Interpolation Using Sequents and Their Generalizations Roman - PowerPoint PPT Presentation

Interpolation Using Sequents and Their Generalizations Roman Kuznets Embedded Computing Systems / Theory and Logic TU Wien PhDs in Logic X May 2, 2018 Roman Kuznets Interpolation using sequents and beyond 1 / 54

Craig Interpolation Definition A logic L has the Craig Interpolation Property (CIP) if, whenever L ⊢ A → B , there exists an interpolant C such that C only uses that common language of A and B L ⊢ A → C L ⊢ C → B These three statements are abbreviated as A → B ← − C Motivation logical (connections to Beth definability, Robinson joint consistency) mathematical (connections to algebra) applied (applications in computer science) Roman Kuznets Interpolation using sequents and beyond 2 / 54

Craig Interpolation: Example Example (Classical propositional logic CPL) CPL ⊢ p ∧ ¬ q → ( q → r ) One of the interpolants is ¬ q . Indeed, CPL ⊢ p ∧ ¬ q → ¬ q and CPL ⊢ ¬ q → ( q → r ) In fact, in this case ¬ q is a Lyndon interpolant: not only q but its polarity is common among p ∧ ¬ q and q → r . Roman Kuznets Interpolation using sequents and beyond 3 / 54

The Role of Constants What are interpolants of q → ( p → p ) and ¬ ( q → q ) → p ? Definition Craig Interpolation A logic without constants in the language has CIP if the interpolation property holds for every valid A → B such that neither B nor ¬ A is valid by itself. Roman Kuznets Interpolation using sequents and beyond 4 / 54

Craig Interpolation: Proof Methods Craig, 1957: Sequent calculus for predicate logic modulo propositional one. Proof theoretic method is the most robust constructive way of proving interpolation. Constructive means that the interpolant C is constructed by an algorithm, given a “derivation of A → B ” Roman Kuznets Interpolation using sequents and beyond 5 / 54

Craig Interpolation: First Proper Proof-Theoretic Proofs Maehara(1960, in Japanese/German) Maehara and Takeuti (1961) for predicate infinite language Sch¨ utte (1962, in German) for intuitionistic predicate logic Fitting (1983) for several modal logics Roman Kuznets Interpolation using sequents and beyond 6 / 54

Craig or Sequent Interpolation: naively From Formulas to Sequents... C interpolates A → B if � A → C and � C → B and CL ↓ C interpolates A → B if � A ⇒ C and � C ⇒ B and CL ↓ C interpolates Γ ⇒ ∆ if � Γ ⇒ C and � C ⇒ ∆ and CL CL is the common language condition Example (Problem) p ⇒ p ⇒ p → p Roman Kuznets Interpolation using sequents and beyond 7 / 54

Craig or Sequent Interpolation: naively From Formulas to Sequents... C interpolates A → B if � A → C and � C → B and CL ↓ C interpolates A → B if � A ⇒ C and � C ⇒ B and CL ↓ C interpolates Γ ⇒ ∆ if � Γ ⇒ C and � C ⇒ ∆ and CL CL is the common language condition ...naively Already Maehara noted that formulas move between the antecedent Γ and succedent ∆ . These moves make preserving the CL problematic. Roman Kuznets Interpolation using sequents and beyond 8 / 54

Craig Interpolation: First Proper Proof-Theoretic Proofs Maehara(1960, in Japanese/German) Maehara and Takeuti (1961) for predicate infinite language Sch¨ utte (1962, in German) for intuitionistic predicate logic Fitting (1983) for several modal logics Roman Kuznets Interpolation using sequents and beyond 9 / 54

These Are Not the Sides You’re Looking for Omitting the CL as always present, C interpolates Γ ⇒ ∆ if � Γ ⇒ C and � C ⇒ ∆ ↓ C interpolates Γ; Π ⇒ Σ; ∆ if = ⇒ � A for some A ∈ Γ or � B for some B ∈ Σ � C and = ⇒ � A for some A ∈ Π or � B for some B ∈ ∆ � C ↓ C interpolates Γ; Π ⇒ Σ; ∆ if � C = ⇒ � Γ ⇒ Σ and = ⇒ � Π ⇒ ∆ � C Roman Kuznets Interpolation using sequents and beyond 10 / 54

Proof-theoretic Method It is not entirely correct to say that Craig interpolation is proved by induction on the sequent derivation. In reality, one finds a suitable generalization of CIP and proves that by induction on the split version of the sequent derivation. Roman Kuznets Interpolation using sequents and beyond 11 / 54

Splitting Sequent Rules Γ , A , B ⇒ ∆ Γ , A ∧ B ⇒ ∆ ւ ց Γ , A , B ; Π ⇒ Σ; ∆ Γ; A , B , Π ⇒ Σ; ∆ Γ , A ∧ B ; Π ⇒ Σ; ∆ Γ; A ∧ B , Π ⇒ Σ; ∆ Γ ⇒ A , ∆ Γ ⇒ B , ∆ Γ ⇒ A ∧ B , ∆ ւ ց Γ; Π ⇒ Σ , A ; ∆ Γ; Π ⇒ Σ , B ; ∆ Γ; Π ⇒ Σ; A , ∆ Γ; Π ⇒ Σ; B , ∆ Γ; Π ⇒ Σ , A ∧ B ; ∆ Γ; Π ⇒ Σ; A ∧ B , ∆ Roman Kuznets Interpolation using sequents and beyond 12 / 54

Splitting Initial Sequents Γ; A , Π ⇒ Σ , A ; ∆ Γ , A ; Π ⇒ Σ , A ; ∆ տ ր Γ , A ⇒ A , ∆ ւ ց Γ; A , Π ⇒ Σ; A , ∆ Γ , A ; Π ⇒ Σ; A , ∆ Roman Kuznets Interpolation using sequents and beyond 13 / 54

General Plan of Finding an Interpolant Given a theorem A → B 1 Find a sequent derivation of A ⇒ B 2 Split the endsequent to be A ; ⇒ ; B 3 Propagate this split from the endsequent to the leaves turning the derivation into a split derivation 4 Find interpolants for each split initial sequent in a leaf 5 Propagate interpolants from the leaves to the endsequent using rules for transforming interpolants for each rule 6 Process the interpolant for the endsequent A ; ⇒ ; B into the best form suitable for A → B Roman Kuznets Interpolation using sequents and beyond 14 / 54

Interpolating Initial Sequents Γ; A , Π ⇒ Σ , A ; ∆ ← − ¬ A Γ , A ; Π ⇒ Σ , A ; ∆ ← − ⊥ տ ր ւ ց Γ; A , Π ⇒ Σ; A , ∆ ← − ⊤ Γ , A ; Π ⇒ Σ; A , ∆ ← − A Roman Kuznets Interpolation using sequents and beyond 15 / 54

Typical Interpolant Transformation Γ , A , B ; Π ⇒ Σ; ∆ ← − C Γ; A , B , Π ⇒ Σ; ∆ ← − C Γ , A ∧ B ; Π ⇒ Σ; ∆ ← − C Γ; A ∧ B , Π ⇒ Σ; ∆ ← − C Roman Kuznets Interpolation using sequents and beyond 16 / 54

Cutting to the Chase Γ; Π ⇒ Σ; A , ∆ ← − C 1 Γ; A , Π ⇒ Σ; ∆ ← − C 2 Γ; Π ⇒ Σ; ∆ ← − ? Analyticity Cut rule is problematic because it violates the subformula property. Roman Kuznets Interpolation using sequents and beyond 17 / 54

Interpolant Transformation for Two-Premise Rules Γ; Π ⇒ Σ; A , ∆ ← − C 1 Γ; Π ⇒ Σ; B , ∆ ← − C 2 Γ; Π ⇒ Σ; A ∧ B , ∆ ← − C 1 ∧ C 2 Γ; Π ⇒ Σ , A ; ∆ ← − C 1 Γ; Π ⇒ Σ , B ; ∆ ← − C 2 Γ; Π ⇒ Σ , A ∧ B ; ∆ ← − C 1 ∨ C 2 Roman Kuznets Interpolation using sequents and beyond 18 / 54

Interpolant Transformations for Structural Rules Γ; Π ⇒ Σ; ∆ ← − C Γ , Γ ′ ; Π , Π ′ ⇒ Σ , Σ ′ ; ∆ , ∆ ′ ← − C Γ; Π ⇒ Σ; A , A , ∆ ← − C Γ; Π ⇒ Σ; A , ∆ ← − C Γ; Π ⇒ Σ; A , B , ∆ ← − C Γ; Π ⇒ Σ; B , A , ∆ ← − C Roman Kuznets Interpolation using sequents and beyond 19 / 54

First Result Theorem CPL has Craig and Lyndon interpolation. Roman Kuznets Interpolation using sequents and beyond 20 / 54

Going Modal Γ ⇒ B � Γ ⇒ � B ւ ց Γ; Π ⇒ B ; Γ; Π ⇒ ; B � Γ; � Π ⇒ � B ; � Γ; � Π ⇒ ; � B Γ; Π ⇒ B ; ← − C Γ; Π ⇒ ; B ← − C � Γ; � Π ⇒ � B ; ← − � C � Γ; � Π ⇒ ; � B ← − � C Roman Kuznets Interpolation using sequents and beyond 21 / 54

Kripke Models Definition A Kripke model M = ( W , R , V ) is a triple, where W � = ∅ R ⊆ W × W V : Prop → 2 W M , w � p iff w ∈ V ( p ) M , w � ⊥ M , w � ¬ A iff M , w � A M , w � A ∧ B iff M , w � A and M , w � B M , w � A ∨ B iff M , w � A or M , w � B M , w � A → B iff M , w � A or M , w � B M , w � � A iff M , v � A for all v s.t. wRv M , w � � A iff M , v � A for some v s.t. wRv Roman Kuznets Interpolation using sequents and beyond 22 / 54

Going Modal Γ ⇒ B � Γ ⇒ � B ւ ց Γ; Π ⇒ B ; Γ; Π ⇒ ; B � Γ; � Π ⇒ � B ; � Γ; � Π ⇒ ; � B Γ; Π ⇒ B ; ← − C Γ; Π ⇒ ; B ← − C � Γ; � Π ⇒ � B ; ← − � C � Γ; � Π ⇒ ; � B ← − � C Roman Kuznets Interpolation using sequents and beyond 23 / 54

Un-sequent-ial Modal Logics Relatively few modal logics have cut-free sequent systems. Solutions Abandon the ship proof theory Use cut analytically (shh! it’s not actually a problem) Use sequents but for a computable pre-image of your logic Extend sequent formalism Roman Kuznets Interpolation using sequents and beyond 24 / 54

The Reach of Sequents Modal cube from the Stanford Encyclopedia of Philosophy CIP and LIP for normal modal logics axiomatized by any combination of axioms d, t, b, 4, and 5. S4 S5 ◦ ◦ TB ◦ ◦ T D4 ◦ ◦ D45 ◦ D5 ◦ ◦ DB D K4 ◦ ◦ ◦ B5 K45 ◦ K5 ◦ ◦ K KB 6 via sequents 6 via analytic cuts using advanced calculi Roman Kuznets Interpolation using sequents and beyond 25 / 54

Finding Hypersequent Generalizations of CIP Hypersequents (Mints, 1968; Pottinger, 1983; Avron, 1987) Modal logics: n � ı h (Γ 1 ⇒ ∆ 1 | · · · | Γ n ⇒ ∆ n ) = � ı s (Γ i ⇒ ∆ i ) i =1 Roman Kuznets Interpolation using sequents and beyond 26 / 54