Horn Fragments of Halpern-Shoham Interval Temporal Logic Agi Kurucz Department of Informatics , King’s College London Joint work with D. Bresolin, E. Mu˜ noz-Velasco, V. Ryzhikov, G. Sciavicco and M. Zakharyaschev ACM Transactions on Computational Logic, vol. 18(3) (2017), 22:1-22:39
Allen’s interval relations Time: linear order ( T, ≤ ) Intervals: i = � x, y � with x ≤ y i ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ j i A j After j i B j Begins j i E j Ends j i D j During j i L j Later j i O j Overlaps j i ¯ A j j i ¯ B j j i ¯ E j j i ¯ D j j i ¯ L j j i ¯ O j Agi Kurucz — Logic Colloquium 2018, Udine 1
Allen’s interval relations – 2D representation ( T, ≤ ) i ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ � ✻ � L ¯ ¯ � j D B O A i A j � j � i B j ¯ j E E � y s i E j � j D i D j � j � ¯ i L j O B � j i O j � j � i ¯ A j ¯ � j A i ¯ B j � j ¯ i ¯ � L E j � j i ¯ D j � j i ¯ L j � j � i ¯ O j ✲ x ( T, ≤ ) ¯ ( T, ≤ ) × exp ( T, ≤ ) BE -fragment = ‘expanding’ modal product Agi Kurucz — Logic Colloquium 2018, Udine 2
Halpern-Shoham interval temporal logic HS : propositional multi-modal logic over the 12 Allen relations formulas: ϕ ::= ⊤ | ⊥ | p | ¬ ϕ | ϕ 1 ∧ ϕ 2 | � R � ϕ | [ R ] ϕ p ∈ Variables , R ∈ { L , B , A , D , O , E , ¯ L , ¯ B , ¯ A , ¯ D , ¯ O , ¯ E } models: where ν : Intervals → 2 Variables M = ( T, ≤ , ν ) truth-relation: M , i | = � R � ϕ ⇐ ⇒ M , j | = ϕ for some interval j with i R j M , i | = [ R ] ϕ ⇐ ⇒ M , j | = ϕ for all intervals j with i R j Variants: • discrete, dense, finite, . . . linear orders • open, closed, semi-closed intervals • ‘punctual’ � x, x � intervals allowed/disallowed • ‘reflexive/irreflexive’ interpretation of the Allen relations E.g.: � x 1 , y 1 � B � x 2 , y 2 � ⇐ ⇒ x 1 = x 2 and y 2 ≤ y 1 � x 1 , y 1 � B � x 2 , y 2 � ⇐ ⇒ x 1 = x 2 and y 2 < y 1 is not satisfiable with reflexive semantics p ∧ [ R ] ¬ p Agi Kurucz — Logic Colloquium 2018, Udine 3
The satisfiability problem of HS Task: Given a formula ϕ and a class C of linear orders, are there some model M based on a timeline from C and interval i such that M , i | = ϕ ? Undecidable over any ‘unbounded’ class of linear orders Halpern–Shoham 1991 ‘Taming’ attempts so far: • constraining the underlying linear orders • relativisations • softening the semantics • restricting the set of available modal operators • coarsening the interval relations • restricting the nesting of modal operators Agi Kurucz — Logic Colloquium 2018, Udine 4
Horn fragments of HS clausal normal form of HS -formulas: ::= λ | ¬ λ | ∀ ( λ 0 ∧ · · · ∧ λ n → λ n +1 ∨ · · · ∨ λ n + m ) | ϕ 1 ∧ ϕ 2 ϕ R ( ψ ∧ [ R ] ψ ∧ [¯ ∀ ψ = � where R ] ψ ) λ ::= ⊤ | ⊥ | p | � R � λ | [ R ] λ HS horn : ϕ ::= λ | ∀ ( λ 0 ∧ · · · ∧ λ n → λ ) | ϕ 1 ∧ ϕ 2 HS core : ϕ ::= λ | ∀ ( λ 0 → λ 1 ) | ∀ ( λ 0 ∧ λ 1 → ⊥ ) | ϕ 1 ∧ ϕ 2 HS ✷ horn and HS ✷ core : NO diamond operators � R � in λ Agi Kurucz — Logic Colloquium 2018, Udine 5
What can we express? Some syntactic sugar: for ψ = λ 1 ∧ · · · ∧ λ k • ∀ ( ψ → ¬ λ ) ≡ ∀ ( ψ ∧ λ → ⊥ ) � n � � ψ → λ ′ 1 ∧ · · · ∧ λ ′ ψ → λ ′ • ∀ n ) ≡ i =1 ∀ i ) � � • ∀ ψ → [ R ]( λ ′ 1 ∧· · ·∧ λ ′ n → λ ) ≡ ∀ ( ψ → [ R ] p ) ∧ ∀ ( p ∧ λ ′ 1 ∧· · ·∧ λ ′ n → λ ) � ψ → � R � ( λ ′ 1 ∧· · ·∧ λ ′ � ∀ ( ψ → � R � p ) ∧ ∀ ( p ∧ λ ′ 1 ∧· · ·∧ λ ′ • ∀ n → λ ) ≡ n → λ ) ∀ ( λ → [¯ ∀ ( � R � λ ∧ ψ → λ ′ ) R ]( ψ → λ ′ )) • ≡ Advanced courses cannot be given during the morning sessions. ∀ ( � ¯ D � MorningSession ∧ AdvancedCourse → ⊥ ) ∀ ( � ¯ B � LectureDay ∧ � A � Lunch ↔ MorningSession ) Teaching is both downward and upward hereditary. ∀ ( teaching → [ D ] teaching ) ([ D ] � O � teaching ∨� ¯ � � ∀ D � teaching ) ∧� B � teaching ∧� E � teaching → teaching Agi Kurucz — Logic Colloquium 2018, Udine 6
Our results on the satisfiability problem irreflexive semantics reflexive semantics HS horn undecidable ∗ HS core undecidable ∗ PS PACE -hard ∗ decidable? discrete: undecidable HS ✷ PT IME -complete ∗∗ horn dense: PT IME -complete discrete: PS PACE -hard HS ✷ in PT IME ∗∗ decidable? core dense: in PT IME ∗ over any ‘unbounded’ class of linear orders ∗∗ over any nonempty class of linear orders Note: propositional Datalog is PT IME -complete its core fragment is NL OG S PACE -complete Agi Kurucz — Logic Colloquium 2018, Udine 7
Proving PS PACE lower bounds In both HS ✷ • over discrete linear orders with irreflexive semantics core HS core • over arbitrary linear orders with arbitrary semantics it is possible • to identify/generate an infinite (or unbounded finite) sequence of ‘units’ • to pass polynomial-size information from one unit to the next PS PACE -hardness ❀ � HS ✷ � • over dense linear orders OR with reflexive semantics is in PT IME core � over arbitrary linear orders with irreflexive semantics • HS core � is even undecidable Agi Kurucz — Logic Colloquium 2018, Udine 8
PS PACE lower bounds – the tricks in HS ✷ discrete linear orders with irreflexive semantics: • over core identifying units: ∀ ( unit ↔ [ E ] ⊥ ) µ µ passing poly-size info to next unit: [ λ 1 & λ 2 ⇒ λ ] ← λ ∀ ( λ 1 → [ A ] µ ) ∀ ( λ 2 → [ A ] [ ¯ E ] µ ) λ 1 , λ 2 ∀ ([ A ] [ ¯ A ] µ → λ ) in HS core • over arbitrary linear orders with arbitrary semantics: unit ∧ ∀ ( unit → � A � unit ) generating units: ∀ ( unit → ¬ [ D ] unit ) µ 2 ‘compressing’ info to be passed: [ λ 1 & λ 2 ⇒ λ ] µ µ 1 ∀ ( λ 1 → � A � µ 1 ) ∧ ∀ ( λ 2 → � A � µ 2 ) ∀ ( µ 2 → ¬� ¯ B � µ 1 ) ∀ ( µ 1 → µ ∧ [ ¯ B ] µ ) ∧ ∀ ( µ 2 → [ B ] µ ) ← λ λ 1 , λ 2 ∀ ([ A ] µ → λ ) Agi Kurucz — Logic Colloquium 2018, Udine 9
Proving undecidability: encoding ‘grid-based’ problems • generating the ω × ω -grid (or arbitrarily large finite ‘initial segment’ of it) • passing poly-size info to right- and up-neighbours ❀ encoding • tilings • Turing machine computations • counter machine computations • Post correspondence problem • . . . Agi Kurucz — Logic Colloquium 2018, Udine 10
Often ‘half-grid’ is enough nw ω × ω = { ( i, j ) : i ≤ j < ω } the nw ω × ω tiling problem: given a finite set T t = ( left ( t ) , right ( t ) , up ( t ) , down ( t )) of tile types . . decide whether there exists τ : nw ω × ω → T such that, for all i ≤ j < ω , . up ( τ ( i, j )) = down ( τ ( i, j + 1)) and left ( τ ( i, j )) = right ( τ ( i + 1 , j )) whenever i < j . . (Berger 1966): the ω × ω tiling problem is undecidable the nw ω × ω tiling problem is ❀ undecidable . Agi Kurucz — Logic Colloquium 2018, Udine 11
Often ‘half-grid’ is enough Turing machine empty tape on nw ω × ω : computations starting with � $ C B � $ A B � $ A � steps $ ↑ → tape Agi Kurucz — Logic Colloquium 2018, Udine 12
Diagonal encoding of nw ω × ω HS -models are kind of grid-like BUT HS has NO ‘right-neighbour’ and ‘up-neighbor’ operators not even over discrete orders with irreflexive semantics Idea: Harel 1985, Halpern–Shoham 1991, Marx–Reynolds 1999, Reynolds–Zakharyaschev 2001 (1 , 4) (2 , 4) (3 , 4) (0 , 4) 10 (0 , 3) 6 7 8 9 (0 , 2) 3 4 5 (0 , 1) 1 2 (0 , 0) 0 • generate an infinite sequence of ‘units’ • right-neighbour of each non-diagonal unit is the next unit in the sequence • use ‘local’ pointers to access the units representing the up-neighbour of each grid-location Agi Kurucz — Logic Colloquium 2018, Udine 13
Diagonal encoding of nw ω × ω – version 1 diagonal Halpern–Shoham 1991 wall ↓ ւ line 4 → 10 line 3 → 6 7 8 9 line 2 → 3 4 5 line 1 → 1 2 0 • start of line 1 is 1 , and up of (0) = 1 • start of line i +1 is the end of line i + 1 , for all i > 0 • every line starts with some n on the wall and ends with some m on the diagonal • if n is in line i then up of ( n ) is in line i +1 • if m < n then up of ( m ) < up of ( n ) • if n > 0 is on the wall then there is m with up of ( m ) = n • if n is neither on the wall nor on the diagonal then there is m with up of ( m ) = n Agi Kurucz — Logic Colloquium 2018, Udine 14
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