On the computational complexity of spatial logics with - PowerPoint PPT Presentation

On the computational complexity of spatial logics with connectedness constraints Roman Kontchakov School of Computer Science and Information Systems , Birkbeck , London http://www.dcs.bbk.ac.uk/ ∼ roman joint work with Ian Pratt-Hartmann, Frank Wolter and Michael Zakharyaschev

Motivation Connectedness • is one of the most fundamental concepts of topology (any textbook in the field contains a substantial chapter on connectedness) • in spatial KR&R, the distinction between connected and disconnected regions is recognized as indispensable for various modelling and representation tasks So far only sporadic attempts have been made to investigate the computational complexity of spatial logics with connectedness constraints Topological Methods in Logic Tbilisi 5.06.08 1

S 4 u : syntax and semantics terms: subsets of T τ ◦ τ − | | τ 1 ∩ τ 2 | | τ ::= v i τ complement interior closure topological model M = ( T, · M ) formulas: true or false T a topological space ::= τ 1 = τ 2 | ¬ ϕ | ϕ 1 ∧ ϕ 2 ϕ · M a valuation τ M = τ M e.g., M | = τ 1 = τ 2 iff 1 2 NB. This definition is as expressive as the ‘standard’ one A space is called Aleksandrov if arbitrary intersections of open sets are open Aleksandrov spaces = = = Kripke frames F = ( W, R ) , R is a quasi-order on W (Shehtman 99, Areces et. al 00): Sat ( S 4 u , A LL ) = Sat ( S 4 u , A LEK ) , and this set is PS PACE -complete NB. Sat ( S 4 u , A LL ) � = Sat ( S 4 u , C ON ) (in contrast with S 4 ) Topological Methods in Logic Tbilisi 5.06.08 2

Connectedness A topological space is connected iff it is not the union of two non-empty, disjoint, open sets Example: ( v 1 � = 0 ) ∧ ( v 2 � = 0 ) ∧ ( v 1 ∪ v 2 = 1 ) ∧ ( v − 1 ∩ v 2 = 0 ) ∧ ( v 1 ∩ v − 2 = 0 ) is satisfiable in a topological space T iff T is not connected X ⊆ T is connected in T just in case either it is empty, or the topological space X (with the subspace topology) is connected A maximal connected subset of X is called a component of X An Aleksandrov space induced by F = ( W, R ) is connected iff F is connected (i.e., between any two points x, y ∈ W there is a path along the relation R ∪ R − 1 ) t ❞ t ❏ ❪ ✣ ✡ ❏ ❪ ✣ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ t t Topological Methods in Logic Tbilisi 5.06.08 3

S 4 u over connected topological spaces (Shehtman 99): Sat ( S 4 u , C ON ) = Sat ( S 4 u , C ON A LEK ) = Sat ( S 4 u , R n ) , n ≥ 1 , and this set is PS PACE -complete Example: generating all numbers from 0 to 2 n − 1 : ❝ ◗ ❦ ❝ ◗ ❦ ❝ ◗ ❦ ❝ ◗ ❦ ❝ ◗ ❦ ❝ ❦ ◗ ❝ ◗ ❦ ❝ ◗ ◗ ◗ ◗ ◗ ◗ ◗ 0 1 2 3 4 5 6 7 ✻ ✻ ✻ ✻ ✻ ✻ ✻ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ 1 2 3 4 5 6 7 ❝ ❝ ❝ ❝ ❝ ❝ ❝ 0 and 2 n − 1 are non-empty: v n ∩ · · · ∩ v 1 � = 0 , v n ∩ · · · ∩ v 1 � = 0 m + 1 , for 0 ≤ m < 2 n − 1 : the closure of m m m can share points only with m + 1 m + 1 ( v j ∩ v k ) − ⊆ v j , ( v j ∩ v k ) − ⊆ v j , for n ≥ j > k ≥ 1 ( v k ∩ v k − 1 ∩ · · · ∩ v 1 ) − ⊆ ( v k ∩ v i ) ∪ ( v k ∩ v i ) , for n ≥ k > i ≥ 1 2 n − 1 is a closed set: ( v n ∩ · · · ∩ v 1 ) − ⊆ v n ∩ · · · ∩ v 1 Topological Methods in Logic Tbilisi 5.06.08 4

S 4 u c = S 4 u + connectedness predicate (1) S 4 u c -formulas: ϕ ::= τ 1 = τ 2 | c ( τ ) | ¬ ϕ | ϕ 1 ∧ ϕ 2 = c ( τ ) iff τ M is connected in T M | ↓ one occurrence of c Theorem. Sat ( S 4 u c 1 , A LL ) is PS PACE -complete m � � � Proof. Let ψ = ( τ 0 = 0 ) ∧ ( τ i � = 0 ) ∧ c ( σ ) ∧ ( σ � = 0 ) (conjunct of a full DNF) i =1 τ 0 ◦ 1. guess a type (Hintikka set) t t t σ containing σ and (all points with σ are to be connected to t t t σ ) and expand the tableau branch by branch τ 0 ◦ 2. for each i , guess a type t t t τ i containing τ i and and expand the tableau branch by branch ❆ ✁ σ ❆ ✁ – if σ appears in the tableau r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ σ ❆ ✁ ❆ ✁ then we construct a path to t t t σ ❆ ✁ r ❆ ✁ ❆ ✁ ❪ ❏ (by “divide and conquer”) σ σ σ σ ❜ ❜ ❜ ❜ ❜ ❆ ✁ ❆ ✁ � ✒ ❅ ■ � ✒ ❅ ■ � ✒ ■ ❅ � ✒ ■ ❅ ❏ � ❅ � ❅ � ❅ � ❅ ❆ ✁ ❆ ✁ ❏ � ❅ � ❅ � ❅ � ❅ t t t t τ i t t σ ❜ ❜ ❜ σ ❜ σ ❜ σ ❜ σ path of length 2 | ψ | Topological Methods in Logic Tbilisi 5.06.08 5

S 4 u c = S 4 u + connectedness predicate (2) Theorem. Sat ( S 4 u c, A LL ) is in E XP T IME m k � � � � Proof. Let ψ = ( τ 0 = 0 ) ∧ ( τ i � = 0 ) ∧ c ( σ i ) ∧ ( σ i � = 0 ) (conjunct of a full DNF) i =1 i =1 The proof is by reduction to PDL with converse and nominals [ De Giacomo 95 ] Let α and β be atomic programs and ℓ i a nominal, for each σ i the S 4 -box is simulated by [ α ∗ ] : • τ † is the result of replacing in τ each sub-term ϑ ◦ with [ α ∗ ] ϑ the universal box is simulated by [ γ ] , where γ = ( β ∪ β − ∪ α ∪ α − ) ∗ • m k ψ ′ = [ γ ] ¬ τ † � � � � γ � τ † � � γ � ( ℓ i ∧ σ † i ) ∧ [ γ ]( σ † i → � ( α ∪ α − ; σ † i ?) ∗ � ℓ i ) 0 ∧ i ∧ i =1 i =1 ψ ′ is satisfiable iff ψ is satisfiable NB. Matching lower bound to follow. . . Topological Methods in Logic Tbilisi 5.06.08 6

S 4 u cc = S 4 u + component counting predicates c ≤ k ( τ ) S 4 u cc -formulas: ϕ ::= τ 1 = τ 2 | | ¬ ϕ | ϕ 1 ∧ ϕ 2 = c ≤ k ( τ ) iff τ M has at most k components in T M | reduction to S 4 u c : exponential if k coded in binary! (the v i are fresh variables) � � c ≤ k ( τ ) → � � ∧ • τ = v i c ( v i ) 1 ≤ i ≤ k 1 ≤ i ≤ k � � � τ ∩ v − i ∩ v − ¬ c ≤ k ( τ ) → � � � � � � τ = ∧ v i � = 0 ∧ j = 0 • v i 1 ≤ i ≤ k +1 1 ≤ i ≤ k +1 1 ≤ i<j ≤ k +1 (Pratt-Hartmann 02): Sat ( S 4 u cc, A LL ) = Sat ( S 4 u cc, A LEK ) ; this set is in NE XP T IME Proof. 1. Full S 4 u cc is logspace-reducible to its fragment with no negative occurrences of c ≤ k ( τ ) 2. This fragment of S 4 u cc has exponential fmp (by continuous topological filtration) Topological Methods in Logic Tbilisi 5.06.08 7

S 4 u c in Euclidean spaces satisfiable in R 2 but not in R : • � � � � � � c ( v i ) ∧ v i ∩ v j � = 0 ∧ v 1 ∩ v 2 ∩ v 3 = 0 1 ≤ i ≤ 3 1 ≤ i<j ≤ 3 satisfiable in R 3 but not in R 2 : • � � � � v i ⊆ e ◦ c ( e ◦ � � � � � � ∧ v i � = 0 ∧ e i,j ∩ e k,l = 0 ∧ i,j ) j,k 1 ≤ i ≤ 5 1 ≤ i<j ≤ 5 i ∈{ j,k } { i,j }∩{ k,l } = ∅ • satisfiable in connected spaces but not in R n , for any n ≥ 1 : � ( v − � � ( v 1 ∩ v 2 = 0 ) ∧ i ⊆ v i ) ∧ c ( v i ) ∧ ¬ c ( v 1 ∩ v 2 ) i =1 , 2 Theorem. Sat ( S 4 u cc, R ) is PS PACE -complete Proof. Encoding in temporal logic with S and U over ( R , < ) Topological Methods in Logic Tbilisi 5.06.08 8

Regular closed sets and B X ⊆ T is regular closed if X = X ◦− RC ( T ) regular closed subsets of T RC ( T ) = sets of the form X ◦− , for X ⊆ T RC ( T ) is a Boolean algebra ( RC ( T ) , + , − , ∅ , T ) − X = ( X ) − where X + Y = X ∪ Y and B -terms: | − τ | τ 1 · τ 2 τ ::= r i regular closed sets! B -formulas: | ¬ ϕ | ϕ 1 ∧ ϕ 2 ϕ ::= τ 1 = τ 2 h B is a fragment of S 4 u : B -terms − → S 4 -terms � − − , h ( r i ) = v ◦ h ( τ 1 · τ 2 ) = ( h ( τ 1 ) ∩ h ( τ 2 )) ◦− , � h ( − τ 1 ) = h ( τ 1 ) i Theorem. Sat ( B , R EG ) = Sat ( B , C ON R EG ) = Sat ( B , RC ( R n )) , n ≥ 1 , and this set is NP -complete Proof. Every satisfiable B -formula ϕ is satisfied in a discrete topological space with ≤ | ϕ | points Topological Methods in Logic Tbilisi 5.06.08 9

C = B + contact predicate ↓ Whitehead’s ‘connection’ relation C -formulas: ::= τ 1 = τ 2 | C ( τ 1 , τ 2 ) | ¬ ϕ | ϕ 1 ∧ ϕ 2 ϕ = C ( τ 1 , τ 2 ) iff τ M ∩ τ M M | � = ∅ 1 2 a.k.a. BRCC - 8 r · s = 0 r · ( − s ) = 0 s · ( − r ) = 0 C ( r, − s ) C ( s, − r ) C ( r, s ) . ¬ C ( r, s ) . r r r s r s DC ( r, s ) EC ( r, s ) s TPP ( r, s ) TPPi ( r, s ) s NTPP ( r, s ) NTPPi ( r, s ) EQ ( r, s ) r PO ( r, s ) r s r s r s s r · s � = 0 . r = s . ¬ C ( r, − s ) ¬ C ( s, − r ) ( − r ) · s � = 0 r · ( − s ) � = 0 Topological Methods in Logic Tbilisi 5.06.08 10

Quasi-saw models for C cc Lemma. Every satisfiable C cc -formula is satisfied in a quasi-saw model x 1 x 2 x 3 x 4 x 5 x 6 x 7 depth 0 W 0 ❍ ❨ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❍ ❏ ❪ ✡ ✣ ❏ ❪ ✡ ✣ ✣ ✡ ❏ ❪ ❏ ❪ ✡ ✣ ✻ ❍ ❏ ✡ ❏ ✡ ✡ ❏ ❏ ✡ ❍ ❍ ❏ ✡ ❏ ✡ ✡ ❏ ❏ ✡ ❍ W 1 depth 1 s s s s s s A valuation may be defined only on points of depth 0 and ‘computed’ on points of depth 1 z ∈ τ M ∩ W 1 iff there is x ∈ τ M ∩ W 0 with zRx x 1 x 2 x 3 x 5 x 6 x 7 PPPPPPPP ❝ s ❝ s ❝ ❝ s ❝ ❝ s P ❝ x 4 Topological Methods in Logic Tbilisi 5.06.08 11