket down right. Tully still holds Mamie's hand. In his other hand he has Mamie's parasol. Oh, I don't think you've met before. I don't think we have. Let me introduce you. This is my wife. Pointing to Pamela. And this is... Pointing to Mamie. This is... This is Mrs. Tully. Tully drops Mamie's hand, thunderstruck. What? Pamela doubtfully. Mrs. Tully? Yes, he was married secretly a week ago. Tully, boiling with rage. Oh, I say. Mamie turns her ring round to look like wedding ring and holds hand up conspicuously. I'm sorry to let the cat out of the bag, old man, but it can't be helped. Tully rushes across stage in front of table and arm chair, with Mamie's sunshade raised in a threatening manner. John, John! John, kneeling to Tully, pleadingly. Bertram, Bertram! Tully's face relaxes and develops into a broad smile. Oh, John, John! Giggles. Pamela and Mamie shake hands, center. Curtain. End of Act One.


Hello and welcome to our video summarising everything you need to know about the songs of our cells volume 1 part 2 anthology. My name is Barbara and this is the 15th and final of our 15 part video series where we examine all poems within the songs of our cells volume 1 part 2 anthology. If you want to see the other videos make sure you check out our channel and look through all other 14 videos which will be available for you to read, analyse and walk through when it comes to analysing each poem within this collection. So let's finish off with the final poem in this collection which is the Kraken by Alfred Lord Tennyson. Now this poem is a 15 line variant of an Italian or Petrarchan sonnet. This poem can be divided into two sections that can then be divided further. The Kraken is made up of one octave, in other words a set of 8 lines which can be divided into two sets of four lines and one final set of seven lines. This poem does follow a consistent rhyme scheme of A B A B C D D C E F E G G F E. Now the Kraken describes the slumbering bulk of this creature. It's eventual rise to the surface of the sea resulting in death and it's a really apocalyptic image that's used within the poem. Now this poem begins with the speaker describing how deeply one would have to look in the ocean to find the Kraken. It's in a place that no human can truly go. He continues on to state that the Kraken is not the king of this place but just another feature however it's an enduring feature of this part of the ocean. This is due to the fact that the Kraken has been sleeping an ancient sleep. This become home or rather the Kraken has become home for all of the creatures of the deep and in the final lines of the poem it's revealed that eventually the Kraken will wake up and it will bring all of its power to man and angels alike and then die when it reaches the surface. Now when we look at lines one to four the poem starts with the speaker describing a portion of the sea that's far from the reach or for understanding of humankind. He describes a place that's unfathomable to the human eye and can only be described in the grandest of terms. The speaker then lays out the details of the habitat of the Kraken and the life it lives. The place that this Kraken resides in is located below or under the layer of the upper deep so this is really the deepest, deepest part of the sea. The portion of the sea that's successful to us that is referred to as the upper section is only the start there's so much more to come and as the speaker takes the reader deeper and deeper into the ocean he passes this abysmal sea. The Kraken doesn't reside anywhere close to the surface in fact he's far far beneath all of these layers of the ocean. Once the speaker has taken his readers down through the ocean he comes upon the Kraken and this Kraken is in a deep sleep. It's an estate. This is a state rather that he's inhabited for an innumerable swath of years. His sleep is end to quote from the poem ancient, dreamless and uninvaded. Now we as readers might get the feeling now that the speaker has taken the narrative to this place that his sleep might be coming to an end. We appear perhaps maybe to be invading the Kraken's realm and in the final line of this section the speaker starts his description of what this place is and the besets within it. Now when we look at lines five to eight we get a greater understanding of this world which is so deep below. We're taken to a place that no human has ever been or will ever go. In the realm of the Kraken there's no sun. He is much too deep below the waves for the sun to touch him. Now the next lines in this section emphasise the extended period of time that this beast has been sleeping. He's remained there in the same spot for so long that it's millennia of growth surrounding him. There are huge sponges and sea plants from every form of swell which are all around him and the Kraken is so deep within the ocean that there's only sickly light about him. In other words there's only a very tiny small sliver of light. It's also in the meagre lighting that life has made its home. The Kraken has slept for so long in the same place that creatures of the sea have come to live in its wondrous grotto which is a shortened form of grotto and within the unnumbered and secret parts of its body. This Kraken's mass is becoming nothing more than another part of the ocean on which other creatures have made their home. Now in lines 9 to 15 there's a change that's predicted in the realm of the Kraken. Now this section states the speaker is emphasising that the beast has resided in this place for an endless number of days and there's enormous polypy or growths that winnow or cover its giant arms. Now one might imagine that this Kraken is covered with perhaps a green layer of small green growths. The speaker also describes into quote the hath he lain for ages. The speaker then goes on further to state that the Kraken will continue to lie, batting upon huge sea worms whilst he sleeps at least for a while longer. Now the final three lines of this poem give a hint of what will happen when the time comes for the Kraken to wake up. This will only happen when, to quote from the poem, the latter fire shall heat the deep. The Kraken will wake up and be driven up from the ocean floor either by necessity or by a newly reinvigorated passion. He will, to quote, once by man and by the angels be seen again. However the world in its entirety will see the beast and know its greatness. However we get the sense that it's almost this apocalyptic occasion that might happen which could also result in the end of the world. Now this one act of power and assertion as the Kraken surfaces from the depths of the ocean will come crashing down as, and to quote from the poem, roaring he shall rise and on the surface die. Now this entire poem can be interpreted as a metaphor for any type of impassioned revolution. The Kraken can be seen as a contingent of people who have perhaps a harbored hatred or discontentment or oppressed fear that finally reaches a breaking point and then there's a pure unencumbered emotion that comes roaring to the surface but when it reaches the clear air it then dies. So even the best intentions can fall short when they are met with the challenges of reality and that's one way you can interpret the meaning of this poem. So that's all. If you found this video useful I'd really appreciate it if you gave it a big thumbs up but also consider subscribing to this channel. Also make sure you visit our website www.firstreadtutor.com there you will find a wealth of model answers essays as well as online courses which you can take in a very short period of time to improve your knowledge of both this anthology but also other areas of English especially if you're doing this in very short timed conditions. Thank you so much for listening and make sure you check out our other 14 videos which look at other poems with this anthology.


C'est mon plaisir d'introduire Renko Lajik, qui a sa PSG-99 de Oxford, et presque depuis, il a été professeur de l'Université de Ravwick. Il a fait beaucoup de contributions dans le quartier, donc il a rejeté l'automate avec ce fameux papier sur l'LTL avec 3 quantifiers, et l'algorithme paritien de paritien de paritien de paritien. Et beaucoup de travail aussi sur ce système de vectorisation avec le state, sur le pétrinette, sur le branching de ce type. Il a eu le meilleur papier à l'arrivée de ce dernier stock, pour le breakthrough de l'arrivée dans la complexité. Il a été une longue durée de l'arrivée, que l'arrivée pour le pétrinette n'était pas expérimentaire. L'arrivée est galactique, je ne sais pas où c'est. L'année dernière, ils ont prouvé qu'il n'était pas expérimentaire. Avec ça, aujourd'hui, il parle d'un homme, je ne le sais pas très bien, mais on verra si Fenkel est correct ou non. Merci beaucoup Paul, merci pour l'introduction, et pour l'inutation de parler ici à FSTTCS, c'est très bon d'être à IIT Bombay et d'être en Indie, et c'est très bon que Paul, qui m'a introduit, est de Cachan, et beaucoup de travail que j'ai faits dans les années, est lié à mes collaborations avec plusieurs personnes à Cachan, Stéphane Demri et d'autres. J'ai un peu d'un titre inusuel, qui est qu'Alan est correct, et c'est des exemples de plusieurs conjectures sur les variants traditionnels. Ce sera plus clair, ce qui est le sens de ce titre, mais qu'est-ce qu'il y a sur ce titre? Je pensais que depuis demain, dans l'Université Générale, je pensais que ce serait bien de montrer une map de l'Union Européenne, à la place où mes coauthors et moi viennent. C'est donc basé sur le travail joint avec Jérôme Leroux, qui est ici, ou qui sera ici, et puis avec Philippe Mazowiecki. Jérôme est à Bordeaux, et Philippe Mazowiecki est maintenant à MPI en Zarbrewkin, et Slavomir Lasotta et Ocec Czervinsky de Warsaw. Ok, donc, c'est ce qu'on parle des systèmes traditionnels, mais qu'est-ce que les systèmes traditionnels? Je ne vais pas vraiment les définir, mais j'aimerais clarifier, qu'il y a plusieurs formes qui sont vraiment tous équivalents, ou essentiellement les mêmes. Donc, nous avons des systèmes traditionnels, nous avons des systèmes traditionnels avec des States, des vaches abbreviées, et des vaches abbreviées avec des double S. Ensuite, nous avons des pétrinettes, et puis, peut-être, c'est un terme non-deterministe avec des machines minstres, avec aucun test. Donc, tous ces formes sont basically the same. Il y a des features de la translation, qui sont très facilement translatées, d'une à l'autre. Il y a des features que l'on doit être un peu attentionnées avec la translation, comme quand on va d'autres systèmes traditionnels avec des States, d'autres systèmes traditionnels avec des States, et la dimension, typiquement, augmente par 3, par ajouter 3, et ainsi. Mais, ne faisons pas trop peur sur ces systèmes, parce qu'ils ne sont pas vraiment rélevés pour ce talk. Même si on regarde, et on fixe les dimensions, mais dans ces cas, on regarde les vaches avec double S. Donc, n'hésitez pas. Maintenant, pourquoi nous étudions tous ces systèmes, sur les pétrinettes, ou sur les vaches? Bien, d'un côté, ils ont beaucoup d'applications pour modéliser un analyse de hardware, de softwares, des processus chemiques, des processus biologiques, des processus business, etc. Donc, c'est une bonne raison. L'autre est que, en fait, les pétrinettes arrivent dans différents secteurs, de théorique, de computer science, en étudiant des objets complètement différents. Ils sont très proches de l'héologique. Ils arrivent à l'intérieur des databases. Ils arrivent aussi à l'intérieur de concurrence, etc. Donc, c'est vraiment un modèle fondamental de concurrence. C'est, par contre, un modèle fondamental de concurrence qui a été créé depuis plusieurs années. Il y a beaucoup de outils qui ont été développés et appliqués, etc. Donc, un sujet central dans le théorique de VAS est le problème de l'achat d'achat. C'est un problème de décision. Il s'agit de deux configurations. Une configuration initiale et une configuration finale. Dans la terminologie de la pétrinette, ce sont les marquants initiales et les marquants finales. Est-ce qu'il s'agit de la configuration initiale pour la configuration finale? Ce problème a été fameux pour ses difficultés pendant les années. Qu'est-ce que ça veut dire? En fait, même avant, le problème de décision de l'achat d'achat en 1976, Lipton a montré que c'est difficile pour l'exponentialité. Et puis, il y avait des proofs partiaux. Je ne suis pas vraiment sûr de ce qu'un proof partiaux est. Mais il y avait des proofs partiaux et c'est vraiment un proof complet. Le premier temps où le problème a été décider en 1981 par le maire. Et puis, un an plus tard, c'était un papier stock en 1981, je crois. Mais un an plus tard, le stock Cosseraggio a donné des proofs partiaux. Mais quand je dis un proof simplifié, n'oubliez pas que ce proof est toujours difficile à comprendre. Et, pour exemple, un whole book a été écrit juste présenté par Rotenauer, juste présenté par Cosseraggio. Donc, dix ans plus tard, Lambert a développé le proof Cosseraggio et a utilisé un peu plus de travail et a utilisé pour montrer des résultats plus basés sur ce que Cosseraggio a fait. Et puis, dans une série de papiers, 2009, 2011, 2012 par Jérôme Le Roux, il a fait beaucoup plus de progress. Et en fait, comparé avec un algorithme de Cosseraggio pour le problème de richability pour nous, comparé avec cet algorithme, qui est un algorithme très compliqué, Jérôme a en fait étendu un algorithme très trivial, mais c'était basé sur un travail complexe qui a établi que, si la richability s'étendu, il y a une définition définie dans le algorithme de presse qui sépare l'initiel de la configuration finie. Donc, l'algorithme trivial consiste en deux algorithmes semi-algorithmes qui circulent en parallèle. L'une s'étendu et l'autre s'étendu c'est pour un formulaire de presse qui donne l'invariant. Donc, c'était un bon progrès. Et puis, en 2015, 2019, Leroux et Schmitz ont réussi d'obtenir l'accarmanie, qui, en un sens, c'est la plus plus non primitive d'une function. Ils ont réussi d'obtenir, en termes de function d'accarmanie, le temps d'obtenir d'un algorithmme pour VAS. Et, en fait, l'obtenir qu'ils ont obtenu est primitive récursive pour la dimension fixe. Plus précisément, si la dimension est D, c'est en FD, plus un petit constat. Et puis, en 2019, après beaucoup de années, en relation à le travail de Lipton, en 1976, ce qu'on a réussi d'obtenir, c'est d'obtenir un plus bas qui est en termes d'un tower d'exponential. Donc, ça montre que le problème n'est pas élémentaire. Maintenant, selon votre point de vue, vous pensez, ah, c'est tout résoluant maintenant parce qu'on a un très bas bas et l'oppère bas est galactique. Mais, en fait, pour les personnes qui sont dans ces choses, un énorme gaffe reste entre un tower d'exponentials et la function d'accarmanie. Donc, en fait, en fait, on ne comprend pas le problème d'exponential pour la pétrinette d'aujourd'hui. Donc, ce que je vais faire aujourd'hui n'est pas exactement ce n'est pas élémentaire bas bas mais un peu, je pense, des observations d'observations que et un peu plus de travail qui ont eu cet effort. Et j'espère que je vais vous intéresser dans la pétrinette d'aujourd'hui et j'espère que je vais aussi mettre un peu de lumière sur comment nous avons obtenu la pétrinette d'aujourd'hui qui, je pense, est intéressante et ça va me donner de retour au titre de la discussion. Donc, j'ai dit que je ne vais pas définir la pétrinette d'aujourd'hui mais ici, c'est une présentation possible de la pétrinette de la pétrinette donc, il y a des variables x et y ici juste deux variables, x et y et ils sont des numéros naturels ok? Donc, qu'est-ce que ce programme fait? Il fait x equals x plus 1 pardonne-moi, je vais utiliser ce genre de notation x plus equals 1 donc, c'est, on commence à 0 donc, quand on va regarder des programmes comme ça l'idée c'est que tous les variables ont commencé à 0 et on est intéressé à savoir si le programme peut exécuter et terminer avec tous les variables contre 0 donc, c'est comme les initiales et les finales configurations consistent de tous les 0s dans les variables ok? et c'est sans loss de généralité donc, je vais me dire ce programme fait donc, premièrement, il juste x plus 1 et ensuite il répète un nombre non-deterministe un nombre de fois peut-être 0, peut-être 1, 2 fois peut-être beaucoup de fois x equals x minus 1 et y equals y plus 2 et ensuite il répète un nombre de fois y minus 1 et x equals x plus 1 donc, decrement y by 1 et increment x by 1 et ensuite, il répète donc, qu'est-ce que l'exemple fait ici? bien, dans les deux dimensions parce qu'il y a deux variables, x et y on a commencé à 0, 0 et ensuite, x equals x plus 1 nous sommes ici à 1, 0 et ensuite ce qui peut arriver est qu'on arrive à 0, 2 et puis 2, 0 et donc donc, après des paires de ces loops vous voyez c'est basically doublant x c'est doublant x et et en en poursuivant y et celui-ci est transférer la valeur de la vague du y à x donc, après les zigs-ags on peut en arriver à 2 pour power n après doublant la valeur initiale qui était 1 n times on peut arriver à 2 pour power n Et puis, les restes n'ont des paires de loops qui font l'opposite. Elles peuvent halver la valeur de la fin de temps. C'est la partie de la ronde que l'on a détruit dans la ronde. Nous pouvons donc rentrer vers où nous avons commencé. Et ensuite, nous prenons une de x et nous sommes en arrière à l'origine. C'est un exemple de ronde de 0,0 à 0,0 de ce programme. On a un petit programme qui s'appelle Linear in N. Mais ce ronde est exponentiellement long. Ces loops sont répétées souvent exponentiellement. C'est assez intéressant. Il semble qu'une petite vache peut avoir une très longue ronde exponentielle, déjà en dimension 2. Est-ce vrai que ce programme, le plus long ronde de 0,0 à 0,0 est exponentiellement long ? Est-ce vrai ? Qu'est-ce que tu penses ? Paul est correct. Ce n'est pas vrai parce que ces loops, comme je l'ai dit, n'ont pas besoin de rentrer tout à l'heure. Elles peuvent tous s'éteindre. Elles peuvent tous exécuter 0 fois. Et ensuite, nous avons juste x et x qui sont incrementés ici et décrémentés là-bas. Et c'est comme un ronde trivial. C'est très court. En fait, il s'agit d'un exemple de flatvas. Qu'est-ce que c'est un flatvas ? C'est un flatvas avec des loops non nestées. J'espère que c'est clair que ce programme n'avait pas de loops non nestées. Flatvas sont une classe importante pour nous. Ils arrivent en pratique, c'est plus facile d'analyser. C'était un flatvas. En fait, il se trouve que si la dimension est juste 1 ou 2, si la dimension est moins qu'à 2, et si ces constants, comme juste 1 et 2 ici, les constants qui sont ajoutés et subtractés aux counters, si ils sont mis en unerie, le problème de richability peut être soulevé en NL, c'est en NL pour les dimensions 1 et 2, et en fait, les langues les plus courtes, lesquelles se trouvent à la richability, c'est un polynomial. Et cela requiert un travail pour montrer que, pour la dimension 2, cela a été fait dans un papier en 2016. Et pour la dimension 1, cela a été fait dans un papier en 1975. Quand ces constants, comme 1 et 2 ici, peuvent être plus grandes et succincts, puis, la richability, c'est en NL pour les dimensions 1, c'est par une reduction straightforward du problème du sub-sum, par exemple, et les langues les plus courtes de la richabilité exponentielle, c'est très facile de voir. Alors, qu'est-ce que la question naturelle d'arriver de cette table? Bien, qu'est-ce que sur la dimension 3? Ok? Et flat-vas avec les constants dans la dimension 3? Et cela a été un problème ouvert pour plusieurs années, et vous savez, nous avons conjecturé que la complexité NL pour la complexité et un peu de polynomial sur les langues les plus courtes de la dimension 3 n'est pas le cas. C'est mon premier contre-example que je vais vous présenter aujourd'hui. Et puis je vous présenterai 2 plus. J'ai 3 contre-examples pour y aller. C'est le premier. C'est plutôt une simple équation que nous vous savez, nous sommes habitués d'une school primaire peut-être où quand nous multipliez ces fractions ok? Ce que nous avons, c'est juste N plus 1. Je veux dire que c'est N plus 1 sur 1, mais vous savez, tout sort de cancelations d'exception pour N plus 1 et 1. Et ici, nous avons N plus 1 factorial divided by N factorial. Ok? So, what is, how is that equation going to help us? Well, we can actually kind of write it as a program, which is very similar to the program that I showed you before. But now we have 3 variables x, y and z. Ok? And hopefully a picture will help. So, now I'm kind of looking at a run of this program in three-dimensional space, where I have, you know, x and y and z. So, at the beginning I set x and z to 1, and then I, some number of times I increment x and z. Or in other words, at the beginning what I do is I set x and z to an arbitrary positive integer. Let's call it capital M. Ok? So the beginning, what this run looks like, it goes from the origin to M in x and z, but y stays at 0. So this is what it does in that first phase. And then how do I implement that equation from then on? Well, what I then do, this kind of looks complicated, but what I then do is starting from this point where x and z are M and y is 0, I kind of zigzag between, if you like, between the floor and the floor is the x-z plane and this wall, which is the y-z plane. So I zigzag between them, keeping z constant at M. So z stays at M. But then I zigzag, how do I zigzag? Well, I zigzag by repeatedly essentially multiplying multiplying x multiplying x by by these fractions. So my first zigzag multiply, you know, it keeps decrementing x by N and incrementing y by N plus 1. It does that many times and then it transfers the value back from y back to x and then the next zigzag does the multiplication by N over N minus 1 and so on. And then where I end up where I end up at is I end up back on the floor so y is 0, x is now M times N plus 1 because I multiply by all those fractions and so in the end I have multiplied by N plus 1. But z is still M, z is constant, it's still equal to M. And then in the last phase what I do is I go back from there to the origin but not at this slope 1-1 but at the slope N plus 1-1. What have I shown? Why is this interesting? It's interesting because the only way in which I can start with all variable 0 and end up with all variable 0 is by having a perfect picture like this. Why is that? These loops, as we know they don't have to execute as many times as they possibly can and these multiplications where the fractions don't have to be exact because the loops are nondeterministic but all these fractions are greater than 1. And so if any loop if there is any remainder when dividing and multiplying by these numbers if any loop doesn't execute completely we'll end up with x being strictly smaller than M times N plus 1. And then we're not going to be able to go back to the origin using this slope N plus 1-1 at the end. So what this construction gives us is a flat vase and in fact it's a linear path scheme for those who know so it's a special kind of flat vase in dimension 3 where the shortest runs witnessing reachability long, y exponentially long because this value capital M it has to be essentially divisible by the least common multiple of 1, 2, 3, 4, 5, 6 up to N because all these multiplications all the multiplications by the fractions can have to be exact and we multiply that capital M by these fractions one by one ok? So it's a nice example which shows that that our conjecture about polynomially long witnesses of reachability for flat vase in dimensions 3 and above is wrong, it's false so we disproved that and unfortunately and also we can we can do some further work and the reachability problem for flat vase is actually NP hard but the best dimension there that we can manage at the moment is 7 I think rather than 3 so we still don't know the complexity status for for this for this problem in dimensions 3, 4, 5, 6 I think ok? so that's my first example let me move on to vase so I'm going to kind of drop the flatness restriction and consider vase so specifically look at this table I'll tell you something that's known about vase and coverability and reachability runs and fix dimensions versus arbitrary dimensions so this is my second topic so here what we have is shortest coverability run what is a coverability run so I told you that that what I was interested in here is runs that start with all variables being 0 and finish with all variables being 0 this corresponds to to the reachability problem but the coverability problem which is easier for vase corresponds to just having runs that start with all variables being 0 but we don't care what values we have at the end so the only important thing is that the end of the program is reached so those are the runs that I call coverability runs so where it's not important to reach the origin at the end but just to finish so how long can its shortest coverability runs be well back in 76 the same work that I already cited by Lipton it shows that if the dimension of the vase is not fixed then actually doubly exponentially long runs may be needed even for even for coverability and Rakov showed the corresponding upper bound and in fact those two papers together established that as far as coverability runs shortest coverability runs are concerned they are exactly doubly exponential for general vase for shortest reachability runs the work of Le Rouen Schmitz and our work from this year show the shortest the headline results of those works are about the complexity of the reachability problem but actually if you look a bit further into them they show bounds about lengths of shortest reachability runs and they are again Akhermanian and non elementary but what about vase in fixed dimension so if the dimension is fixed then what do we know about coverability runs well in fact Rossier and Yann back in 1986 they showed that in fact singly exponential runs suffice if we're interested in coverability if we don't care whether we get all zeros so to speak at the end but for reachability runs has been known I mean how short how short for vase in fixed dimension can these runs be and the natural conjecture has been to well at least to explore whether they are perhaps also exponential I mean there was no evidence to suggest that in fixed dimensional vase shortest reachability runs need to be more than exponentially long in other words perhaps all the dimension is the only parameter that requires these high bounds that we have when the dimension is not fixed but again this turns out not to be the case so I'm going to show you an example where already in dimension 4 so vase in dimension 4 has I'll give you a family of vases in dimension 4 where the shortest reachability runs are doubly exponentially long ok already in dimension 4 now to present you that looking numbers at first the previous one my first countre example was in terms of that n plus 1 factorial divided by n factorial now it gets a bit a bit more hairy but actually you can look at this in detail but I'll tell you what's important about these numbers so what we have here is n plus 1 equations so a bunch of fractions raised to some powers and multiplied equals some fraction and then again some powers of fractions multiplied equals some fractions and so on and the last one is quite small and quite simple and the previous one and so on so what is n plus 1 here we have 2 to 2n plus 1 divided by 2 to 2n here we have the the opposite fraction so 1 over that and that same fraction repeats here and the powers are 2 to n 2 to n minus 1 and down to 2 to 1, 2 to 0 so in fact all of these powers sum up to 2 to n minus 1 so what we get is just the original fraction and the rest are the same so each of these equations is easy to see and then what we actually do to get the equation we really want to work with is we multiply all of these equations so you can think of you can think of kind of the columns here that have the same exponents the exponent here is 2 to n minus 1 2 to n minus 1 and this column is 2 to n minus 2 and so on here it's 2 to 1 and here it's 2 to 0 those are the exponents so when we multiply all of those equations together what we get is basically it's a product consisting I mean consisting of n plus 1 terms so i range from 0 up to n each is some fraction raised to the power 2 to n minus i and then all of them I mean all of them multiply together that product ends up being equal as some further fraction SN over RN and what is important is that all of these numbers SNi, RNi, SN and RN they are exponentially large in n I mean they are all of the form 2 to 2n plus 2 to something multiplied some small number of times and so on so they are all exponentially in n and they are raised to these exponential powers but somehow the product of all those fractions on the left is just a single fraction exponential number over an exponential number and moreover what is significant is that each of these fractions is greater than 1 and also they are ordered so as i range from 0 up to n these fractions get progressively smaller but they are all greater than 1 and of course SN over RN is also greater than 1 so it's a fascinating fact that numbers like this exist for all n so these are indeed numbers that are just a little bit larger than 1 raised to these huge powers and then the end result is a simple fraction exponential divided by exponential ok so how are those numbers going to help us with constructing a kind of a nasty example involving four dimensional vase well the vase is quite simple in fact and as before we are just going to kind of program this equation that's all that we are doing programming this equation and we are going to have four variables x, y, z and w which is to say the the vase is four dimensional so what will this program do at the start it will initialize x to some positive multiple of RN what is RN, it's this denominator here so x initially will be some positive multiple of RN and the z will count that positive multiple so x will be equal to z times RN after this initial stage and then this four loop it's not actually kind of in the vase strictly speaking it's syntactic sugar it's a macro so what it means is that what it's body is to be inline n plus 1 times for the corresponding values of i but forget about that if you want so this four loop goes from 0 to n it literally runs through this product and then what we are going to be doing is multiplying x repeatedly by all these fractions so how are we going to do that well we use that pair of loops with x and y so here we repeatedly subtract the denominator from x and add the numerator to y and here we transfer the value of y back to x so those two loops should do one multiplication by a fraction s and i divided by r and i we use w to count the loop through that's the inner loop that runs from 1 up to 2 to n minus i because we need to multiply with the same fraction 2 to n minus i times we add to w w is initially 0 we add to w 2 to n minus i and then whenever we multiply by a single fraction we decrement w and at the end what we do is at the end we repeatedly subtract sn the numerator from x and 1 from z so remember that x was initialised by some number of times z times rn when then we multiply that x by all of these guys what we should get according to this equation is what it's exactly sn times z and in this last loop we kind of test that's the case and the point is that since at the end we need to have 0s in all the variables that test that test indeed checks whether x is equal to z times sn now can anybody see some kind of really suspicious things here this looks like well at first sight maybe it's promising but it also kind of looks dodgy so so what is what is a concern here yeah so this is I mean that is an issue it's not such a big problem because these constants rn, rni, sni and so on you know they're a little bit complicated but they can be computed fairly easily and hardcoded into the program another concern is that these loops they're just repeating non deterministically many times there's no guarantee how many times they will repeat but like in the previous example that we've seen if any of them repeats for fewer times then it possibly can and indeed if any of the multiplications by the fractions are not exact if there are any remainders that will necessarily cause the final value of x just before that final loop to be strictly smaller than sn times z and then this cannot decrease x and z down to zero but there is one more thing that's really worrying here which is that I told you that the variable w is used to count in a loop that the outer loop goes from 0 to n but the inner loop goes from 1 up to 2 to n minus i but there there was like no test that w here ends up being decremented to zero I mean we never tested that so how can this possibly work so the beautiful thing here is that this relies on this ordering of the fractions and we start by multiplying with the largest fraction and then a slightly smaller fraction and then a slightly smaller fraction and so on so indeed what can happen in this program is that we don't multiply by the first fraction as many times as we can we kind of save some w to be used to multiply by the second fraction than is prescribed and so on but the thing is that the second and third and so on fractions are smaller so if that happens that can only result in a smaller value of x at the end than is required so in fact for this program to run from all variables 0 and terminate with all variables 0 the only way it can do that is that it kind of perfectly perfectly computes this product of fractions and in turn that is only possible that is only possible if the initial value of of x is divisible by a doubly exponentially large number, why doubly exponential because if you look at just the kind of what happens when i is equal to 0 we are multiplying by this fraction an exponential number of times ok and so the initial value of x has to be divisible by rn0 power 2 to n which is a doubly exponentially large quantity so this vass indeed has it's a 4 dimensional vass and it's shortest reachability witnesses are doubly exponential ok now that completes my second example and my third one is going to be for a slightly different kind of system and that's again I'm afraid in this area there are multiple names for the same thing push down vass or grammar vass there are other names too so how can we think of those systems they can be written down as programs like this one except we have also calls and returns to procedures that's one way to think about those systems so we have a push down stack so I'll give you a small example of such a program to kind of warm up imagine that and this will also explain why the difference between unary and binary encoded constants is not relevant when we work with grammar vass so imagine that I want to add this binary number these are binary digits bk-1 bk-2 up to b1 b0 to x to my counter x so this may be in a succinctly given grammar vass well what I can do is I can actually instead of doing that I can call this special procedure p0 and p0 what does it do well it adds this binary digit b0 to x and then it calls p1 twice and then what p1 does is it adds b1 to x and calls p2 twice and so on it adds the most significant binary digit to bk-1 so it's a kind of an easy example but you see that one can do quite a lot when one has a push down stack in conjunction with being a vass now I'll show you another example but before that let me try to summarize what is known for grammar vass for grammar vass I'll focus on dimension 1 so when there is only one variable so these are kind of unbelievably simple systems in some sense there's only one natural value variable and a push down stack I mean what could be easier than that so in dimension 1 the coverability problem is known to be next pace so Leroux Sutra and Totske show that in iCalp 15 and it's known to be p-space hard and we can't close this gap between p-space and x-space the reference here is to the MSC thesis of a smart guy at Warsaw Julius Straszinski who worked on this with several of us and obtained this p-space hardness as his master's result in Warsaw so this is known for the coverability problem for the reachability problem and this is the problem that has been you know whose tackling has been advocated by Alain Frankel for many years this is still open we don't know whether the reachability problem is even decidable all we have is this p-space in terms of the complexity we don't know nothing else about that for dimension 1 for kind of arbitrary dimension it turns out that reachability is reducible to coverability interestingly but at the cost of one further increasing the dimension by adding one more variable and at FSTTCS also back in 2011 maybe somebody remembers where FSTTCS in 2011 was Atting and Ganti showed that when the grammars or when the behavior of the stack is restricted to finite index a finite index grammar is where so a grammar of index k is where every word can be derived using at most k non-terminals along the way ok so if the grammars are restricted to being finite index then actually these systems are and they are kind of not difficult but very nice and insightful translations between these systems and vast hierarchical zero test and decidability of reachability for the latter has been shown by Reinhardt in his habilitation thesis and by by Bonnet in his PhD thesis but has never been publish and I am aware of maybe three four five people who have ever read this proof of decidability so the rest of us kind of I mean we we know and we like those colleagues and we believe them but you know I'm not it's really difficult to read that that stuff but without the finite index restriction we know nothing about the you know whether the problem is decidable it's conjectured to be decidable so so that's kind of an embarrassing state of affairs regarding our knowledge of grammar grammar vast but back to showing you an example this is a simple grammar vast it's just one procedure q so what does it do this is a non deterministic choice so we either increment x call q twice decrement x in return or we subtract 2 to power n and we add 2 to power n in return remember that the values of the variables are restricted to be non negative so this this decrement of 2 to n can only succeed if the current value of x is 2 to n or larger ok so if kind of in terms of pictures if we think of runs of this grammar vast we get trees like this where you know we start at the top and then we traverse the tree and we add all the numbers as we go along but at every point the current sum has to be non negative I mean those are basically runs about what one dimensional grammar vast kind of do so for this particular grammar vast it's runs can be depicted as trees of this kind where we kind of do plus 1, plus 1, plus 1, plus 1 and then when we reach a leaf at the bottom we do minus 2 to n plus 2 to n and then as we go up we are subtracting 1 as we are going down we are adding 1 and basically the only way that we can start with x equals 0 and return with x equals 0 is to have a binary tree whose depth is at least 2 to n so then the size of the tree is doubly exponential it has to be but actually it turns out that in polynomial space we can kind of handle G vast of this kind it turns out and it's not very difficult that we can compute we can compute the maximum increase of x and how large x has to be to achieve that maximum increase and so on all in polynomial space so this kind of example although it looks bad at first it's not so difficult to deal with and this in our ongoing work to close the gap this gap between x-space and p-space actually this is not such a big problem but I'll show an example that we don't know how to deal with now and it uses the same numbers s, s, n, i r, n, i, s, n r, n and I'm not going to show you the grammar vast written as a program but I'm going to show you what shape of trees we get as it runs so at the start I'm afraid it's probably not very easy to see from the back but I'll tell you what this does at the start we kind of this is the top of the tree so what's happening is as we are traversing the tree in the you know say in fix ordering at the start we are repeatedly adding r, n to x that was our denominator on the right hand side of that big equation we are repeatedly adding r, n but to come back up at the end so there will be some big computation down here but to come back up and reach 0 at the end we need the same number of times to subtract s, n and s, n is larger than r, n right so this this is we call it a borrow loop so we are borrowing r, n, some number of times but with a promise that will pay back s, n the same number of times right and s, n is larger than r, n so then what happens below in this example is firstly what we do is we generate 2 to n copies and this is easy to do with the grammar of us you know size basically n we generate 2 to n copies of the following kind of behavior which is a long loop that keeps subtracting r, n, 0 on the left and then adding on the right s, n, 0 the same number of times so this is like a multiplication by that first fraction s, n, 0 divided by r, n, 0 and we do that 2 to n times then after that we do a similar thing here we branch to width 2 to n minus 1 and we do it with the second fraction we are implementing that equation ok and at the end we do it just once and the fraction is s, n, n divided by r, n, n ok so so basically what's happening here and these loops can be called invest loops why invest because we are here we are paying we are paying r, n, i on the left s, n, i on the right and s, n, i is larger so we are investing something smaller to get something larger many times so the only way that so to speak all of these investments can generate enough to pay back this borrowing at the start the only way to do that right and if you remember the only way to do that is to repeat these loops doubly exponentially many times right that's needed for the for the exactness by the way the ratio of of the non terminal that appears here is s, n over r, n so that's the maximum multiplicative constant that that the value of the variable going in on the left can be enlarged by when we come back to when we complete that computation and come back up on the right it's s, n divided by r, n so what we have here is an example I haven't given you that as a program but it's not difficult to write that it's an example of a jiva's a gramma vass in dimension 1 where the shortest reachability or even coverability even coverability runs have this kind of doubly exponential depth and one of my key contributions to this work was to come up with the name for this kind of run which is a long neck here and lots of long tentacles at the bottom something else that we came up we came up against in a work on gramma vass is in fact the ABC conjecture and and that came up with studying what happens in situations where so to speak the investments at the bottom are not quite enough to repay the borrowing at the top but there is some kind of small gap and studying how small that gap could be and for those of you who are not familiar with the ABC conjecture I would encourage you to look at for example the Wikipedia page about the ABC conjecture and where it discusses towards the end briefly the current state of affairs where it seems that it's not proved yet there was a claim apparently a credible claim to begin with that it was proved but it turns out that that has not really been accepted yet so it's one of those difficult mathematical problems which has ended up from our point of view in an unexpected way being related to gramma vass in dimension 1 and that is further testimony than Alain Fonquel was right to advocate the study of one-dimensional gramma vass because not only is there this apparent link with the ABC conjecture but also finding these giraffe octopus examples has then led us by realizing that actually we can use some of the ideas encountered there on vass it has led us to the non-elementary lower bound for the reachability problem for vass so thank you very much why the past tense in the title of your book past tense in the title of your book that's a good question I suppose it's from the point of view of this of this result of the non-elementary lower bound for the reachability problem for vass but you're completely right that it shouldn't really have been in the past tense because as I think I have conveyed and this is probably my main message is that we still know very little I mean all these very simple to state problems that we just have no idea whether they're even decidable let alone the complexity and this grammar vass in dimension 1 are an example and it seems that they do have deep connections in unexpected ways to other such questions I know that you have a student comment sur le cause avage du algorithm for reachability yes that's right we have an implementation of the coseraggio algorithm and we we thought that it would not be good for anything other than other than studying the coseraggio algorithm because at least for some people they really understand things when they're written when they're programmed with proper comments and so on but actually it turns out and this is surprising that on some benchmarks this tool outperforms state of the art coverability checkers and this is in cases in negative cases where we don't have coverability and what's involved here is a straightforward reduction from coverability to reachability so it seems that coseraggio's algorithm at least in more testing remains to be done but at least in cases where coverability fails or rather where the system is safe the coseraggio's algorithm can quickly discover some invariance that in effect show that the system is safe which we didn't expect and do you know the complexity of this sub part of the coseraggio algorithm not yet, no it's a good question not a good question you have the title of i know it is initial experimental results but i'll be interested to talk offline ok, thank you is there any way to relate this to decidability of some kind of arithmetical theory extending prasburger with experimentation and other things is there a way to rephrase some of these questions as what are also known to be extremely hard problems regarding decidability of arithmetical theories to be honest i don't know of the top of my head it could be worth exploring there are rich connections between the reachability problem for vas quite a number of problems in other areas to an extent where sylvain schmitz has proposed regarding the reachability problem or rather the class of all problems that have reductions to and from the reachability problem is a complexity class now when we go to grammar vas i'm not aware of so many connections but this is a good question when we go to branching vas which is another well-known extension of vas reachability is still not known to be decidable then there are connections to database theory to linear logic but as far as arithmetic theories go i'm not aware at the moment but there could be just maybe my ignorance