The Ultimate Undecidability Result for the HalpernShoham Logic
Jerzy Marcinkowski, Jakub Michaliszyn
Instytut Informatyki University of Wrocªaw
June 24, 2011 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... 24.06.2011 2 / 26 13 relative positions of intervals can be expressed using ≤
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 13 relative positions of intervals can be expressed using ≤
Before and after.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 13 relative positions of intervals can be expressed using ≤
Meet and met by.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 13 relative positions of intervals can be expressed using ≤
Overlaps and overlapped by.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 13 relative positions of intervals can be expressed using ≤
Starts and nishes.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 13 relative positions of intervals can be expressed using ≤
Contains and during.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 13 relative positions of intervals can be expressed using ≤
Started by and nished by.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 13 relative positions of intervals can be expressed using ≤
Equals.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 3 / 26 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... 24.06.2011 4 / 26 What can we do with those relations?
HalpernShoham logic.
⟨before⟩p ∧ ⟨during⟩(r ∧ ⟨after⟩q)
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 5 / 26 Interval temporal logics labeling : I(D) → P(Var), where I(D) = {[a, b]∣a, b ∈ D ∧ a ≤ b}
Formally - the models
Any total order D = ⟨D, ≤⟩ Set of propositional variables Var Classic temporal logics labeling : D → P(Var)
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 6 / 26 Formally - the models
Any total order D = ⟨D, ≤⟩ Set of propositional variables Var Classic temporal logics labeling : D → P(Var) Interval temporal logics labeling : I(D) → P(Var), where I(D) = {[a, b]∣a, b ∈ D ∧ a ≤ b}
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 6 / 26 HalpernShoham (LICS'86) undecidability (of the satis ability 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, BE¯, B¯E¯, AAD¯ , A¯D¯ B¯,.... The positive side: decidability of BB¯, AA¯, ABB¯L¯ (next presentation), . . . The decidability of BBD¯ DL¯ L¯ over the class of dense structures.
The HalpernShoham Logic HalpernShoham logic contains 12 operators A, A¯, B, B¯, D, D¯ , E, E¯, L, L¯, O, O¯
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 7 / 26 Lodaya ('00) undecidability using begins and ends. Further results ('07-'10) undecidability of O, BE¯, B¯E¯, AAD¯ , A¯D¯ B¯,.... The positive side: decidability of BB¯, AA¯, ABB¯L¯ (next presentation), . . . The decidability of BBD¯ DL¯ L¯ over the class of dense structures.
The HalpernShoham Logic HalpernShoham logic contains 12 operators A, A¯, B, B¯, D, D¯ , E, E¯, L, L¯, O, O¯ HalpernShoham (LICS'86) undecidability (of the satis ability problem for the formulae of the HS logic) using after, begins and ends.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 7 / 26 Further results ('07-'10) undecidability of O, BE¯, B¯E¯, AAD¯ , A¯D¯ B¯,.... The positive side: decidability of BB¯, AA¯, ABB¯L¯ (next presentation), . . . The decidability of BBD¯ DL¯ L¯ over the class of dense structures.
The HalpernShoham Logic HalpernShoham logic contains 12 operators A, A¯, B, B¯, D, D¯ , E, E¯, L, L¯, O, O¯ HalpernShoham (LICS'86) undecidability (of the satis ability 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... 24.06.2011 7 / 26 The positive side: decidability of BB¯, AA¯, ABB¯L¯ (next presentation), . . . The decidability of BBD¯ DL¯ L¯ over the class of dense structures.
The HalpernShoham Logic HalpernShoham logic contains 12 operators A, A¯, B, B¯, D, D¯ , E, E¯, L, L¯, O, O¯ HalpernShoham (LICS'86) undecidability (of the satis ability 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, BE¯, B¯E¯, AAD¯ , A¯D¯ B¯,....
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 7 / 26 The decidability of BBD¯ DL¯ L¯ over the class of dense structures.
The HalpernShoham Logic HalpernShoham logic contains 12 operators A, A¯, B, B¯, D, D¯ , E, E¯, L, L¯, O, O¯ HalpernShoham (LICS'86) undecidability (of the satis ability 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, BE¯, B¯E¯, AAD¯ , A¯D¯ B¯,.... The positive side: decidability of BB¯, AA¯, ABB¯L¯ (next presentation), . . .
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 7 / 26 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, A¯, B, B¯, D, D¯ , E, E¯, L, L¯, O, O¯ HalpernShoham (LICS'86) undecidability (of the satis ability 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, BE¯, B¯E¯, AAD¯ , A¯D¯ B¯,.... The positive side: decidability of BB¯, AA¯, ABB¯L¯ (next presentation), . . . The decidability of BBD¯ DL¯ L¯ over the class of dense structures.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 7 / 26 The HalpernShoham Logic HalpernShoham logic contains 12 operators A, A¯, B, B¯, D, D¯ , E, E¯, L, L¯, O, O¯ HalpernShoham (LICS'86) undecidability (of the satis ability 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, BE¯, B¯E¯, AAD¯ , A¯D¯ B¯,.... The positive side: decidability of BB¯, AA¯, ABB¯L¯ (next presentation), . . . The decidability of BBD¯ DL¯ L¯ over the class of dense structures.
The Logic of Subintervals The logic of subintervals contains only one operator (D): ⟨D⟩' is satis ed if ' is satis ed in some subinterval. [D]' is satis ed if ' is satis ed in all subintervals.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 7 / 26 The logic of re exive 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.
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... 24.06.2011 8 / 26 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.
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 re exive 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... 24.06.2011 8 / 26 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.
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 re exive 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... 24.06.2011 8 / 26 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 re exive 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... 24.06.2011 8 / 26 each nice period of my life contains an unpleasant fragment there is no error while printing
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... 24.06.2011 9 / 26 there is no error while printing
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... 24.06.2011 9 / 26 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... 24.06.2011 9 / 26 Pre xes 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).
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 re exive 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... 24.06.2011 10 / 26 The logic of subintervals is undecidable over the class of discrete orders (this presentation).
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 re exive 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. Pre xes 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... 24.06.2011 10 / 26 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 re exive 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. Pre xes 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... 24.06.2011 10 / 26 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... 24.06.2011 11 / 26 An overview of the proof
The logic of subintervals is undecidable
over the class of all nite structures. How we imagine that triangle structures
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 13 / 26 How we imagine that triangle structures
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 14 / 26 How we imagine that triangle structures
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 15 / 26 Problem 1 symmetry
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 16 / 26 Problem 1 symmetry
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 16 / 26 ≤1: [D]0
1: ≤1 ∧ ¬0
Our little library of formulae
0: [D]⊥
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 17 / 26 1: ≤1 ∧ ¬0
Our little library of formulae
0: [D]⊥
≤1: [D]0
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 17 / 26 1: ≤1 ∧ ¬0
Our little library of formulae
0: [D]⊥
≤1: [D]0
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 17 / 26 Our little library of formulae
0: [D]⊥
≤1: [D]0
1: ≤1 ∧ ¬0
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 17 / 26 Problem 1 symmetry
Now we can deal with the symmetry
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Each point-interval is labeled with exactly one of s0, s1, s2, and each interval labeled with s0, s1, or s2 is an point-interval.
[D](0 ⇔ s1 ∨ s2 ∨ s0) ∧ [D]¬(s0 ∧ s1) ∧ [D]¬(s1 ∧ s2) ∧ [D]¬(s0 ∧ s2)
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Each interval with length 2 contains intervals labeled with s0, s1, and s2.
[D](2 ⇒ ⟨D⟩s0 ∧ ⟨D⟩s1 ∧ ⟨D⟩s2)
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Each interval with length 2 contains intervals labeled with s0, s1, and s2.
[D](2 ⇒ ⟨D⟩s0 ∧ ⟨D⟩s1 ∧ ⟨D⟩s2)
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Each interval with length 2 contains intervals labeled with s0, s1, and s2.
[D](2 ⇒ ⟨D⟩s0 ∧ ⟨D⟩s1 ∧ ⟨D⟩s2)
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 Problem 1 symmetry
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 18 / 26 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.
The source of the undecidability
Regularity + ability to measure ≥ undecidability
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 19 / 26 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... 24.06.2011 19 / 26 Second step regularity
We can encode any nite automaton.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 20 / 26 Second step regularity
We can encode any nite automaton.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 20 / 26 Second step regularity
We can encode any nite automaton.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 20 / 26 Third step the cloud.
Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 21 / 26 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... 24.06.2011 21 / 26 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... 24.06.2011 22 / 26 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... 24.06.2011 22 / 26 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... 24.06.2011 22 / 26 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... 24.06.2011 22 / 26 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... 24.06.2011 22 / 26 Toy example
We already know how to encode regularity. Consider the regular language de ned 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... 24.06.2011 23 / 26 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... 24.06.2011 24 / 26 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... 24.06.2011 24 / 26 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... 24.06.2011 24 / 26 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... 24.06.2011 24 / 26 Is the logic of subintervals decidable over the class of all structures? (open)
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... 24.06.2011 25 / 26 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... 24.06.2011 25 / 26 Thank you! 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 ABBL¯