On dynamic topological and metric logics Roman Kontchakov - PowerPoint PPT Presentation

On dynamic topological and metric logics Roman Kontchakov Department of Computer Science , King’s College London http://www.dcs.kcl.ac.uk/staff/romanvk joint work with Boris Konev, Frank Wolter and Michael Zakharyaschev

Dynamic systems . ‘space’ + f x f 2 ( x ) f ( x ) . AiML 2004 11.09.04 1

Dynamic systems f f . . ‘space’ + f x x f 2 ( x ) f 2 ( x ) f ( x ) y f 2 ( ) f ( x ) y f ( ) y . . 0 1 2 AiML 2004 11.09.04 1

Dynamic systems f f . . ‘space’ + f x x f 2 ( x ) f 2 ( x ) f ( x ) y f 2 ( ) f ( x ) y f ( ) y . . 0 1 2 Temporal logic to describe and reason about behaviour of dynamic systems: • variables p are interpreted by sets of points, i.e., point x is in p : x ∈ p x ∈ ✷ F p • x always stays in p : x ∈ ✷ F ✸ F p • x occurs in p infinitely often: • . . . AiML 2004 11.09.04 1

Dynamic topological logic 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 ) f : T → T ⇒ f − 1 ( X ) open ) a total continuous function ( X open AiML 2004 11.09.04 2

Dynamic topological logic 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 ) f : T → T ⇒ f − 1 ( X ) open ) a total continuous function ( X open Dynamic topo-logic DT L • propositional variables p, q, . . . • the Booleans ¬ , ∧ and ∨ • modal (topological) operators I and C � , ✷ F and ✸ F • temporal operators AiML 2004 11.09.04 2

Dynamic topological logic 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 ) f : T → T ⇒ f − 1 ( X ) open ) a total continuous function ( X open Dynamic topo-logic DT L V a valuation : • propositional variables p, q, . . . subsets of T − , ∩ and ∪ • the Booleans ¬ , ∧ and ∨ I and C • modal (topological) operators I and C V ( � ϕ ) = f − 1 ( V ( ϕ )) � , ✷ F and ✸ F • temporal operators AiML 2004 11.09.04 2

Dynamic topological logic 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 ) f : T → T ⇒ f − 1 ( X ) open ) a total continuous function ( X open Dynamic topo-logic DT L V a valuation : • propositional variables p, q, . . . subsets of T − , ∩ and ∪ • the Booleans ¬ , ∧ and ∨ I and C • modal (topological) operators I and C V ( � ϕ ) = f − 1 ( V ( ϕ )) � , ✷ F and ✸ F • temporal operators ∞ ∞ � f − n ( V ( ϕ )) � f − n ( V ( ϕ )) V ( ✷ F ϕ ) = V ( ✸ F ϕ ) = and n =1 n =1 AiML 2004 11.09.04 2

Classes of dynamic topological structures Topological spaces T = � T, I � • 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 • . . . AiML 2004 11.09.04 3

Classes of dynamic topological structures Topological spaces T = � T, I � • 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 • . . . Functions f : T → T • continuous • homeomorphisms: continuous bijections with continuous inverses • . . . AiML 2004 11.09.04 3

Known results DT L � — subset of DT L containing no ‘infinite’ operators ( ✷ F and ✸ F ) 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. Log � {� F , f �} � Log � {� R , f �} NB. (Slavnov 2003, Kremer & Mints 2003) AiML 2004 11.09.04 4

Known results DT L � — subset of DT L containing no ‘infinite’ operators ( ✷ F and ✸ F ) 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. Log � {� F , f �} � Log � {� R , f �} NB. (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 } , Log � {� R , f � | f a homeomorphism } coincide, have the fmp , are finitely axiomatisable , and so decidable. AiML 2004 11.09.04 4

Known results DT L � — subset of DT L containing no ‘infinite’ operators ( ✷ F and ✸ F ) 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. Log � {� F , f �} � Log � {� R , f �} NB. (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 } , Log � {� R , f � | f a homeomorphism } coincide, have the fmp , are finitely axiomatisable , and so decidable. Open problem : axiomatisations and algorithmic properties of the full DT L ? AiML 2004 11.09.04 4

Homeomorphisms: bad news Theorem 1. No logic from the list below is recursively enumerable : • 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. AiML 2004 11.09.04 5

Homeomorphisms: bad news Theorem 1. No logic from the list below is recursively enumerable : • 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. NB. All these logics are different . AiML 2004 11.09.04 5

Continuous maps: some good news Finite iterations : • arbitrary finite flows of time finite change assumption • (the system eventually stabilises) AiML 2004 11.09.04 6

Continuous maps: some good news Finite iterations : • arbitrary finite flows of time finite change assumption • (the system eventually stabilises) Theorem 2. The two topo-logics Log ∗ {� F , f �} Log ∗ {� F , f � | F an Aleksandrov space } and 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. AiML 2004 11.09.04 6

Continuous maps: some good news Finite iterations : • arbitrary finite flows of time finite change assumption • (the system eventually stabilises) Theorem 2. The two topo-logics Log ∗ {� F , f �} Log ∗ {� F , f � | F an Aleksandrov space } and 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. However: Theorem 3. The two topo-logics Log ∗ {� F , f � | f a homeomorphism } and Log ∗ {� F , f � | F an Aleksandrov space , f a homeomorphism } coincide but are not recursively enumerable . AiML 2004 11.09.04 6

Dynamics in metric spaces f f . x f 2 ( x ) f ( x ) y f 2 ( y ) f ( y ) . 0 1 2 AiML 2004 11.09.04 7

Dynamics in metric spaces f f . x f 2 ( x ) f ( x ) y f 2 ( y ) f ( y ) . 0 1 2 where d : W × W → R + is a metric, A metric space D = � W, d � , induces T d = � W, I d � : the topological space I d X = { x ∈ W | ∃ δ > 0 ∀ y ∈ W ( d ( x, y ) < δ → y ∈ X ) } AiML 2004 11.09.04 7

Logics of metric spaces D = � W, d � a metric space • propositional variables p, q, . . . • the Booleans ¬ , ∧ and ∨ • interior I and closure C operators • universal ∀ and existential ∃ modalities metric operators ∃ ≤ a and ∀ ≤ a , for a ∈ Q + • AiML 2004 11.09.04 8

Logics of metric spaces D = � W, d � V a valuation : a metric space • propositional variables p, q, . . . subsets of W • the Booleans ¬ , ∧ and ∨ − , ∩ and ∪ I d and C d • interior I and closure C operators • universal ∀ and existential ∃ modalities metric operators ∃ ≤ a and ∀ ≤ a , for a ∈ Q + • . V ( ∃ ≤ a ϕ ) = { x ∈ W | ∃ y ∈ V ( ϕ ) such that d ( x, y ) ≤ a } a V ( ϕ ) a V ( ∀ ≤ a ϕ ) = { x ∈ W | ∀ y ∈ V ( ϕ ) such that d ( x, y ) ≤ a } . AiML 2004 11.09.04 8