On Hypersequents and Labelled Sequents Translating Labelled Sequent Proofs to Hypersequent Proofs Robert Rothenberg 1 2 1School of Computer Science University of St Andrews 2Interactive Information, Ltd Edinburgh Workshop in Honour of Roy Dyckhoff St Andrews, 18-19 November 2011 Extensions of Gentzen-style Sequent Calculi Extensions to Gentzen-style sequent calculi obtained by changing to specific syntactic features [Paoli] in order to control proof search for non-classical logics, such as: I Labelled Systems I Multiple Sequents (e.g. higher-order sequents, hypersequents) I Multi-sided Sequents I Multi-arrow Sequents (e.g. sequents of relations) I Multi-comma Systems (e.g. Display Logics) I Deep Inference Systems (e.g. Calculus of Structures) Many systems are hybrids of these, such as nested sequents or relational hypersequents. Why Compare Formalisms? I Interface vs implementation (automated proof assistants) I Translating proofs of meta properties. I Novel and interesting rules obtained from other formalisms. I Formal criteria for comparing formalisms. I Illuminate the meaning of particular syntactic features. I Use abstraction to conceive of new extensions? (akin to juggling notation...) I Develop a hierarchy of the strength of proof systems. Why Compare Labelled Sequents and Hypersequents? I Folklore about relationship, but no published formal comparison beyond specific calculi (mainly for S5). I There are labelled and hypersequent calculi for overlapping sets of logics. (Here we look at some Int∗ logics.) I A comparison of the rules for some logics suggests a relationship. Labelled Systems I First labelled systems apparently introduced by [Kanger, 1957] for S5 and [Maslov, 1967] for Int. I The language of formulae is extended with a language of annotations to control inference, e.g. Γ)∆;Ay x R Γ)∆; A where y is fresh for the conclusion. I Additional kinds of formulae based on labels may be used for controlling inference, e.g. Rxy. I Easily obtained using the relational semantics of a logic. Syntax of Labelled Sequents x I Formulae in a sequent are annotated with labels, e.g. A . x1 xn x1 xn Γ1 ;:::; Γn )∆1 ;:::; ∆n I Sequents may also contain relational formulae which indicate a relationship between labels , e.g. Rxy. x1 xn x1 xn Rxi1 xj1 ;:::; Rxik xjk ; Γ1 ;:::; Γn )∆1 ;:::; ∆n I In some calculi, labels may be complex expressions, or may contain variables. I . relational formulae may be n-ary, occur on either side, or even be “first class" and combined with formulae, e.g. Rxy ^ (A _ B)x. The Simple Relational Calculus G3I I A labelled calculus with atomic labels and binary relations. I A fragment of the calculus G3I from [Negri, 2005]: Rxy; Σ; P x; Γ)∆;P y Rxy; Σ; (A⊃B)x; Γ)∆;Ay Rxy; Σ; (A⊃B)x;By; Γ)∆ L⊃ Rxy; Σ; (A⊃B)x; Γ)∆ Rxy;^ Σ; Ay; Γ)∆;By R⊃ Σ; Γ)∆; (A⊃B)x The rules for ^, _ and ? are standard. I The pure relational rules (or \ordering rules"): Rxx; Σ; Γ)∆ Rxz; Rxy; Ryz; Σ; Γ)∆ refl trans Σ; Γ)∆ Rxy; Ryz; Σ; Γ)∆ A Similar Calculus for BiInt [Pinto & Uustalu, 2009] give a similar calculus for BiInt, with (aside from the dual of ⊃) contraction as a primitive rule and replacing the axiom with Σ; Ax; Γ)∆;Ax Rxy; Σ; Ax;Ay; Γ)∆ Rxy; Σ; Γ)∆;Ax;Ay L R Rxy; Σ; Ax; Γ)∆ mono Rxy; Σ; Γ)∆;Ay mono The mono rules are derivable in G3I using cut, e.g.: . Rxy; Σ; Ax; Γ)∆;Ay Rxy; Σ; Ax;Ay; Γ)∆ cut Rxy; Σ; Ax; Γ)∆ Geometric Rules I A geometric rule is a G3-style rule of the form [z=¯^ y¯]Σ1; Σ0; Γ)∆ ::: [z=¯^ y¯]Σn; Σ0; Γ)∆ Σ0; Γ)∆ where the variables z¯^ do not occur free in the conclusion, and each Σi is a multiset of atoms. I Geometric rules can be added to G3-style calculi without affecting admissibility of cut, weakening or contraction. [Negri 2005] [Simpson 1994]. I A geometric implication [Palmgren 2002?] is a formula of the form 8x:¯ (A⊃B), without ⊃; 8 in subformulae of A; B. They are constructively equivalent to: 8x:¯ ((P10 ^:::^Pk0 )⊃9y:¯ ((P11 ^:::^Pk1 )_:::_(P1n ^:::^Pkn ))) I Frame conditions of many logics in Int∗ are geometric implications. Extending G3I for Geometric Intermediate Logics I Adding rules that correspond to frame conditions of logics. I Adding the \directedness" rule yields a calculus for Jan: Rxz;^ Ryz;^ Rwx; Rwy; Σ; Γ)∆ dir Rwx; Rwy; Σ; Γ)∆ I Adding the \linearity rule" yields a calculus for GD: Rxy; Σ; Γ)∆ Ryx; Σ; Γ)∆ lin Σ; Γ)∆ I Adding the \symmetry" rule yields a calculus for Cl: Rxy; Ryx; Σ; Γ)∆ sym Rxy; Σ; Γ)∆ I Weakening, contraction and cut admissibility is preserved. Hypersequents I Attributed to [Avron] although similar calculi occur in earlier work by [Beth], [Sambin & Valentini], [Pottinger]. I A hypersequent is a non-empty list/multiset of sequents Γ1)∆1 j ::: j Γn)∆n called its components. I A hypersequent H is true in an interpretation I iff one of its components, Γi)∆i 2 H is true in that interpretation, i.e. ( ^^ Γ1 ⊃ __ ∆1) _ ::: _ ( ^^ Γn ⊃ __ ∆n) Syntax of Hypersequents I Internal rules are (structural) rules which have one active component in each premiss, and one principal component in the conclusion. External rules are (structural) rules which are not internal rules. I The standard external rules are 0 0 0 H HjΓ)∆jΓ)∆ HjΓ )∆ jΓ)∆jH EW EC EP HjΓ)∆ HjΓ)∆ HjΓ)∆jΓ0)∆0jH0 where H; H0 denote the side components. I The hyperextention of a sequent calculus is its extension as a hypersequent calculus by adding hypercontexts to rules and the standard external rules. A Hyperextention of a Calculus for Int HjΓ)∆; ? Ax L? R? Γ;P)P; ∆ Γ; ?)∆ HjΓ)∆ HjΓ;A)∆ HjΓ;B)∆ HjΓ)A; ∆ HjΓ)B; ∆ L_ R_1 R_2 HjΓ;A _ B)∆ HjΓ)A _ B; ∆ HjΓ)A _ B; ∆ HjΓ)∆;A HjΓ;B)∆ HjΓ;A)B L⊃ R⊃ HjΓ;A⊃B)∆ HjΓ)A⊃B; ∆ HjΓ)∆ HjΓ; Γ0; Γ0)∆; ∆0; ∆0 W C HjΓ; Γ0)∆; ∆0 HjΓ; Γ0)∆; ∆0 plus the dual ^ rules and standard external rules and (hyperextended) cut. Extensions for Some Intermediate Logics I Adding the LQ rule yields a calculus for Jan: HjΓ1; Γ2) LQ HjΓ1) jΓ2) I Adding the communication rule yields a calculus for GD: HjΓ1; Γ2)∆1 HjΓ1; Γ2)∆2 Com HjΓ1)∆1jΓ2)∆2 I Adding the split rule yields a calculus for Cl: HjΓ1; Γ2)∆1; ∆2 S HjΓ1)∆1jΓ2)∆2 The Labelled and Hypersequent Rules Look Similar Hypersequent Rule Relational Rule HjΓ1; Γ2) Rxz;^ Ryz;^ Rwx; Rwy;Σ; Γ)∆ HjΓ1)jΓ2) Rwx; Rwy;Σ; Γ)∆ HjΓ1; Γ2)∆1 HjΓ1; Γ2)∆2 Rxy; Σ; Γ)∆ Ryx; Σ; Γ)∆ HjΓ1)∆1jΓ2)∆2 Σ; Γ)∆ HjΓ1; Γ2)∆1; ∆2 Rxy; Ryx; Σ; Γ)∆ HjΓ1)∆1jΓ2)∆2 Rxy; Σ; Γ)∆ Components roughly correspond to labels, and relational formula roughly correspond to subset relations. Translation of Labelled Sequents to Hypersequents I We want a translation of proofs in labelled systems like G3I∗ to (familiar) hypersequent systems. I Each label corresponds to a component. I Relations are translated using monotonicity: Rxy is translated by including the antecedent (r. succedent) of the component for x (r. y) as a subset of the antecedent (r. succedent) of the component for y (r. x). e.g., Rxy; Ax;By)Cx;Dy 7! A)C; D j A; B)D The process is called transitive unfolding. I The translation makes an explicit relationship between labels into an implicit relationship between components. Labelled Calculi are More Expressive than Hypersequents I The two labelled sequents, Rxy; Rxz;Γx)Rxy; Ryz;Γx) both translate to the same hypersequent, Γ) j Γ) j Γ) I What do relations mean w.r.t. hypersequents? e.g. The following holds for Int models: Rxy;(A _ B)x; (B ⊃C)y)Ax;Cy but the corresponding hypersequent is not derivable for Int: A _ B)A; C j A _ B; B ⊃C)C Hypersequents and Monotonicity I Ideally, we'd like hypersequent rules to act on multiple components in accordance with monotonicity, just as labelled rules do. I But the following rule is not valid for Int: HjA; Γ)∆; ∆0jA; Γ; Γ0)∆0 L ⊆ HjA; Γ)∆; ∆0jΓ; Γ0)∆0 I A simple counterexample is A)A ^ BjA; B)A ^ B L ⊆ A)A ^ BjB)A ^ B which is valid for GD = Int + (A⊃B) _ (B ⊃A). The Translation Requires Communication Theorem Labelled proofs in G3I∗ (that do not contain ordering rules with principal relational formulae) can be translated into hypersequent proofs in a corresponding calculus augmented with the Com rule, HjΓ)∆; ∆0jΓ; Γ0)∆0 HjΓ; Γ0)∆jΓ0)∆; ∆0 Com HjΓ)∆jΓ0)∆0 I Labelled rules and proofs for some logics Int∗ can be translated into hypersequent proofs for GD∗. I The restriction on ordering rules has to do with the admissibility of cut. A rule such as Ryx; Rxy; Ryz;Γ)∆ Rzy; Rxy; Ryz;Γ)∆ bd Rxy; Ryz;Γ)∆ 2 translates to hypersequent rules with duplicated metavariables in the conclusion, and that may affect cut admissibility.( ?) Translation of Proofs I Note that this work is about translating proofs of arbitrary labelled sequents (with relations) into hypersequents. I The communication rule allows us to derive hypersequent variants of the labelled rules. I We proceed by transitive unfolding the premisses of each labelled inference and then applying the hypersequent variant of the inference rule, to obtain a conclusion that is the transitive unfolding of the conclusion of the labelled inference. I The refl, trans and mono rules are ignored as they are implicit in the translation. (?) Monotonicity Rules Lemma The rules HjA; Γ)∆; ∆0jA; Γ; Γ0)∆0 HjΓ)∆; ∆0;AjΓ; Γ0)∆0;A L⊂ R⊂ HjA; Γ)∆; ∆0jΓ; Γ0)∆0 ∼ HjΓ)∆; ∆0jΓ; Γ0)∆0;A ∼ are derivable using Com.