Sequent Calculus Research Articles

A natural deduction and its corresponding sequent calculus for positive contraction - less relevant logic

Reports on Mathematical Logic » 2017 » Number 52 » A natural deduction and its corresponding sequent calculus for positive contraction - less relevant logic A A A

Proof Analysis of Peirce’s Alpha System of Graphs

Charles Peirce's alpha system $$\mathfrak {S}_\alpha $$S? is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of $$\mathfrak {S}_\alpha $$S? is given in terms of two embedding theorems: the system $$\mathfrak {S}_\alpha $$S? and Brunnler's deep inference system for classical propositional logic can be embedded into each other; and the system $$\mathfrak {S}_\alpha $$S? and Gentzen sequent calculus $$\mathbf {G3c}^*$$G3c? can be embedded into each other.

Paraconsistent double negation as a modal operator

A paraconsistent modal-like logic, , is defined as a Gentzen-type sequent calculus. The modal operator in the modal logic can be simulated by the paraconsistent double negation in . Some theorems for embedding into a Gentzen-type sequent calculus for and vice versa are proved. The cut-elimination and completeness theorems for are also proved.

An alternative Gentzenisation of RW+∘

In this paper, we give a sequent calculus for the positive contraction-less relevant logic and we give a proof that it is cut-free without the use of the truth constant t. Based on , we re-prove the decidability of the logic .

Algebraic proof theory: Hypersequents and hypercompletions

We continue our program of establishing connections between proof-theoretic and order-algebraic properties in the setting of substructural logics and residuated lattices. Extending our previous work that connects a strong form of cut-admissibility in sequent calculi with closure under MacNeille completions of corresponding varieties, we now consider hypersequent calculi and more general completions; these capture logics/varieties that were not covered by the previous approach and that are characterized by Hilbert axioms (algebraic equations) residing in the level P3 of the substructural hierarchy. We provide algebraic foundations for substructural hypersequent calculi and an algorithm to transform P3 axioms/equations into equivalent structural hypersequent rules. Using residuated hyperframes we link strong analyticity in the resulting calculi with a new algebraic completion, which we call hyper-MacNeille.

Circular proofs for the modal mu‐calculus

AbstractInspired by Stirling's tableau proofs [4] we introduce a finite, cut‐free sound and complete sequent calculus for the modal mu‐calculus. Proofs in this system are finite trees in which leaves are either axioms or assumptions that are discharged by a specific rule of the calculus. The discharge rules provide a way to unfold assumptions motivating the name circular proofs. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Certification of Prefixed Tableau Proofs for Modal Logic

Different theorem provers tend to produce proof objects in different formats and this is especially the case for modal logics, where several deductive formalisms (and provers based on them) have been presented. This work falls within the general project of establishing a common specification language in order to certify proofs given in a wide range of deductive formalisms. In particular, by using a translation from the modal language into a first-order polarized language and a checker whose small kernel is based on a classical focused sequent calculus, we are able to certify modal proofs given in labeled sequent calculi, prefixed tableaux and free-variable prefixed tableaux. We describe the general method for the logic K, present its implementation in a prolog-like language, provide some examples and discuss how to extend the approach to other normal modal logics

Sequent calculus for minimal non-normal temporal logic}{Sequent calculus for minimal non-normal temporal logic

The minimal non-normal modal logic C2 is temporalized, and the minimal non-normal temporal logic C2t is obtained. The Hilbert-style axiomatic system HC2t, which is sound and complete, is established for C2t. The logic C2t is more general than the minimal temporal logic Kt. The proof theory of C2t is approached by introducing the labelled sequent calculus GC2t. The calculus GC2t is sound and complete, and involves cut elimination. The decidability of GC2t is obtained, and the proof search can be performed in GC2t.

Omega-inconsistency without cuts and nonstandard models

This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee (1985) for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency and shows thus, pace Cobreros et al.(2013), that the result in McGee (1985) does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic.

Sequent calculus as a compiler intermediate language

The λ-calculus is popular as an intermediate language for practical compilers. But in the world of logic it has a lesser-known twin, born at the same time, called the sequent calculus . Perhaps that would make for a good intermediate language, too? To explore this question we designed Sequent Core, a practically-oriented core calculus based on the sequent calculus, and used it to re-implement a substantial chunk of the Glasgow Haskell Compiler.

A Unified Procedure for Provability and Counter-Model Generation in Minimal Implicational Logic

This paper presents results on the definition of a sequent calculus for Minimal Implicational Propositional Logic (M→) aimed to be used for provability and counter-model generation in this logic. The system tracks the attempts to construct a proof in such a way that, if the original formula is a M→ tautology, the tree structure produced by the proving process is a proof, otherwise, it is used to construct a counter-model using Kripke semantics.

Non-classical Elegance for Sequent Calculus Enthusiasts

In this paper we develop what we can describe as a "dual two-sided" cut-free sequent calculus system for the non-classical logics of truth lp, k 3, stt and a non-reflexive logic ts which is, arguably, more elegant than the three-sided sequent calculus developed by Ripley (The Review of Symbolic Logic 5:354---378, 2012) for the same logics. Its elegance stems from how it employs more or less the standard sequent calculus rules for the various connectives and truth, and the fact that it offers a rather neat connection between derivable sequents and validity in comparison to the calculus developed by Ripley (The Review of Symbolic Logic 5:354---378, 2012).

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 S4LPN based on Artemov–Fitting models.

On Structural Features of the Implication Fragment of Frege’s Grundgesetze

We set out the implication fragment of Frege’s Grundgesetze, clarifying the implication rules and showing that this system extends Absolute Implication, or the implication fragment of Intuitionist logic. We set out a sequent calculus which naturally captures Frege’s implication proofs, and draw particular attention to the Cut-like features of his Hypothetical Syllogism rule.

Proof Certificates for Equality Reasoning

The kinds of inference rules and decision procedures that one writes for proofs involving equality and rewriting are rather different from proofs that one might write in first-order logic using, say, sequent calculus or natural deduction. For example, equational logic proofs are often chains of replacements or applications of oriented rewriting and normal forms. In contrast, proofs involving logical connectives are trees of introduction and elimination rules. We shall illustrate here how it is possible to check various equality-based proof systems with a programmable proof checker (the kernel checker) for first-order logic. Our proof checker's design is based on the implementation of focused proof search and on making calls to (user-supplied) clerks and experts predicates that are tied to the two phases found in focused proofs. It is the specification of these clerks and experts that provide a formal definition of the structure of proof evidence. As we shall show, such formal definitions work just as well in the equational setting as in the logic setting where this scheme for proof checking was originally developed. Additionally, executing such a formal definition on top of a kernel provides an actual proof checker that can also do a degree of proof reconstruction. We shall illustrate the flexibility of this approach by showing how to formally define (and check) rewriting proofs of a variety of designs.

Deductive Argumentation by Enhanced Sequent Calculi and Dynamic Derivations

Logic-based approaches for analyzing and evaluating arguments have been largely studied in recent years, yielding a variety of formal methods for argumentation-based reasoning. The goal of this paper is to provide an abstract, proof theoretical investigation of logical argumentation, where arguments are represented by sequents, conflicts between arguments are represented by sequent elimination rules, and deductions are made by dynamic proof systems extending standard sequent calculi.

Simple Decision Procedure for S5 in Standard Cut-Free Sequent Calculus

In the paper a decision procedure for S5 is presented which uses a cut-free sequent calculus with additional rules allowing a reduction to normal modal forms. It utilizes the fact that in S5 every formula is equivalent to some 1-degree formula, i.e. a modally-flat formula with modal functors having only boolean formulas in its scope. In contrast to many sequent calculi (SC) for S5 the presented system does not introduce any extra devices. Thus it is a standard version of SC but with some additional simple rewrite rules. The procedure combines the proces of saturation of sequents with reduction of their elements to some normal modal form.

Sequent Calculus Representations for Quantum Circuits

When considering a sequent-style proof system for quantum programs, there are certain elements of quantum mechanics that we may wish to capture, such as phase, dynamics of unitary transformations, and measurement probabilities. Traditional quantum logics which focus primarily on the abstract orthomodular lattice theory and structures of Hilbert spaces have not satisfactorily captured some of these elements. We can start from 'scratch' in an attempt to conceptually characterize the types of proof rules which should be in a system that represents elements necessary for quantum algorithms. This present work attempts to do this from the perspective of the quantum circuit model of quantum computation. A sequent calculus based on single quantum circuits is suggested, and its ability to incorporate important conceptual and dynamic aspects of quantum computing is discussed. In particular, preserving the representation of phase helps illustrate the role of interference as a resource in quantum computation. Interference also provides an intuitive basis for a non-monotonic calculus.

Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory

Classical (or Boolean) type theory is the type theory that allows the type inference $\sigma \to \bot) \to \bot => \sigma$ (the type counterpart of double-negation elimination), where $\sigma$ is any type and $\bot$ is absurdity type. This paper first presents a denotational semantics for a simplified version of Parigot's lambda-mu calculus, a premier example of classical type theory. In this semantics the domain of each type is divided into infinitely many ranks and contains not only the usual members of the type at rank 0 but also their negative, conjunctive, and disjunctive shadows in the higher ranks, which form an infinitely nested Boolean structure. Absurdity type $\bot$ is identified as the type of truth values. The paper then presents a new deduction system of classical type theory, a sequent calculus called the classical type system (CTS), which involves the standard logical operators such as negation, conjunction, and disjunction and thus reflects the discussed semantic structure in a more straightforward fashion.

Gentzen-Style Sequent Calculus for Semi-intuitionistic Logic

Fil: Castano, Diego Nicolas. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca. Instituto de Matematica Bahia Blanca. Universidad Nacional del Sur. Departamento de Matematica. Instituto de Matematica Bahia Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matematica; Argentina