Algebraic proof theory for substructural logics: Cut-elimination and completions

We carry out a unified investigation of two prominent topics in proof theory and order algebra: cut-elimination and completion, in the setting of substructural logics and residuated lattices.We introduce the substructural hierarchy — a new classification of logical axioms (algebraic equations) over full Lambek calculus FL, and show that a stronger form of cut-elimination for extensions of FL and the MacNeille completion for subvarieties of pointed residuated lattices coincide up to the level N₂ in the hierarchy. Negative results, which indicate limitations of cut-elimination and the MacNeille completion, as well as of the expressive power of structural sequent calculus rules, are also provided.Our arguments interweave proof theory and algebra, leading to an integrated discipline which we call algebraic proof theory.     

Order-enriched categorical models of the classical sequent calculus

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.     

Proof complexity of intuitionistic implicational formulas

We study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of intuitionistic extended Frege (EF) or substitution Frege (SF) proofs up to a polynomial. On the other hand, EF proofs in the implicational fragment of IPC polynomially simulate full intuitionistic logic for implicational tautologies. The results also apply to other fragments of other superintuitionistic logics under certain conditions.In particular, the exponential lower bounds on the length of intuitionistic EF proofs by Hrube? (2007), generalized to exponential separation between EF and SF systems in superintuitionistic logics of unbounded branching by Je?ábek (2009), can be realized by implicational tautologies.     

Uniform interpolation and the existence of sequent calculi

This paper presents a uniform and modular method to prove uniform interpolation for several intermediate and intuitionistic modal logics. The proof-theoretic method uses sequent calculi that are extensions of the terminating sequent calculus $\mathsf{G4ip}$ for intuitionistic propositional logic. It is shown that whenever the rules in a calculus satisfy certain structural properties, the corresponding logic has uniform interpolation. It follows that the intuitionistic versions of $\mathsf{K}$ and $\mathsf{KD}$ (without the diamond operator) have uniform interpolation. It also follows that no intermediate or intuitionistic modal logic without uniform interpolation has a sequent calculus satisfying those structural properties, thereby establishing that except for the seven intermediate logics that have uniform interpolation, no intermediate logic has such a sequent calculus.     

Bounded linear-time temporal logic: A proof-theoretic investigation

Propositional and first-order bounded linear-time temporal logics (BLTL and FBLTL, respectively) are introduced by restricting Gentzen type sequent calculi for linear-time temporal logics. The corresponding Robinson type resolution calculi, RC and FRC for BLTL and FBLTL respectively are obtained. To prove the equivalence between FRC and FBLTL, a temporal version of Herbrand theorem is used. The completeness theorems for BLTL and FBLTL are proved for simple semantics with both a bounded time domain and some bounded valuation conditions on temporal operators. The cut-elimination theorems for BLTL and FBLTL are also proved using some theorems for embedding BLTL and FBLTL into propositional (first-order, respectively) classical logic. Although FBLTL is undecidable, its monadic fragment is shown to be decidable.     

The bounded proof property via step algebras and step frames

The paper introduces semantic and algorithmic methods for establishing a variant of the analytic subformula property (called ‘the bounded proof property’, bpp) for modal propositional logics. The bpp is much weaker property than full cut-elimination, but it is nevertheless sufficient for establishing decidability results. Our methodology originated from tools and techniques developed on one side within the algebraic/coalgebraic literature dealing with free algebra constructions and on the other side from classical correspondence theory in modal logic. As such, our approach is orthogonal to recent literature based on proof-theoretic methods and, in a way, complements it.     

Labeled sequent calculus for justification logics

Justification logics are modal-like logics that provide a framework for reasoning about justifications. This paper introduces labeled sequent calculi for justification logics, as well as for combined modal-justification logics. Using a method due to Sara Negri, we internalize the Kripke-style semantics of justification and modal-justification logics, known as Fitting models, within the syntax of the sequent calculus to produce labeled sequent calculi. We show that all rules of these systems are invertible and the structural rules (weakening and contraction) and the cut rule are admissible. Soundness and completeness are established as well. The analyticity for some of our labeled sequent calculi are shown by proving that they enjoy the subformula, sublabel and subterm properties. We also present an analytic labeled sequent calculus for $\mathsf{S4LPN}$ based on Artemov–Fitting models.     

Reasoning about proof and knowledge

In previous work [15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.     

Classical proof forestry

Classical proof forests are a proof formalism for first-order classical logic based on Herbrand’s Theorem and backtracking games in the style of Coquand. First described by Miller in a cut-free setting as an economical representation of first-order and higher-order classical proof, defining features of the forests are a strict focus on witnessing terms for quantifiers and the absence of inessential structure, or ‘bureaucracy’.This paper presents classical proof forests as a graphical proof formalism and investigates the possibility of composing forests by cut-elimination. Cut-reduction steps take the form of a local rewrite relation that arises from the structure of the forests in a natural way. Yet reductions, which are significantly different from those of the sequent calculus, are combinatorially intricate and do not exclude the possibility of infinite reduction traces, of which an example is given.Cut-elimination, in the form of a weak normalisation theorem, is obtained using a modified version of the rewrite relation inspired by the game-theoretic interpretation of the forests. It is conjectured that the modified reduction relation is, in fact, strongly normalising.     

Distributive-Lattice Semantics of Sequent Calculi with Structural Rules

The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables (viz., metavariables for finite sets of formulas), upon the basis of the conception of model introduced in (Fuzzy Sets Syst 121(3):27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models (called its semantics) that a rule is derivable in the calculus iff it is sound with respect to each model of the semantics. We then show how semantics of admissible rules of such calculi can be found with using a method of free models. Next, our universal approach is applied to sequent calculi for many-valued logics with equality determinant. Finally, we exemplify this application by studying sequent calculi for some of such logics.      

Cut-Elimination Theorem for the Logic of Constant Domains

The logic CD is an intermediate logic (stronger than intuitionistic logic and weaker than classical logic) which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD (which is same as LK except that (→) and (?–) rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD . In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD , saying that all “cuts” except some special forms can be eliminated from a proof in LD . From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD : fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD . Mathematics Subject Classification : 03B55. 03F05.     

Prefixed tableaus and nested sequents

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that Gentzen sequent calculi and tableaus are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.     

Admissible rules in the implication-negation fragment of intuitionistic logic

Uniform infinite bases are defined for the single-conclusion and multiple-conclusion admissible rules of the implication-negation fragments of intuitionistic logic and its consistent axiomatic extensions (intermediate logics). A Kripke semantics characterization is given for the (hereditarily) structurally complete implication-negation fragments of intermediate logics, and it is shown that the admissible rules of this fragment of form a PSPACE-complete set and have no finite basis.     

A minimal classical sequent calculus free of structural rules

Gentzen’s classical sequent calculus has explicit structural rules for contraction and weakening. They can be absorbed (in a right-sided formulation) by replacing the axiom P,¬P by Γ,P,¬P for any context Γ, and replacing the original disjunction rule with Γ,A,B implies Γ,A∨B.This paper presents a classical sequent calculus which is also free of contraction and weakening, but more symmetrically: both contraction and weakening are absorbed into conjunction, leaving the axiom rule intact. It uses a blended conjunction rule, combining the standard context-sharing and context-splitting rules: Γ,Δ,A and Γ,Σ,B implies Γ,Δ,Σ,A∧B. We refer to this system as minimal sequent calculus.We prove a minimality theorem for the propositional fragment : any propositional sequent calculus S (within a standard class of right-sided calculi) is complete if and only ifS contains (that is, each rule of is derivable in S). Thus one can view as a minimal complete core of Gentzen’s .     

The relation between intuitionistic and classical modal logics

Intuitionistic propositional logicInt and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded as just fragments of classical modal logics containingS4. The main aim of this paper is to construct a similar correspondence between intermediate logics augmented with modal operators—we call them intuitionistic modal logics—and classical polymodal logics We study the class of intuitionistic polymodal logics in which modal operators satisfy only the congruence rules and so may be treated as various sorts of □ and ◇. Supported by the Alexander von Humboldt Foundation. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 121–155, March–April, 1997.     

Infinitary first-order categorical logic

We present a unified categorical treatment of completeness theorems for several classical and intuitionistic infinitary logics with a proposed axiomatization. This provides new completeness theorems and subsumes previous ones by Gödel, Kripke, Beth, Karp and Joyal. As an application we prove, using large cardinals assumptions, the disjunction and existence properties for infinitary intuitionistic first-order logics.     

Cut-free formulations for a quantified logic of here and there

A predicate extension SQHT⁼ of the logic of here-and-there was introduced by V. Lifschitz, D. Pearce, and A. Valverde to characterize strong equivalence of logic programs with variables and equality with respect to stable models. The semantics for this logic is determined by intuitionistic Kripke models with two worlds (here and there) with constant individual domain and decidable equality. Our sequent formulation has special rules for implication and for pushing negation inside formulas. The soundness proof allows us to establish that SQHT⁼ is a conservative extension of the logic of weak excluded middle with respect to sequents without positive occurrences of implication. The completeness proof uses a non-closed branch of a proof search tree. The interplay between rules for pushing negation inside and truth in the “there” (non-root) world of the resulting Kripke model can be of independent interest. We prove that existence is definable in terms of remaining connectives.     

Some remarks on lengths of propositional proofs

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m ^d+1) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. Supported in part by NSF grant DMS-9205181     

Sequent calculus proof theory of intuitionistic apartness and order relations

Contraction-free sequent calculi for intuitionistic theories of apartness and order are given and cut-elimination for the calculi proved. Among the consequences of the result is the disjunction property for these theories. Through methods of proof analysis and permutation of rules, we establish conservativity of the theory of apartness over the theory of equality defined as the negation of apartness, for sequents in which all atomic formulas appear negated. The proof extends to conservativity results for the theories of constructive order over the usual theories of order. Received: 4 December 1997