Analytic Calculi for Substructural Logics: Theory and Applications - PowerPoint PPT Presentation

Analytic Calculi for Substructural Logics: Theory and Applications Agata Ciabattoni Vienna University of Technology agata@logic.at Proof Theory & Universal Algebra Computation

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Substructural logics include ▸ intuitionistic logic, ▸ intermediate logics, ▸ relevance logics, ▸ linear logic, ▸ fuzzy logics, ▸ ... lack the properties expressed by sequent calculus structural rules useful for reasoning about natural language, vagueness, resources, dynamic data structures, algebraic varieties, concurrency ... 1 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III This Talk Theory Systematic and automated introduction of sequent and hypersequent calculi with Kazushige Terui & Nikolaos Galatos (LICS 2008, Algebra Universalis 2011, APAL 2012, APAL 2017) 2 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III This Talk Theory Systematic and automated introduction of sequent and hypersequent calculi Chapter I with Kazushige Terui & Nikolaos Galatos (LICS 2008, Algebra Universalis 2011, APAL 2012, APAL 2017) Applications Extraction of concurrent λ -calculi Chapter III with Federico Aschieri & Francesco A. Genco (LICS 2017, Submitted 2018) From hypersequent calculi to natural deduction systems with Francesco A. Genco Chapter II (TOCL 2018) 3 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Substructural Logics Substructural logics Defined as axiomatic extensions of Full Lambek calculus FL subvarieties of (pointed) residuated lattices RL Algebraization For any set A ∪ { A , B } of formulas, A ⊢ FL + B A iff ε [A] ⊧ RL + ε ( B ) ε ( A ) where ε (−) is the equation corresponding to − . Example: G¨ odel logic obtained by adding ( α → β ) ∨ ( β → α ) to intuitionistic logic ( FL + exchange, weakening and contraction) or equivalently 1 ≤ ( x → y ) ∨ ( y → x ) 4 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Substructural Logics Substructural logics Defined as axiomatic extensions of Full Lambek calculus FL subvarieties of (pointed) residuated lattices RL Example: G¨ odel logic obtained by adding ( α → β ) ∨ ( β → α ) to intuitionistic logic ( FL + exchange, weakening and contraction) or equivalently 1 ≤ ( x → y ) ∨ ( y → x ) to Heyting algebras ( RL + commutativity, integrality and idempotency) 4 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Why analytic calculi? Substructural logics Defined as axiomatic extensions of Full Lambek calculus FL subvarieties of (pointed) residuated lattices RL Their applicability/usefulness strongly depends on the availability of Analytic calculi useful for establishing various properties of logics key for developing automated reasoning methods 5 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Sequent Calculus Sequents (Gentzen 1934) A 1 ,..., A n ⇒ B 1 ,..., B m Axioms: E.g. A ⇒ A , � ⇒ A Rules: Structural E.g. Γ , B , A ⇒ Π Γ , A , A ⇒ Π Γ ⇒ Π Γ , A , B ⇒ Π ( e , l ) ( c , l ) Γ , A ⇒ Π ( w , l ) Γ , A ⇒ Π Logical (left and right) Cut Γ ⇒ A Σ , A ⇒ Π Cut Γ , Σ ⇒ Π 6 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Sequent Calculus Sequents (Gentzen 1934) A 1 ,..., A n ⇒ B Axioms: E.g. A ⇒ A , � ⇒ A Rules: Structural E.g. Γ , B , A ⇒ Π Γ , A , A ⇒ Π Γ ⇒ Π Γ , A , B ⇒ Π ( e , l ) ( c , l ) Γ , A ⇒ Π ( w , l ) Γ , A ⇒ Π Logical (left and right) Cut Γ ⇒ A Σ , A ⇒ Π Cut Γ , Σ ⇒ Π 6 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Sequent Calculus – the cut rule Γ ⇒ A A ⇒ Π Cut Γ , Σ ⇒ Π key to prove completeness w.r.t. Hilbert systems A A → B modus ponens B bad for proof search Cut-elimination theorem Each proof using Cut can be transformed into a proof without Cut. 7 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Sequent Calculus – state of the art Cut-free sequent calculi have been successfully used to prove consistency, decidability, interpolation, . . . to give syntactic proofs of algebraic properties for which (in particular cases) semantic methods are not known or do not work well Many useful and interesting logics have no cut-free sequent calculus 8 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Some extensions of the sequent calculus A large range of generalizations of sequent calculus have been introduced hypersequent calculus (Avron, Mints, Pottinger) display calculus (Belnap) nested sequents (Br¨ unnler, Fitting) deep inference (Guglielmi) bunched calculi (Dunn, Mints, . . . ) labelled systems (Gabbay, Negri, Vigan´ o, . . . ) systems of rules (Negri) many placed sequents (TU Vienna) ... 9 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Defining analytic calculi: state of the art The definition of analytic calculi is usually logic-tailored. Steps: (i) choosing a framework (ii) looking for the “right” inference rule(s) (iii) proving cut-elimination 10 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Systematic introduction of (hyper)sequent calculi Chapter I 11 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III The base logic: FLe FLe ≈ commutative Full Lambek calculus FLe ≈ intuitionistic logic without weakening and contraction FLe ≈ intuitionistic Linear Logic without exponentials Algebraic semantics: A (bounded pointed) commutative residuated lattice is P = ⟨ P , ∧ , ∨ , ⊗ , → , ⊺ , 0 , 1 , �⟩ 1 ⟨ P , ∧ , ∨⟩ is a lattice with ⊺ greatest and � least 2 ⟨ P , ⊗ , 1 ⟩ is a commutative monoid. 3 For any x , y , z ∈ P , x ⊗ y ≤ z ⇐ ⇒ y ≤ x → z 4 0 ∈ P . 12 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Sequent calculus for commutative FL FLe A , B , Γ ⇒ Π Γ ⇒ A ∆ ⇒ B ⊗ r A ⊗ B , Γ ⇒ Π ⊗ l Γ , ∆ ⇒ A ⊗ B Γ ⇒ A B , ∆ ⇒ Π A , Γ ⇒ B Γ ⇒ A → B → r → l Γ , A → B , ∆ ⇒ Π A , Γ ⇒ Π B , Γ ⇒ Π Γ ⇒ A i Γ ⇒ A 1 ∨ A 2 ∨ r ∨ l 0 ⇒ 0 l A ∨ B , Γ ⇒ Π A i , Γ ⇒ Π Γ ⇒ A Γ ⇒ B ∧ r Γ ⇒ ⊺ ⊺ r A 1 ∧ A 2 , Γ ⇒ Π ∧ l Γ ⇒ A ∧ B Γ ⇒ Γ ⇒ Π Γ ⇒ 0 0 r ⇒ 1 1 r � , Γ ⇒ Π � l 1 , Γ ⇒ Π 1 l 13 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III (Commutative) Substructural Logics defined by adding Hilbert axioms to the sequent calculus FLe (or algebraic equations to commutative residuated lattices). From axioms to rules: example A , A , Γ ⇒ Π ( c ) Contraction: α → α ⊗ α A , Γ ⇒ Π Γ ⇒ Π Γ , A ⇒ Π ( w , l ) Weakening l: α → 1 Γ ⇒ Γ ⇒ A ( w , r ) Weakening r: 0 → α Equivalence between rules and axioms 14 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III (Commutative) Substructural Logics defined by adding Hilbert axioms to the sequent calculus FLe (or algebraic equations to commutative residuated lattices). From axioms to rules: example A , A , Γ ⇒ Π ( c ) Contraction: α → α ⊗ α A , Γ ⇒ Π Γ ⇒ Π Γ , A ⇒ Π ( w , l ) Weakening l: α → 1 Γ ⇒ Γ ⇒ A ( w , r ) Weakening r: 0 → α Equivalence between rules and axioms ⊢ FLe +( axiom ) ⊢ FLe +( rule ) = For which axioms can we do it? 14 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Algebraic Proof Theory (AC, N. Galatos and K. Terui – APAL 2012, APAL 2017) Which Hilbert axioms can be transformed into rules that preserve cut-elimination? Which algebraic equations over residuated lattices are preserved by algebraic completions? A completion of an algebra A is a complete algebra B (i.e. it has arbitrary ⋁ and ⋀ ) such that A ⊆ B . 15 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Classification Formulas are classified according to the polarity of their connectives w.r.t. a calculus (e.g., FLe ) (J.-M. Andreoli, 1992) Positive polarity: rule introducing the connective on the left is invertible Γ , A ⇒ Π Γ , B ⇒ Π ∨ l E.g. Γ , A ∨ B → Π Negative polarity: rule introducing the connective/quantifier on the right is invertible A , Γ ⇒ B Γ ⇒ A → B → r E.g. 16 / 51

Introduction Chapter I Curry-Howard Correspondence Chapter II Chapter III Substructural Hierarchy Definition (AC, Galatos and Terui, LICS 2008) The classes P n , N n of positive and negative axioms/equations are: P 0 ∶∶ = N 0 ∶∶ = atomic formulas P n + 1 ∶∶ = N n ∣ P n + 1 ∨ P n + 1 ∣ P n + 1 ⊗ P n + 1 ∣ 1 ∣ � N n + 1 ∶∶ = P n ∣ P n + 1 → N n + 1 ∣ N n + 1 ∧ N n + 1 ∣ 0 ∣ ⊺ 17 / 51