Sequent Calculus Research Articles

Structural Focalization

Focusing, introduced by Jean-Marc Andreoli in the context of classical linear logic [Andreoli 1992], defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and grouping sequences of non-invertible rules. A focused sequent calculus is defined relative to some nonfocused sequent calculus; focalization is the property that every nonfocused derivation can be transformed into a focused derivation. In this article, we present a focused sequent calculus for propositional intuitionistic logic and prove the focalization property relative to a standard presentation of propositional intuitionistic logic. Compared to existing approaches, the proof is quite concise, depending only on the internal soundness and completeness of the focused logic. In turn, both of these properties can be established (and mechanically verified) by structural induction in the style of Pfenning's structural cut elimination without the need for any tedious and repetitious invertibility lemmas. The proof of cut admissibility for the focused system, which establishes internal soundness, is not particularly novel. The proof of identity expansion, which establishes internal completeness, is a major contribution of this work.

A Survey of Nonstandard Sequent Calculi

The paper is a brief survey of some sequent calculi (SC) which do not follow strictly the shape of sequent calculus introduced by Gentzen. We propose the following rough classification of all SC: Systems which are based on some deviations from the ordinary notion of a sequent are called generalised; remaining ones are called ordinary. Among the latter we distinguish three types according to the proportion between the number of primitive sequents and rules. In particular, in one of these types, called Gentzen’s type, we have a subtype of standard SC due to Gentzen. Hence by nonstandard ones we mean all these ordinary SC where other kinds of rules are applied than those admitted in standard Gentzen’s sequent calculi. We describe briefly some of the most interesting or important nonstandard SC belonging to the three abovementioned types.

Proof analysis beyond geometric theories: from rule systems to systems of rules

A class of axiomatic theories with arbitrary quantifier alternations is identified and a conversion to normal form is provided in terms of generalized geometric implications . The class is also characterized in terms of Glivenko classes as those first-order formulas that do not contain implications or universal quantifiers in the negative part. It is shown how the methods of proof analysis can be extended to cover such axioms by means of conversion to systems of rules . The structural properties for the resulting extensions of sequent calculus are established and a generalization of the first-order Barr theorem is shown to follow as an immediate application. The method is also applied to obtain complete labelled proof systems for logics defined through their relational semantics. In particular, the method provides analytic proof systems for all the modal logics in the Sahlqvist fragment.

A multi-focused proof system isomorphic to expansion proofs

The sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps---such as instantiating a block of quantifiers---by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical first-order logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means for abstracting away low-level details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that collect together all possible parallel foci are canonical. Moreover, if we start with a certain focused sequent proof system, such proofs are isomorphic to expansion proofs---a well known, minimalistic, and parallel generalization of Herbrand disjunctions---for classical first-order logic. This technique appears to be a systematic way to recover the "essence of proof" from within sequent calculus proofs.

Interaction and Depth against Nondeterminism in Proof Search

Deep inference is a proof theoretic methodology that generalizes the standard notion of inference of the sequent calculus, whereby inference rules become applicable at any depth inside logical expressions. Deep inference provides more freedom in the design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, however with the cost of a greater nondeterminism than in the sequent calculus. In this paper, we show that the nondeterminism in proof search can be reduced without losing the shorter proofs and without sacrificing proof theoretic cleanliness. For this, we exploit an interaction and depth scheme in the logical expressions. We demonstrate our method on deep inference systems for multiplicative linear logic and classical logic, discuss its proof complexity and its relation to focusing, and present implementations.

UNIFORM INTERPOLATION IN SUBSTRUCTURAL LOGICS

AbstractUniform interpolation property of a given logic is a stronger form of Craig’s interpolation property where both pre-interpolant and post-interpolant always exist uniformly for any provable implication in the logic. It is known that there exist logics, e.g., modal propositional logic S4, which have Craig’s interpolation property but do not have uniform interpolation property. The situation is even worse for predicate logics, as classical predicate logic does not have uniform interpolation property as pointed out by L. Henkin.In this paper, uniform interpolation property of basic substructural logics is studied by applying the proof-theoretic method introduced by A. Pitts (Pitts, 1992). It is shown that uniform interpolation property holds even for their predicate extensions, as long as they can be formalized by sequent calculi without contraction rules. For instance, uniform interpolation property of full Lambek predicate calculus, i.e., the substructural logic without any structural rule, and of both linear and affine predicate logics without exponentials are proved.

A simple sequent calculus for nominal logic

Nominal logic is a variant of first-order logic that provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the new-quantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made to develop convenient rules for reasoning with the new-quantifier, but we a rgue that none of these attempts is completely satisfactory. In this article we develop a new sequent calculus for nominal logic in which the rules for the newquantifier are much simpler than in previous attempts. We als o prove several structural and metatheoretic properties, including cut-elimination, consistenc y, and equivalence to Pitts’ axiomatization of nominal logic.

Cut Rule for the Resolution Method

In this work, we extend the resolution method with a new rule similar to the cut rule in Gentzen sequent calculi. We obtain upper and lower bounds on the complexity of refutations with this counterpart of the cut rule and without it. The complexity of refutations is also compared to the complexity of sequent proofs with cuts by formulas that do not contain implications, conjunctions, and ∃ quantifiers and in which negation may occur only immediately before elementary formulas. Bibliography: 3 titles.

A cut-free sequent calculus for relevant logic RW

A cut-free sequent calculus for relevant logic RW

Extracting BB′IW Inhabitants of Simple Types From Proofs in the Sequent Calculus $${LT_\to^{t}}$$ L T → t for Implicational Ticket Entailment

The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by combinators over the base { B, B′, I, W } is decidable too. Type-assignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a combinator for a theorem of \({T_\to}\) . Here we describe an algorithm to extract an inhabitant from a sequent calculus proof—without translating the proof into another proof system.

On Hilbert's Axiomatics of Propositional Logic

In this paper I will consider the axioms for propositional logic which were presented by Hilbert in his conferences during the year 1922, and those which were presented by Hilbert and Bernays in the book Grundlagen der Mathematik, I (1934). I will describe a general procedure in order to translate Hilbert's axioms into rules on sequents and I will show that, following this procedure, Hilbert's axioms become particular cases of (derived or primitive) rules of Gentzen's Sequent Calculus and contain ideas which will be focused and developed in Gentzen's Sequent Calculus and also in more recent logical investigations.

Towards a More General Concept of Inference

The main objective of this paper is to sketch unifying conceptual and formal framework for inference that is able to explain various proof techniques without implicitly changing the underlying notion of inference rules. We base this framework upon the so-called two-dimensional, i.e., deduction to deduction, account of inference introduced by Tichý in his seminal work The Foundation’s of Frege’s Logic (1988). Consequently, it will be argued that sequent calculus provides suitable basis for such general concept of inference and therefore should not be seen just as technical tool, but philosophically well-founded system that can rival natural deduction in terms of its “naturalness”.

Proof search for propositional abstract separation logics via labelled sequents

Abstract separation logics are a family of extensions of Hoare logic for reasoning about programs that mutate memory. These logics are "abstract" because they are independent of any particular concrete memory model. Their assertion languages, called propositional abstract separation logics, extend the logic of (Boolean) Bunched Implications (BBI) in various ways. We develop a modular proof theory for various propositional abstract separation logics using cut-free labelled sequent calculi. We first extend the cut-fee labelled sequent calculus for BBI of Hou et al to handle Calcagno et al's original logic of separation algebras by adding sound rules for partial-determinism and cancellativity, while preserving cut-elimination. We prove the completeness of our calculus via a sound intermediate calculus that enables us to construct counter-models from the failure to find a proof. We then capture other propositional abstract separation logics by adding sound rules for indivisible unit and disjointness, while maintaining completeness and cut-elimination. We present a theorem prover based on our labelled calculus for these logics.

A proof system for separation logic with magic wand

Separation logic is an extension of Hoare logic which is acknowledged as an enabling technology for large-scale program verification. It features two new logical connectives, separating conjunction and separating implication, but most of the applications of separation logic have exploited only separating conjunction without considering separating implication. Nevertheless the power of separating implication has been well recognized and there is a growing interest in its use for program verification. This paper develops a proof system for full separation logic which supports not only separating conjunction but also separating implication. The proof system is developed in the style of sequent calculus and satisfies the admissibility of cut. The key challenge in the development is to devise a set of inference rules for manipulating heap structures that ensure the completeness of the proof system with respect to separation logic. We show that our proof of completeness directly translates to a proof search strategy.

A new sequent calculus for automated proof search

In this paper, a new sequent calculus is presented. This sequent calculus is suitable for automated proof search of natural deduction. Also such calculus represents convenient theoretical basis for the proof

Semantic minimalism for logical constants

In [31], I defended a minimalist account of meaning for logical constants as a way to ward off Quine’s meaning variance charge against deviant logics. Its key idea was that some deviant propositional logics share with classical logic the operational meanings of all their connectives, as encoded in their sequent calculus operational rules, yet validate different sequents than classical logic — therefore, we can have genuine rivalry between logics without meaning variance. In his [19], Ole Hjortland levelled several objections at this view. The aim of this paper is to address these criticisms, highlighting at the same time the role played by logical consequence in this version of semantic minimalism.

Speech Acts, Categoricity, and the Meanings of Logical Connectives

In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulae. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap (1943) called categoricity. We show that categorical systems can be given for any finite many-valued logic using n-sided sequent calculus. These systems are understood as a further development of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion, and with Mark Twain, that denial is more than a river in Egypt.

Decidability of Logic of Correlated Knowledge

Terminating procedure GS-LCK-PROC of the proof search in the sequent calculus GS-LCK of logic of correlated knowledge is presented in this paper. Also decidability of logic of correlated knowledge is proved, where GS-LCK-PROC is a decision procedure.

From User Requirements to Software Specifications: An Approach Based on Problem Transformation

This paper aims at deriving software specification descriptions from elicited user requirements and domain descriptions. It provides an approach to transforming user requirements into software specifications in a smooth and logical way. Based on previous in-depth research on Problem Frames, the study adopts Hoare's Communicating Sequential Processes (CSP) and Lai's weakest-environment calculus to transform an entire problem diagram. The derived software specifications are abstract models resembling program code, whose correctness can be verified by the model checker FDR. This paper provides foundational work for embedded software development, i.e., deriving software code from requirements descriptions, automating document transformation and validation, etc. The theory presented in this paper, together with the FDR model checker tool, may help to improve the efficiency and accuracy in embedded software development.

Herbrand-Confluence

We consider cut-elimination in the sequent calculus for classical first-order logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cut-free proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbrand-disjunction of a cut-free proof. Our main theorem is a confluence result for a natural class of proofs: all (possibly infinitely many) normal forms of the non-erasing reduction lead to the same Herbrand-disjunction.