The Ultimate Undecidability Result for the HalpernShoham Logic

Transcription

1 The Ultimate Undecidability Result for the HalpernShoham Logic Jerzy Marcinkowski, Jakub Michaliszyn Instytut Informatyki University of Wrocªaw June 24, 2011

2 Intervals Given an arbitrary total order D,. An interval: [a, b] such that a, b D and a b. What relative positions of two intervals can be expressed using? Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

3 13 relative positions of intervals can be expressed using Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

4 13 relative positions of intervals can be expressed using Before and after. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

5 13 relative positions of intervals can be expressed using Meet and met by. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

6 13 relative positions of intervals can be expressed using Overlaps and overlapped by. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

7 13 relative positions of intervals can be expressed using Starts and nishes. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

8 13 relative positions of intervals can be expressed using Contains and during. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

9 13 relative positions of intervals can be expressed using Started by and nished by. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

10 13 relative positions of intervals can be expressed using Equals. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

11 What can we do with those relations? Allen's algebra. xy.x before y z.z meet y z met by x Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

12 What can we do with those relations? HalpernShoham logic. before p during (r after q) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

13 Formally - the models Any total order D = D, Set of propositional variables Var Classic temporal logics labeling γ : D P(Var) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

14 Formally - the models Any total order D = D, Set of propositional variables Var Classic temporal logics labeling γ : D P(Var) Interval temporal logics labeling γ : I(D) P(Var), where I(D) = {[a, b] a, b D a b} Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

15 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, Ā, B, B, D, D, E, Ē, L, L, O, Ō Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

16 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, Ā, B, B, D, D, E, Ē, L, L, O, Ō HalpernShoham (LICS'86) undecidability (of the satisability problem for the formulae of the HS logic) using after, begins and ends. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

17 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, Ā, B, B, D, D, E, Ē, L, L, O, Ō HalpernShoham (LICS'86) undecidability (of the satisability problem for the formulae of the HS logic) using after, begins and ends. Lodaya ('00) undecidability using begins and ends. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

18 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, Ā, B, B, D, D, E, Ē, L, L, O, Ō HalpernShoham (LICS'86) undecidability (of the satisability problem for the formulae of the HS logic) using after, begins and ends. Lodaya ('00) undecidability using begins and ends. Further results ('07-'10) undecidability of O, BĒ, BĒ, AĀD, Ā D B,.... Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

19 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, Ā, B, B, D, D, E, Ē, L, L, O, Ō HalpernShoham (LICS'86) undecidability (of the satisability problem for the formulae of the HS logic) using after, begins and ends. Lodaya ('00) undecidability using begins and ends. Further results ('07-'10) undecidability of O, BĒ, BĒ, AĀD, Ā D B,.... The positive side: decidability of B B, AĀ, AB B L (next presentation),... Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

20 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, Ā, B, B, D, D, E, Ē, L, L, O, Ō HalpernShoham (LICS'86) undecidability (of the satisability problem for the formulae of the HS logic) using after, begins and ends. Lodaya ('00) undecidability using begins and ends. Further results ('07-'10) undecidability of O, BĒ, BĒ, AĀD, Ā D B,.... The positive side: decidability of B B, AĀ, AB B L (next presentation),... The decidability of B BD DL L over the class of dense structures. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

21 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, Ā, B, B, D, D, E, Ē, L, L, O, Ō HalpernShoham (LICS'86) undecidability (of the satisability problem for the formulae of the HS logic) using after, begins and ends. Lodaya ('00) undecidability using begins and ends. Further results ('07-'10) undecidability of O, BĒ, BĒ, AĀD, Ā D B,.... The positive side: decidability of B B, AĀ, AB B L (next presentation),... The decidability of B BD DL L over the class of dense structures. The Logic of Subintervals The logic of subintervals contains only one operator (D): D φ is satised if φ is satised in some subinterval. [D]φ is satised if φ is satised in all subintervals. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

22 Logic of subintervals results The logic of subintervals is PSPACEcomplete over the class of dense orders (D. Bresolin, V. Goranko, A. Montanari, P. Sala, 2007). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

23 Logic of subintervals results The logic of subintervals is PSPACEcomplete over the class of dense orders (D. Bresolin, V. Goranko, A. Montanari, P. Sala, 2007). The logic of reexive subintervals is PSPACEcomplete over the class of (nite) discrete orders (A. Montanari, I. Pratt-Hartmann, P. Sala, 2010). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

24 Logic of subintervals results The logic of subintervals is PSPACEcomplete over the class of dense orders (D. Bresolin, V. Goranko, A. Montanari, P. Sala, 2007). The logic of reexive subintervals is PSPACEcomplete over the class of (nite) discrete orders (A. Montanari, I. Pratt-Hartmann, P. Sala, 2010). The logic of subintervals is in EXPSPACE over the class of discrete orders if we allow to remove intervals from models (follows from T. Schwentick, T. Zeume, 2010). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

25 Logic of subintervals results The logic of subintervals is PSPACEcomplete over the class of dense orders (D. Bresolin, V. Goranko, A. Montanari, P. Sala, 2007). The logic of reexive subintervals is PSPACEcomplete over the class of (nite) discrete orders (A. Montanari, I. Pratt-Hartmann, P. Sala, 2010). The logic of subintervals is in EXPSPACE over the class of discrete orders if we allow to remove intervals from models (follows from T. Schwentick, T. Zeume, 2010). The problem they left open: the complexity of the logic of subintervals over the class of discrete orders. Remark: No nontrivial lower bounds were known. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

26 What can we express using the subinterval relation? each morning I spend a while thinking of you Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

27 What can we express using the subinterval relation? each morning I spend a while thinking of you each nice period of my life contains an unpleasant fragment Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

28 What can we express using the subinterval relation? each morning I spend a while thinking of you each nice period of my life contains an unpleasant fragment there is no error while printing Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

29 Logic of subintervals results The logic of subintervals is PSPACEcomplete over the class of dense orders (D. Bresolin, V. Goranko, A. Montanari, P. Sala, 2007). The logic of reexive subintervals is PSPACEcomplete over the class of (nite) discrete orders (A. Montanari, I. Pratt-Hartmann, P. Sala, 2010). The logic of subintervals is in EXPSPACE over the class of discrete orders if we allow to remove intervals from models (follows from T. Schwentick, T. Zeume, 2010). The problem they left open: the complexity of the logic of subintervals over the class of discrete orders. Remark: No nontrivial lower bounds where known. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

30 Logic of subintervals results The logic of subintervals is PSPACEcomplete over the class of dense orders (D. Bresolin, V. Goranko, A. Montanari, P. Sala, 2007). The logic of reexive subintervals is PSPACEcomplete over the class of (nite) discrete orders (A. Montanari, I. Pratt-Hartmann, P. Sala, 2010). The logic of subintervals is in EXPSPACE over the class of discrete orders if we allow to remove intervals from models (follows from T. Schwentick, T. Zeume, 2010). The problem they left open: the complexity of the logic of subintervals over the class of discrete orders. Remark: No nontrivial lower bounds where known. Prexes and subintervals are enough to make the HS logic undecidable over the class of discrete orders (E. Kiero«ski, J. Marcinkowski, J. Michaliszyn, ICALP 2010). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

31 Logic of subintervals results The logic of subintervals is PSPACEcomplete over the class of dense orders (D. Bresolin, V. Goranko, A. Montanari, P. Sala, 2007). The logic of reexive subintervals is PSPACEcomplete over the class of (nite) discrete orders (A. Montanari, I. Pratt-Hartmann, P. Sala, 2010). The logic of subintervals is in EXPSPACE over the class of discrete orders if we allow to remove intervals from models (follows from T. Schwentick, T. Zeume, 2010). The problem they left open: the complexity of the logic of subintervals over the class of discrete orders. Remark: No nontrivial lower bounds where known. Prexes and subintervals are enough to make the HS logic undecidable over the class of discrete orders (E. Kiero«ski, J. Marcinkowski, J. Michaliszyn, ICALP 2010). The logic of subintervals is undecidable over the class of discrete orders (this presentation). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

32 Our undecidability result Assumptions Assumption In our paper In this presentation Order All discrete All nite Do we allow point intervals ([a, a])? Whatever Yes Subinterval relation or superinterval relation? Does not matter Subinterval Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

33 An overview of the proof The logic of subintervals is undecidable over the class of all nite structures.

34 How we imagine that triangle structures Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

35 How we imagine that triangle structures Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

36 How we imagine that triangle structures Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

37 Problem 1 symmetry Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

38 Problem 1 symmetry Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

39 Our little library of formulae λ 0 : [D] Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

40 Our little library of formulae λ 0 : [D] λ 1 : [D]λ 0 Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

41 Our little library of formulae λ 0 : [D] λ 1 : [D]λ 0 Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

42 Our little library of formulae λ 0 : [D] λ 1 : [D]λ 0 λ 1 : λ 1 λ 0 Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

43 Problem 1 symmetry Now we can deal with the symmetry Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

44 Problem 1 symmetry Each point-interval is labeled with exactly one of s 0, s 1, s 2, and each interval labeled with s 0, s 1, or s 2 is an point-interval. [D](λ 0 s 1 s 2 s 0 ) [D] (s 0 s 1 ) [D] (s 1 s 2 ) [D] (s 0 s 2 ) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

45 Problem 1 symmetry Each interval with length 2 contains intervals labeled with s 0, s 1, and s 2. [D](λ 2 D s 0 D s 1 D s 2 ) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

46 Problem 1 symmetry Each interval with length 2 contains intervals labeled with s 0, s 1, and s 2. [D](λ 2 D s 0 D s 1 D s 2 ) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

47 Problem 1 symmetry Each interval with length 2 contains intervals labeled with s 0, s 1, and s 2. [D](λ 2 D s 0 D s 1 D s 2 ) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

48 Problem 1 symmetry Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

49 Problem 1 symmetry Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

50 Problem 1 symmetry Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

51 Problem 1 symmetry Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

52 The source of the undecidability Regularity + ability to measure undecidability Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

53 The source of the undecidability Regularity + ability to measure undecidability Actually, with D we only have very limited ability to measure. One of the technical lemmas is that this limited ability already leads to the undecidability. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

54 Second step regularity We can encode any nite automaton. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

55 Second step regularity We can encode any nite automaton. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

56 Second step regularity We can encode any nite automaton. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

57 Third step the cloud. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

58 Third step the cloud. What remains to be explained: 1 How to write a formula saying that a propositional variable p is a cloud. 2 How to use this cloud. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

59 Third step the cloud. There exists an interval labeled with p. Intervals labeled with p do not contain each other. Any interval that contains an interval with p, contains two such intervals (one with e and one with e). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

60 Third step the cloud. There exists an interval labeled with p. Intervals labeled with p do not contain each other. Any interval that contains an interval with p, contains two such intervals (one with e and one with e). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

61 Third step the cloud. There exists an interval labeled with p. Intervals labeled with p do not contain each other. Any interval that contains an interval with p, contains two such intervals (one with e and one with e). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

62 Third step the cloud. There exists an interval labeled with p. Intervals labeled with p do not contain each other. Any interval that contains an interval with p, contains two such intervals (one with e and one with e). Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

63 Third step the cloud. There exists an interval labeled with p. Intervals labeled with p do not contain each other. Any interval that contains an interval with p, contains two such intervals (one with e and one with e). 1 D p 2 [D](p [D] p) 3 [D]( D p D (p e) D (p e)) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

64 Toy example We already know how to encode regularity. Consider the regular language dened by (ac bc ). We want to force that each maximal block of c has the same length. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

65 Toy example No angel can see both a and b. Each angel has to see a or b. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

66 Toy example No angel can see both a and b. Each angel has to see a or b. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

67 Toy example No angel can see both a and b. Each angel has to see a or b. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

68 Toy example No angel can see both a and b. Each angel has to see a or b. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

69 Summary The logic of subintervals is decidable over the class of dense structures. The logic of subintervals is undecidable over the class of discrete structures. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

70 Summary The logic of subintervals is decidable over the class of dense structures. The logic of subintervals is undecidable over the class of discrete structures. Is the logic of subintervals decidable over the class of all structures? (open) Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result / 26

71 Thank you!

72 Thank you! Coming next: Davide Bresolin, Angelo Montanari, Pietro Sala and Guido Sciavicco. What's decidable about Halpern and Shoham's interval logic? The maximal fragment AB BL