From parametricity to modularity and back in correspondence theory: preliminary considerations Alessandra Palmigiano http://www.appliedlogictudelft.nl SYCO 3 Oxford 27 March 2019
The phenomenon of correspondence F , w � �� p → � p F | = ∀ y , z ( xRy & yRz → xRz )[ w ] iff
The phenomenon of correspondence F , w � �� p → � p F | = ∀ y , z ( xRy & yRz → xRz )[ w ] iff ( ⇒ ) Assume wRy and yRz . To show: w ∈ R − 1 [ z ] . Consider the minimal valuation making the antecedent true at w : V ∗ ( p ) = { z } . If wRy and yRz then F , V ∗ , w � �� p . Hence, F , V ∗ , w � � p , i.e. ] V ∗ = R − 1 [ V ∗ ( p )] = R − 1 [ z ] . w ∈ [ [ � p ]
Correspondence theory ◮ gives syntactic conditions for modal formulas to have a first order correspondent (e.g. Sahlqvist formulas) ◮ Computes algorithmically the first order correspondent of these formulas ◮ Benefits: These formulas generate logics that are strongly complete w.r.t. first-order definable classes of frames.
Correspondence via Duality Correspondence theory arises semantically: ✬✩ Kripke ✫✪ Frames ■ ✍ q Modal First order Correspondence logic logic ✐
Correspondence via Duality An asymmetry: ✬✩ Kripke ✫✪ Frames ■ ✍ Modal First order logic logic Non canonical Canonical interpretation interpretation
Correspondence via Duality Symmetry re-established via duality : ✬✩ ✬✩ q Kripke BAOs ✫✪ ✫✪ Frames ✐ ✒ ■ Modal First order logic logic
Correspondence via Duality Correspondence available not just for modal logic: ✬✩ ✬✩ q Algebra Spaces ✫✪ ✫✪ ✐ ✒ ■ Model AAL theory q Propositional First order Correspondence logic logic ✐
Correspondence via Duality Correspondence available not just for modal logic: ✬✩ ✬✩ q Algebra Spaces ✫✪ ✫✪ ✐ ✒ ■ Model AAL theory q Propositional First order Correspondence logic logic ✐ ◮ specific correspondences as logical reflections of dualities ◮ dual characterizations as instances of unified correspondence
Unified correspondence Display calculi Hybrid logics [GMPTZ18] [CR17] MV-logics Normal (D)LE-logics [BCM19] [CP12, CP19] Polarity-based and Mu-calculi graph-based semantics [CFPS15, CGP14, CC17] [CFPPW] Regular DLE-logics Sahlqvist via translation Kripke frames with [CPZ19] impossible worlds Constructive canonicity [PSZ17a] [CP16, CCPZ] Finite lattices and Jónsson-style vs monotone ML Sambin-style canonicity Canonicity via [FPS] [PSZ17b] pseudo-correspondence [CPSZ]
Main tools of unified correspondence Parametric Sahlqvist theory ◮ Definition of Sahlqvist formulas/sequents for all signatures of normal (D)LE-logics ◮ in terms of the order-theoretic properties of the algebraic interpretation of logical connectives The algorithm ALBA (also parametric ) ◮ computes the first-order correspondent of normal DLE- terms/inequalities. ◮ reduction steps sound on complex algebras of relational structures (perfect normal DLEs)
Normal DLE-logics (D)LE: (Distributive) Lattice Expansions: A = ( L , F A , G A ) (distributive) lattice signature + operations of any finite arity. Additional operations partitioned in families f ∈ F and g ∈ G . Normality : In each coordinate, ◮ f -type operations preserve finite joins or reverse finite meets ; ◮ g -type operations preserve finite meets or reverse finite joins . Examples ◮ Distributive Modal Logic: F := { � , � } and G := { � , � } ◮ Bi-intuitionistic modal logic: F := { � , > } and G := { � , →} ◮ Full Lambek calculus: F := {◦} and G := { /, \} ◮ Lambek-Grishin calculus: F := {◦ , / ⊕ , \ ⊕ } and G := {⊕ , / ◦ , \ ◦ } ◮ . . . Relational/complex algebra semantics ◮ f -type operations have residuals f ♯ i in each coordinate i ; ◮ g -type operations have residuals g ♭ h in each coordinate h .
Inductive inequalities − + ≤ + f , − g − g , + f , + ∨ , −∧ + ∨ , −∧ + g , − f ... + g , − f ... + p + p
Examples: reflexivity and transitivity ∀ p [ � p ≤ p ] iff ∀ p ∀ j ∀ m [( j ≤ � p & p ≤ m ) ⇒ j ≤ m ] (generators) iff ∀ p ∀ j ∀ m [( � j ≤ p & p ≤ m ) ⇒ j ≤ m ] (adjunction) ∀ j ∀ m [ � j ≤ m ⇒ j ≤ m ] iff (Ackermann) iff ∀ j [ j ≤ � j ] (Inv. Ackermann)
Examples: reflexivity and transitivity ∀ p [ � p ≤ p ] iff ∀ p ∀ j ∀ m [( j ≤ � p & p ≤ m ) ⇒ j ≤ m ] (generators) iff ∀ p ∀ j ∀ m [( � j ≤ p & p ≤ m ) ⇒ j ≤ m ] (adjunction) ∀ j ∀ m [ � j ≤ m ⇒ j ≤ m ] iff (Ackermann) iff ∀ j [ j ≤ � j ] (Inv. Ackermann) ∀ p [ �� p ≤ � p ] ∀ p ∀ j ∀ m [( j ≤ p & � p ≤ m ) ⇒ �� j ≤ m ] iff (generators) iff ∀ j ∀ m [ � j ≤ m ⇒ �� j ≤ m ] (Ackermann) ∀ j [ �� j ≤ � j ] iff (Inv. Ackermann)
Examples: reflexivity and transitivity ∀ p [ � p ≤ p ] iff ∀ p ∀ j ∀ m [( j ≤ � p & p ≤ m ) ⇒ j ≤ m ] (generators) iff ∀ p ∀ j ∀ m [( � j ≤ p & p ≤ m ) ⇒ j ≤ m ] (adjunction) ∀ j ∀ m [ � j ≤ m ⇒ j ≤ m ] iff (Ackermann) iff ∀ j [ j ≤ � j ] (Inv. Ackermann) ∀ p [ �� p ≤ � p ] ∀ p ∀ j ∀ m [( j ≤ p & � p ≤ m ) ⇒ �� j ≤ m ] iff (generators) iff ∀ j ∀ m [ � j ≤ m ⇒ �� j ≤ m ] (Ackermann) ∀ j [ �� j ≤ � j ] iff (Inv. Ackermann) Modularity: One reduction, many translations! On Kripke frames: ∀ j [ j ≤ � j ] ∀ x [∆[ { x } ] ⊆ R [ { x } ]] i.e. ∆ ⊆ R � ∀ x [ R − 1 [ R − 1 [ { x } ]] ⊆ R − 1 [ { x } ]] ∀ j [ �� j ≤ � j ] i.e. R ; R ⊆ R � But how about more general semantic contexts?
Questions Conceptual questions ◮ can we connect the meaning of the first-order correspondents in the ‘Boolean contexts’ to the meaning of those in other contexts? ◮ can we characterize or capture (or even define) meaning-preservation across contexts? Technical questions ◮ is there an automated way to syntactically generate fist-order correspondents from those on the Boolean context? ◮ more broadly, is there an automated way to syntactically generate fist-order correspondents relative to a more general semantic context from those relative to a more restricted context?
Case Studies
CS1: Polarity-based semantics of LE-logics Formal contexts ( A , X , I ) are abstract representations of databases: X I A A : set of Objects X : set of Features I ⊆ A × X . Intuitively, aIx reads: object a has feature x Formal concepts: “rectangles” maximally contained in I
Formulas as formal concepts ( abcd , ∅ ) p q y x z ( ab , x ) ( cd , z ) V ( q ) X V ( p ) � I ( bc , y ) A ( b , xy ) ( c , yz ) a c b d p pq q ( ∅ , xyz ) Let P = ( A , X , I ) and P + be the complex algebra of P . Models: M := ( P , V ) with V : Prop → P + V ( p ) := ([ [ p ] ] , ( [ p ] )) M , a � p membership: iff a ∈ [ [ p ] ] M M , x ≻ p x ∈ ( [ p ] ) M description: iff
Semantics of modal formulas Enriched formal contexts: F = ( A , X , I , { R i | i ∈ Agents } ) R i ⊆ A × X and ∀ a (( R ↑ [ a ]) ↓↑ = R ↑ [ a ]) and ∀ x (( R ↓ [ x ]) ↑↓ = R ↓ [ x ]) ⊤ y x z a = x d = z X y � I A c b a c b d ⊥ � i ϕ : concept ϕ according to agent i V ( � i ϕ ) = � i V ( ϕ ) = ( R ↓ )] , ( R ↓ )]) ↑ ) i [( [ ϕ ] i [( [ ϕ ] M , a � � i ϕ iff for all x ∈ X , if M , x ≻ ϕ , then aR i x M , x ≻ � i ϕ for all a ∈ A , if M , a � � i ϕ , then aIx iff
Epistemic interpretation Reflexivity aka Factivity ∀ p [ � p ≤ p ] iff ∀ j [ j ≤ � j ] ∀ a [ a ↑↓ ⊆ R ↓ [ a ↑ ]] iff ∀ a [ a ∈ R ↓ [ a ↑ ]] ( R ↓ [ a ↑ ] Galois-stable) iff iff R i ⊆ I Agent i ’s attributions are factually correct! Transitivity aka Positive introspection ∀ p [ � p ≤ �� p ] iff ∀ m [ � m ≤ �� m ] ∀ x [ R ↓ [ x ↓↑ ] ⊆ R ↓ [( R ↓ [ x ↓↑ ]) ↑ ]] iff ∀ x [ R ↓ [ x ] ⊆ R ↓ [( R ↓ [ x ]) ↑ ]] ( R ↓ [ a ↑ ] Galois-stable) iff R ⊆ R ; R iff If agent i recognizes object a as an x -object, then i must also attribute to a all the features shared by x -objects according to i .
CS2: Graph-based semantics of LE-logics One-sorted structures G = ( Z , E ) , with E reflexive: ab ac cb � a c b ca bc ba G + := ( Z , Z , E c ) +
CS2: Graph-based semantics of LE-logics One-sorted structures G = ( Z , E ) , with E reflexive: ab ac ac cb � a c b ca bc ba G + := ( Z , Z , E c ) + Representation. States: maximally disjoint filter-ideal pairs ( F , I ) ; F ∩ I ′ = ∅ ( F , I ) E ( F ′ , I ′ ) iff
CS2: Graph-based semantics of LE-logics One-sorted structures G = ( Z , E ) , with E reflexive: ab ab ac cb � a c b ca bc ba G + := ( Z , Z , E c ) + Representation. States: maximally disjoint filter-ideal pairs ( F , I ) ; F ∩ I ′ = ∅ ( F , I ) E ( F ′ , I ′ ) iff ∀ z ′ [ zR � z ′ ⇒ M , z ′ ⊁ ψ ] M , z � � ψ iff ∀ z ′ [ z ′ Ez ⇒ M , z ′ � � ψ ] M , z ≻ � ψ iff
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