Coalgebraic Correspondence Theory and Gaifman Locality Tadeusz Litak a , Dirk Pattinson b , and Lutz Schr¨ oder a a Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg b Australian National University, Canberra ALCOP 2015, Delft Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 1 ALCOP 2015, Delft
Introduction Modal logic is invariant under bisimulation. Modal logic is a fragment of FOL: � φ ˆ = ∀ y . xRy → φ ( y ) ◮ Van Benthem: Modal logic is the bisimulation-invariant fragment of FOL. ◮ Rosen: This remains true over finite structures. Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 2 ALCOP 2015, Delft
Modal Logic beyond � and � ◮ Probabilistic modal logic ◮ Frames: Markov chains ( X , ( P x ) x ∈ X ) ◮ Operators: L p ‘with probability at least p ’ ◮ Graded modal logic ◮ Frames: Multigraphs ( X , f : X × X → N ∪{ ∞ } ) ◮ Operators: � k ‘in more than k successors’ ◮ Conditional logic ◮ Frames: e.g. selection function frames ( X , f : X ×P ( X ) → P ( X )) ◮ Operators: ⇒ ‘if . . . then normally . . . ’ ◮ Neighbourhood logic ◮ Frames: Neighbourhood frames ( X , R ⊆ X ×P ( X )) ◮ Operators: � Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 3 ALCOP 2015, Delft
Modal Logic beyond � and � ◮ Probabilistic modal logic ◮ Frames: Markov chains ( X , ( P x ) x ∈ X ) ◮ Operators: L p ‘with probability at least p ’ ◮ Graded modal logic ◮ Frames: Multigraphs ( X , f : X × X → N ∪{ ∞ } ) ◮ Operators: � k ‘in more than k successors’ ◮ Conditional logic ◮ Frames: e.g. selection function frames ( X , f : X ×P ( X ) → P ( X )) ◮ Operators: ⇒ ‘if . . . then normally . . . ’ ◮ Neighbourhood logic ◮ Frames: Neighbourhood frames ( X , R ⊆ X ×P ( X )) ◮ Operators: � What about FO correspondence theory for these? Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 3 ALCOP 2015, Delft
Coalgebraic Modal Logic Similarity type Λ φ , ψ ::= ⊥ | φ ∧ ψ | ¬ φ | ♥ φ ( ♥ ∈ Λ) . Interpret over functor T : Set → Set by predicate liftings [[ ♥ ]] X : P ( X ) → P ( TX ) . Semantics: satisfaction relation | = over T -coalgebras ξ : X → TX , x | = ♥ φ : ⇐ ⇒ ξ ( x ) ∈ [[ ♥ ]] X [[ φ ]] where [[ φ ]] = { y ∈ X | y | = φ } . ◮ This covers all examples above, and more. Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 4 ALCOP 2015, Delft
Coalgebraic Predicate Logic Generalize Chang’s modal FO language (1973) to coalgebraic modalities: φ ::= ⊥ | ¬ φ | φ 1 ∧ φ 2 | x = y | P ( � x ) | ∀ x . φ | x ♥⌈ y : φ ⌉ ◮ Model = FO-model + T -coalgebra ◮ Pure CPL: without P ( � x ) ◮ M , v | = x ♥⌈ y : φ ⌉ ξ ( v ( x )) ∈ [[ ♥ ]] { c ∈ X | M , v [ y �→ c ] | = φ } iff Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 5 ALCOP 2015, Delft
The Standard Translation ST x ( ♥ φ ) = x ♥⌈ x : ST x φ ⌉ . CML = Single-variable quantifier-free CPL Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 6 ALCOP 2015, Delft
Examples ◮ Kripke semantics ( TX = P X ×P V ): Standard FO correspondence language = ˆ x � ⌈ z : z = y ⌉ xRy ◮ Neighbourhoods ( T = Q◦Q op ): Chang’s modal FO language ◮ Graded ML ( T = bags): local counting quantifiers ∃ x k y . φ = ˆ x � k − 1 ⌈ y : φ ⌉ (Axiomatize FO with counting: ¬∃ x 2 y . y = z ) ◮ Similarly for probabilistic ML ( T = distributions), w x y ( φ ) ≥ p = ˆ x L p ⌈ y : φ ⌉ Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 7 ALCOP 2015, Delft
Outline of Otto’s Proof of Rosen’s Theorem ◮ Assume w.l.o.g. finitely many propositional variables. ◮ Note that invariance of φ under disjoint sums implies locality, via Gaifman locality. ◮ Use local unravellings to reduce to locally tree-like structures. ◮ Combine this to prove that φ is already ∼ k -invariant. ◮ Conclude that φ is equivalent to a (finite) modal formula of depth k . Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 8 ALCOP 2015, Delft
Recall: Gaifman’s Theorem Gaifman graph of a FO structure: x − − − y iff x and y are in some basic relation � Gaifman distance, Neighbourhoods N M d ( u ) . Definition: A formula φ ( x ) is Gaifman d -local if for u , w ∈ M , d ( u ) ∼ N M = N M d ( w ) = ⇒ ( M , u | = φ ( x ) ⇐ ⇒ M , w | = φ ( x )) Gaifman’s theorem: Every φ ( x ) ∈ FOL is Gaifman local. Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 9 ALCOP 2015, Delft
Gaifman distance in CPL Wrong idea: “ x − − − y if x ♥⌈ y : φ ⌉ and φ ( z ) ” E.g. in probabilistic logic xL 1 ⌈ y : ⊤⌉ and ⊤ ( z ) , so x − − − z for all x , z . Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 10 ALCOP 2015, Delft
Solution: Support ◮ A ⊆ X is a support of t ∈ TX iff t ∈ TA ⊆ TX . ◮ Then by naturality of predicate liftings, t ∈ [[ ♥ ]] X [[ φ ]] iff t ∈ [[ ♥ ]] A ([[ φ ]] ∩ A ) ◮ Supporting Kripke frame R for ξ : X → TX : R ( x ) = { y | xRy } support of ξ ( x ) Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 11 ALCOP 2015, Delft
Gaifman Locality for Support CPL ◮ Pure support CPL = Pure CPL plus binary predicate supp interpreted by supporting Kripke frame ◮ Inherit Gaifman theorem by translating into multisorted FO language ♥ ⊆ s × n ∈ ⊆ s × n supp ⊆ s × s . Neighbourhood compatibility: Isomorphic nbhds (nearly) remain isomorphic Theorem (Gaifman theorem for pure support-CPL) : Pure support-CPL is Gaifman local Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 12 ALCOP 2015, Delft
The Coalgebraic van Benthem/Rosen Theorem Infinitary version: Λ separating, φ ( x ) ∈ FOL (Λ) ≈ -invariant (over finite models) = ⇒ φ ( x ) equivalent (over finite models) to some infinitary finite-rank modal formula ψ ( x ) . Finitary version: Same with ψ ( x ) finitary for finite Λ . ◮ The finitary version is immediate from the infinitary version. Does the finitary van Benthem/Rosen theorem hold for infinite Λ ? Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 13 ALCOP 2015, Delft
Known Instances ◮ The classical van Benthem/Rosen theorem ◮ The van Benthem theorem for neighbourhood logic (Hansen/Kupke/Pacuit 2009) Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 14 ALCOP 2015, Delft
Conclusion ◮ Coalgebraic predicate logic: FOL over T -coalgebras. ◮ Have proved a coalgebraic van Benthem/Rosen theorem. ◮ Nagging open problem: for infinite signatures, want to improve to finitary formulas. ◮ Key ingredient: Gaifman locality for CPL ◮ Measure distance via support ◮ Inherit from standard FOL by making neighbourhoods explicit Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 15 ALCOP 2015, Delft
Future Work ◮ Investigation of CPL: ◮ Model theory ◮ Decidable fragments ◮ Sahlqvist theory (working from Dahlqvist/Pattinson 2013) Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 16 ALCOP 2015, Delft
The Classical Correspondence Language ◮ One unary predicate p ( x ) for each propositional variable p ◮ Binary relation R ( x , y ) ◮ No axioms or restrictions on models ◮ Standard translation: ST x ( p ) = p ( x ) ST x ( � φ ) = ∀ y . R ( x , y ) → ST y ( φ ) . ◮ Van Benthem/Rosen: for all φ ( x ) ∈ FOL , TFAE: 1. φ ( x ) bisimulation-invariant (over finite structures) 2. φ ( x ) ↔ ST x ( ψ ) for some modal ψ (over finite structures) ◮ Janin/Walukiewicz: the bisimulation-invariant fragment of MSOL is the µ -calculus. Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 17 ALCOP 2015, Delft
� � � � � � � � Coalgebraic Unravelling Recall: Coalgebraic modal logic captures behavioural equivalence ◮ defined via cospans of morphisms X → • ← Y ◮ in general weaker than bisimilarity (via spans X ← • → Y ). Require bounded behavioural equivalence ≈ k , defined via the terminal sequence ξ � TX X ξ 0 ξ n T ξ n − 1 ξ 1 1 T 1 ... Tn ... Key facts: Lemma: For A , B trees of depth k , A , a ≈ B , b iff A , a ≈ k B , b . Unravelling Lemma: For A , a ex. A , a ≈ B , b s.t. N B 3 k ( b ) tree of depth k . Litak/Pattinson/Schr¨ oder: Coalgebraic Correspondence Theory and Gaifman Locality 18 ALCOP 2015, Delft
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