Representation theorems and the semantics of (semi)lattice based - PDF document

Representation theorems and the semantics of (semi)lattice based logics Viorica Sofronie-Stokkermans Max-Planck-Institut f¨ ur Informatik Saarbr¨ ucken Germany

Overview • Motivation • Connection between different classes of mo • Representation theorems • Examples • Decidability results • Automated theorem proving • Conclusions

Motivation Logical consequence provability relation logical connective ⊢ → Residuation condition if and only if p, q ⊢ r p ⊢ q →

Motivation. Premise combination Structural rules Γ , ∆ ⊢ A Γ , ∆ ⊢ A Γ , X Γ , Y, ∆ ⊢ A ∆ , Γ ⊢ A Γ , X (Weakening) (Exchange) (Contraction) Examples – Relevant logic weakening may not hold – Linear logic weakening, contraction do – Lambek calculus contraction, exchange do

Motivation. Premise combination Logical consequence provability relation logical connective ⊢ → ≤ → Residuation condition if and only if φ, ψ ⊢ γ φ ⊢ ψ → [ φ ] ◦ [ ψ ] ≤ [ γ ] [ φ ] ≤ [ ψ ]

Motivation. Premise combination Structural rules Γ , φ, ∆ ⊢ A Γ , φ, ψ, ∆ ⊢ A Γ , φ, Γ , ψ, φ, ∆ ⊢ A Γ , ψ, φ, ∆ ⊢ A Γ , φ, (Weakening) (Exchange) (Contraction) [ ψ ] ◦ [ φ ] ≤ [ φ ] [ φ ] ◦ [ ψ ] ≤ [ ψ ] ◦ [ φ ] [ φ ∆ , A, ∆ ′ ⊢ B ( φ 1 , φ 2 ) , φ 3 ⊢ A Γ ⊢ A ∆ , Γ , ∆ ′ ⊢ B φ 1 , ( φ 2 , φ 3 ) ⊢ A (Regrouping) (Cut) associativity of ◦ ≤ partial order; ◦ monotone

Definitions ( M, ≤ ) poset; ◦ , → : M 2 → M → is the left residuation associated with ◦ if a ◦ b ≤ c iff → is the right residuation associated with ◦ if b ◦ a ≤ c iff Commutative: ( M, ≤ , ◦ , → , 1) left residuated monoid if x ◦ y = – ( M, ◦ , 1) monoid; ◦ monotone in all arguments Integral: – → left residuation associated with ◦ BCC -algeb ( M, ∨ , ∧ , ◦ , → ) left residuated lattice if – ( M, ∨ , ∧ ) lattice; ◦ join-hemimorphism in both arguments – → left residuation associated with ◦ .

Examples Positive logics [Goldblatt 1974, Dunn 1995] Binary logics • no implication in the language φ ⊢ ψ • algebraic models: lattices with operators Logics based on Heyting algebras Post-style • algebraic models: Heyting algebras with operators p ∧ q ≤ r iff Logics based on residuated (semi)lattices � Lukasiewicz-st • algebraic models: residuated (semi)lattices with operato p ◦ q ≤ r iff

Examples • positive logics [Dunn 1995] SLO LO • (modal) intuitionistic • G¨ odel logics [G¨ odel 1930] DLO • SH n , SHK n logics [Iturrioz RDO • Post logics and generalizations • modal logic, dynamic HAO • relevant logic RL [Urquha BAO • fuzzy logics G¨ odel, � Lukasiewicz, • BCC and related logics • Lambek calculus; linea

� Motivation. Semantics Algebraic models ( A, D ) Var � � Fma ( Var Kripke-style models m : Var ( W, { R W } R ∈ Rel ) meaning Relational models algebras of relations

Motivation. Decidability results Logical calculi ◦ Gentzen-style calculi ◦ natural deduction ◦ hypersequent calculi [Avron 1991] Semantics ◦ Algebraic semantics ◦ Kripke-style semantics ◦ Relational semantics Automated theorem proving ◦ embedding into FOL + resolution ◦ tableau methods ◦ natural deduction; labelled deductive systems

Connections between classes of Algebraic models � � ������������������� � � � � representation theorems � representation � � � � � (algebras of sets) � (algeb � � � � � � � Kripke models Relational

� � Algebraic and Kripke-style semantics Algebraic models Kripke-style models D (C) A R E (i) E ( K ) ⊆ P ( K ) algebra of subsets of K (ii) i : A → E ( D ( A )) injective homomorphism ( K, m ) K ∈ R ; m : Var → E ( K Kripke-style models r | = a Theorem If A , R satisfy (C)(i,ii) then A | = φ iff

� � Algebraic and relational semantics Algebraic models Relational models D (C) A R E (i) E ( K ) algebra of relations (ii) i : A → E ( D ( A )) injective homomorphism K ∈ R ; f : Var → E ( K ( K, f ) Relational models a = | a Theorem If A , R satisfy (C)(i,ii) then = φ iff A |

� Representation theorems Natural Dualities: V = ISP ( P ) A ∼ → Hom Rel ( D ( A ) , P ) P ’alter-ego’ D ( A ) � � �������������������� � � � � � � � � � � � � � � � � � Stone 1940: Bool = ISP ( B 2 ) Priestley 1972: D 01 → OF ( D ( L )) → P ( D ( B )) L ֒ B ֒ η L ( x ) = { F ∈ D ( L η B ( x ) = { F ∈ D ( B ) | x ∈ F } Semilattices: SL = ISP ( S 2 ) ( S, ∧ ) ֒ → ( SF ( D ( S )) , ∩ ) η S ( x ) = { F ∈ D ( S ) | x ∈ F } Lattices: η L : ( L, ∧ , ∨ ) ֒ → ( SF ( D ( L )) , ∩ , ∨ ) η L ( x ) := { F ∈ D ( L ) | x ∈ F }

Example 1. Boolean algeb

Example 2. Distributive lattices

Example 3. Semilattices

Example 4. Lattices

Other representation theorems Boolean algebras with operators General Idea: • A �→ D ( A ) topological • J´ onsson and Tarski (1951) with additional Distributive lattices with operators • Goldblatt (1986), VS (2000) • A ∼ = ClosedSubsets Lattices (with operators) • Urquhart (1978) closed wrt: topological order structure • Allwein and Dunn (1993) ... • Dunn and Hartonas (1997) • operators �→ relations • Hartonas (1997) “Gaggles”, “tonoids” Dunn (1990, 1993)

� � Representation theorems f A : A ε 1 × · · · × A εn → A ε join-hemimorphism f ∈ Σ ε 1 ...εn → ε : D D � Rp Σ � SLSp Σ DLO Σ SLO Σ E E R f ( F 1 , . . . , F n , F ) iff f ( F ε 1 1 , . . . , F εn n ) ⊆ F ε D ( A ) ( R − 1 ( U ε 1 1 , . . . , U εn n )) ε E ( X ) f R ( U 1 , . . . , U n ) = Example x ◦ y ≤ z iff x ≤ y → z has type + 1 , +1 → + 1 R ◦ ( F 1 , F 2 , F 3 ) iff F 1 ◦ F 2 ◦ R → ( F 1 , F 2 , F 3 ) iff F 1 → F c → has type + 1 , − 1 → − 1 2 R → ( F 1 , F 2 , F 3 ) iff R ◦ ( F 3 ,

� � Algebraic and Kripke-style semantics D (C) A SLO E LO (i) E ( K ) ⊆ P ( K ) DLO algebra of subsets (ii) i : A ֒ → E ( D ( A RDO ( K, m ), m : Var → HAO r ( K, m ) | = x φ iff x ∈ BAO DLO Priestley rep η A : A → OF SLO , LO Representation (semi)lattices η A : A → S

Logic Algebraic Kripke-style meaning models models DLO Σ Rp Σ m : V Positive ( L, ∨ , ∧ , 0 , 1 , { f } f ∈ Σ ) ( X, ≤ , { R } R ∈ Σ ) HAO Σ Rp Σ m : V Post-style ( L, ∨ , ∧ , ⇒ , 0 , 1 , { f } f ∈ Σ )( X, ≤ , { R } R ∈ Σ ) BAO Σ BAO Σ m : ( B, ∨ , ∧ , 0 , 1 , ¬ , { f } f ∈ Σ ) ( X, { R } R ∈ Σ ) RDO RSp m : V Lukasiewicz � -style ( L, ∨ , ∧ , 0 , 1 , ◦ , → ) ( X, ≤ , R ◦ ) RSO , RLO RSO , RLO m : V ( S, ∧ , 0 , 1 , ◦ , → ) ( X, ∧ , R ◦ ) ( S, ∨ , ∧ , 0 , 1 , ◦ , → ) ( X, ∧ , R ◦ )

Overview • Motivation • Connection between different classes of mo • Representation theorems • Examples • Decidability results • Automated theorem proving • Conclusions

Class u.w.p. References Lattices PTIME Skolem (1920), Burris decidable Blok, Van Alten (1999) ResLatMon decidable Blok, Van Alten (1999) ResLatIntMon decidable Blok, Van Alten (1999) BCK → Modular Lattices undecidable Freese (1980), Herrmann co-NP complete Bloniarz et al.(1987) D 01 DLO Σ , RDO Σ EXPTIME VS (1999, 2001) decidable Andreka DLSgr ∨ , d subclasses undecidable Urquhart (1995) Heyting Algebras DEXP VS (1999) undecidable Kurucz, Nemeti et al. HASgr ∨ , d Boolean Algebras co-NP complete Cook (1971) undecidable Kurucz, Nemeti et al. ResBoolMon undecidable Kurucz, Nemeti et al. BoolSgr ∨ , d decidable Gyuris (1992) BoolSgr ∨

Decidability results Semantics • Algebraic semantics finite model property (uniform) word problem decidable • Kripke-style semantics finite model property embedding into decidable fragments devise sound and complete decision • Relational semantics relational proof systems Automated theorem proving ◦ embedding into FOL + ATP in first-order logic ◦ tableau methods ◦ natural deduction; labelled deductive systems

Class u.w.p. References Lattices PTIME Skolem (1920), Burris decidable Blok, Van Alten (1999) ResLatMon decidable Blok, Van Alten (1999) ResLatIntMon decidable Blok, Van Alten (1999) BCK → Modular Lattices undecidable Freese (1980), Herrmann co-NP complete Bloniarz et al.(1987) D 01 DLO Σ , RDO Σ EXPTIME VS (1999, 2001) decidable Andreka DLSgr ∨ , d subclasses undecidable Urquhart (1995) Heyting Algebras DEXP VS (1999) undecidable Kurucz, Nemeti et al. HASgr ∨ , d Boolean Algebras co-NP complete Cook (1971) undecidable Kurucz, Nemeti et al. ResBoolMon undecidable Kurucz, Nemeti et al. BoolSgr ∨ , d decidable Gyuris (1992) BoolSgr ∨

Resolution-based methods Advantages • direct encoding • restricted (hence efficient) calculi – ordering, selection – simplification/elimination of redundancies • allow use of efficient implementations (SPASS, Saturate) • in many cases better than equational reasoning AC operators �→ logical operations