On dynamic topological logics Roman Kontchakov School of Computer - PowerPoint PPT Presentation

On dynamic topological logics Roman Kontchakov School of Computer Science and Information Systems , Birkbeck , London http://www.dcs.bbk.ac.uk/ ∼ roman joint work with Boris Konev, Frank Wolter and Michael Zakharyaschev

The Story S. Artemov, J. Davoren and A. Nerode. Topological semantics for hybrid systems. LNCS, vol. 1234, Proceedings of LFCS’97, pp. 1-8, 1997 J. Davoren. Modal logics for continuous dynamics . Ph.D. Thesis, Department of Mathematics, Cornell University, 1998 Ph. Kremer and G. Mints. Dynamic topological logic . Bulletin of Symbolic Logic, 3:371–372, 1997 Annals of Pure and Applied Logic, 131:133–158, 2005 J.M. Davoren and R.P . Gor´ e. Bimodal logics for reasoning about continuous dynamics . Advances in Modal Logic, Volume 3, pp. 91–110. World Scientific, 2002 B. Konev, R. Kontchakov, F . Wolter and M. Zakharyaschev. On dynamic topological and metric logics . Proceedings of AiML 2004, pp. 182–196, Manchester, U.K., September 2004 Oxford 05/08/07 1

Dynamical systems f f . . ‘space’ + f x x f 2 ( x ) f 2 ( x ) f ( x ) y f ( x ) f 2 ( ) y f ( ) y . . 0 1 2 Orb f ( x ) = { f ( x ) , f 2 ( x ) , . . . } — the orbit of x Temporal logic × logic of topology to describe and reason about the (asymptotic) behaviour of orbits Oxford 05/08/07 2

Dynamic topological structures Dynamic topological structure F = � T , f � T = � T, I � a topological space is the universe of T T I is the interior operator on T C is the closure operator on T ( C X = − I − X ) • arbitrary topologies • Aleksandrov: arbitrary (not only finite) intersections of open sets are open — every Kripke frame G = � U, R � , where R is a quasi-order , induces the Aleksandrov topological space � U, I G � : I G X = { x ∈ U | ∀ y ( x R y → y ∈ X ) } — conversely, every Aleksandrov space is induced by a quasi-order Euclidean spaces R n , n ≥ 1 • . . . • f : T → T a continuous function ⇒ f − 1 ( X ) open ) ( X open • continuous • homeomorphisms (continuous bijections with continuous inverses) Oxford 05/08/07 3

Dynamic topological logic DT L V a valuation in �� T, I � , f � Formulas: subsets of T propositional variables p, q, . . . • − , ∩ and ∪ • the Booleans ¬ , ∧ and ∨ I and C • topological (‘modal’) operators I and C temporal operators ◦ , ✷ F and ✸ F V ( ◦ ϕ ) = f − 1 ( V ( ϕ )) • . . f f ∞ � f − n ( V ( ϕ )) V ( ✷ F ϕ ) = = { x ∈ T | Orb f ( x ) ⊆ V ( ϕ ) } ◦ ϕ f − 1 ϕ ϕ n =1 . . ∞ � V ( ✸ F ϕ ) = f − n ( V ( ϕ )) = { x ∈ T | Orb f ( x ) ∩ V ( ϕ ) � = ∅} n =1 ψ → ✷ F ✸ F ϕ Example: every ψ satisfies ϕ infinitely often Oxford 05/08/07 4

Known results: no ‘infinite’ operations � — subset of DT L containing no ‘infinite’ operators ( ✷ F and ✸ F ) DT L � � Artemov, Davoren & Nerode (1997): The two dynamic topo-logics Log � � {� F , f �} and Log � � {� F , f � | F an Aleksandrov space } � � coincide, have the fmp , are finitely axiomatisable , and so decidable � {� F , f �} � Log � � {� R , f �} NB. Log � (Slavnov 2003, Kremer & Mints 2003) � � Kremer, Mints & Rybakov (1997): The three dynamic topo-logics Log � � {� F , f � | f a homeomorphism } , � Log � � {� F , f � | F an Aleksandrov space , f a homeomorphism } , � � {� R n , f � | f a homeomorphism } , n ≥ 1 , Log � � coincide, have the fmp , are finitely axiomatisable , and so decidable Oxford 05/08/07 5

Homeomorphisms vs. continuous mappings T = � U, I � is the Aleksandrov space induced by a quasi-order G = � U, R � f is a homeomorphism f is continuous iff iff ⇔ xRy f ( x ) Rf ( y ) ⇒ ⇒ ⇒ f ( x ) Rf ( y ) xRy a DTM can be unwound into a product model an e-product model . . = = = G 0 G 1 G 2 ⊆ ⊆ ⊆ G 0 G 1 G 2 . . . . . ( lcom ) ( rcom ) ( lcom ) ( rcom ) . . . S4 ⊕ DAlt ⊕ ( ◦ I p ↔ ↔ I ◦ p ) S4 ⊕ DAlt ⊕ ( ◦ I p → → I ◦ p ) ↔ → Oxford 05/08/07 6

DTLs with homeomorphisms No logic from the list below is recursively enumerable : Theorem 1 (AiML 2004) . • Log {� F , f � | f a homeomorphism } , • Log {� F , f � | F an Aleksandrov space , f a homeomorphism } , Log {� R n , f � | f a homeomorphism } , • n ≥ 1 . Proof. By reduction of the undecidable but r.e. Post’s Correspondence Problem to the satisfiability problem (more on the next slide) NB. All these logics are different . Oxford 05/08/07 7

Encoding PCP PCP: given a set of pairs { ( u 1 , v 1 ) , . . . , ( u k , v k ) } of nonempty finite words, decide whether there exists an N ≥ 1 and a sequence i 1 , . . . , i N such that u i 1 · u i 2 · . . . · u i N = v i 1 · v i 2 · . . . · v i N Post (1946): The PCP is undecidable and the set of PCP instances without solutions is not R.E. i 1 i 2 i N . • Aleksandrov space � U, I � (induced by � U, R � ) u i N v i N F I ( ψ 1 → ◦ ψ 2 ) + ✷ ‘local’ formulas u i 2 v i 2 � ✸ F plus I ( L a ↔ R a ) a ∈ A u i 1 v i 1 . arbitrary topological spaces and R n : • the formula requires only a finite number of iterations and thus the completeness results for Log � � {· · · } can be used � Oxford 05/08/07 8

DTLs with continuous mappings No logic from the list below is decidable : Theorem 2. • Log {� F , f �} , • Log {� F , f � | F an Aleksandrov space } , Log {� R n , f �} , • n ≥ 1 . Proof. By reduction of the undecidable ω -reachability problem for lossy channels to the satisfiability problem (more on the next slide) NB. All these logics are different . Oxford 05/08/07 9

Encoding lossy channels backwards Single channel system Q — a set of control states Σ — an alphabet of messages S = � Q, Σ , ∆ � ∆ ⊆ Q ×{ ? , ! }× Σ × Q — a set of transitions send receive � q, ! ,a,q ′ � � q, ? ,a,q ′ � → ℓ � q ′ , w ′ � → ℓ � q ′ , w ′ � � q, w � − − − − − � q, w · a � − − − − − w ′ ⊑ a · w w ′ ⊑ w iff iff backward encoding: loss of messages = introduction of new points . . . .                                      w     w ′ w · a a · w  w ′ w                                       . . . . Oxford 05/08/07 10

Encoding lossy channels: ω -reachability (1) ω -reachability: given a single channel lossy system S and two states q 0 and q rec , decide whether, for every n > 0 , there is a computation δ 1 δ 2 δ 3 � q 0 , ǫ � − → ℓ � q i 1 , w 1 � − → ℓ � q i 2 , w 2 � − → ℓ . . . reaching q rec at least n times ω -reachability is undecidable Schnoebelen (2004): . m m m m . q 0 q 0 q 0 q 0 The ω -reachability problem can be encoded F I ( ψ 1 → ◦ ψ 2 ) + ✷ ✷ F ✸ F m using only ‘local’ formulas plus plus. . . Oxford 05/08/07 11

Encoding lossy channels: ω -reachability (2) . . . m m m m m m m m m m m m . . . q rec q rec q rec q 0 q 0 q rec q 0 q rec q rec q 0 F ( light → ◦ light ) light ∧ ✷ + F ( m → ◦ I ( light → on )) ✷ + F ( C ( light ∧ on ∧ ◦ ¬ on ) → q rec ) ✷ + ✷ + F ( m → I ( light → ¬ on )) ✷ F ( m → I ( light → ◦ S light )) F I (( light ∧ on ∧ ◦ ¬ on ) → ¬ S ( light ∧ on ∧ ◦ ¬ on )) ✷ + Oxford 05/08/07 12

Finite iterations arbitrary finite flows of time • • finite change assumption (the system eventually stabilises) Theorem 3 (APAL 2006) . The two topo-logics Log fin {� F , f �} and Log fin {� F , f � | F an Aleksandrov space } coincide and are decidable , but not in primitive recursive time Proof. By Kruskal’s tree theorem and reduction of the reachability problem for lossy channels ( decidable but not in primitive recursive time ) However: Theorem 4 (AiML 2004) . The two topo-logics Log fin {� F , f � | f a homeomorphism } and Log fin {� F , f � | F an Aleksandrov space , f a homeomorphism } coincide but are not recursively enumerable Oxford 05/08/07 13

Open problems Axiomatisation of DTL over Euclidean spaces (without ✷ F , ✸ F ) • • Are full DTLs with continuous mappings r.e.? • If so, are they finitely axiomatisable? Axiomatisations? • . . . Oxford 05/08/07 14