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The Ultimate Undecidability Result for the Halpern–Shoham Logic

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The Ultimate Undecidability Result for the Halpern–Shoham Logic 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 hD; ≤⟩. An interval: [a; b] such that a; b 2 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. 8xy:x before y ) 9z: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. hbeforeip ^ hduringi(r ^ hafteriq) 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) = f[a; b]ja; b 2 D ^ a ≤ bg Formally - the models Any total order D = hD; ≤⟩ 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 = hD; ≤⟩ Set of propositional variables Var Classic temporal logics labeling : D !P(Var) Interval temporal logics labeling : I(D) !P(Var), where I(D) = f[a; b]ja; b 2 D ^ a ≤ bg Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 6 / 26 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, 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 satisability 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 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... 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 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, 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 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, 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 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, 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 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, 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): hDi' is satised if ' is satised in some subinterval. [D]' is satised if ' is satised in all subintervals. Jakub Michaliszyn (II UWr, PL) The Ultimate Undecidability Result... 24.06.2011 7 / 26 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. 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 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... 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 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... 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 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.
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