A sequent calculus with procedure calls

Mathematical Structures in Computer Science, 2008

X is an untyped continuation-style formal language with a typed subset which provides a Curry-Howard isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X , we will show how elaborate calculi can be embedded, like the λ-calculus, Bloo and Rose's calculus of explicit substitutions λx, Parigot's λµ and Curien and Herbelin's λµμ.

Journal of Functional Programming, 2000

It is well known that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover, normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a one-one map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (λN) and the other to sequent calculus (λL). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the many-one correspondence mentioned above. Moreover, the second type assignment system has a ‘cut-free’ fragment (λLcf). This fragment generates exactly ...

ACM Transactions on Computational Logic, 2007

In this paper we present a cut-free sequent calculus, called SeqS, for some standard conditional logics. The calculus uses labels and transition formulas and can be used to prove decidability and space complexity bounds for the respective logics. We also present CondLean, a theorem prover for these logics implementing SeqS calculi for weaker systems written in SICStus Prolog.

2002

In this paper, a new notion for sequent calculus (à la Gentzen) for Pure Type Systems (PTS) is introduced. This new calculus, $ \mathcal{K} $ , is equivalent to the standard PTS, and it has a cut-free subsystem, $ \mathcal{K}^{{\text{cf}}} $ , that will be proved to hold non-trivial properties such as the structural rules of Gentzen/Kleene: thinning, contraction, and interchange. An interpretation of completeness of the $ \mathcal{K}^{{\text{cf}}} $ system yields the concept of Cut Elimination, (CE), and it is an essential technique in proof theory; thus we think that it will have a deep impact on PTS and in logical frameworks based on PTS.

2017

We show how Leibnitz’s indiscernibility principle and Gentzen’s original work lead to extensions of the sequent calculus to first order logic with equality and investigate the cut elimination property. Furthermore we discuss and improve the nonlengthening property of Lifschitz and Orevkov in [5] and [8].

2013

Computer-aided reasoning plays a great role in computer science and mathematical logic, from logic programing to automated reasoning, via interactive proof assistants, etc. The general aim of this thesis is to design a general framework where various techniques of Computer-aided reasoning can be implemented, so that they can collaborate, be generalised, and implemented in a safe and trusted way. The framework I propose is a sequent calculus called LKp(T), which generalises an older calculus of the literature to the presence of an arbitrary background theory for which we have a decision procedure, like linear arithmetic. The thesis develops the meta-theory of LKp(T), such as its logical completeness. We then show how it specifies a proof-search procedure that can emulate a well-known technique from the field of Satisfiability-modulo-Theories, namely DPLL(T). Finally, clause and connection tableaux are other widely used techniques of automated reasoning, of a rather different nature f...

2007

This paper revisits the results of Barendregt and Ghilezan [3] and generalizes them for classical logic. Instead of λ-calculus, we use here λµ-calculus as the basic term calculus. We consider two extensionally equivalent type assignment systems for λµ-calculus, one corresponding to classical natural deduction, and the other to classical sequent calculus. Their relations and normalisation properties are investigated. As a consequence a short proof of Cut elimination theorem is obtained.

Proceedings 31st IEEE International Symposium on Multiple-Valued Logic, 2001

In the analytic calculus Ê ½ for Gödel logic has been introduced. Ê ½ operates on "sequents of relations". We show constructively how to eliminate cuts from Ê ½ -derivations. The version of the cut rule we consider allows to derive other forms of cut as well as a rule corresponding to the "communication rule" of Avron's hypersequent calculus for ½ . Moreover, we give an explicit description of all the axioms of Ê ½ and prove their completeness.

2013

In this thesis we consider generic tools and techniques for the proof-theoretic investigation of not necessarily normal modal logics based on minimal, intuitionistic or classical propositional logic. The underlying framework is that of ordinary symmetric or asymmetric two-sided sequent calculi without additional structural connectives, and the point of interest are the logical rules in such a system. We introduce the format of a sequent rule with context restrictions and the slightly weaker format of a shallow rule. The format of a rule with context restrictions is expressive enough to capture most normal modal logics in the S5 cube, standard systems for minimal, intuitionistic and classical propositional logic and a wide variety of non-normal modal logics. For systems given by such rules we provide sufficient criteria for cut elimination and decidability together with generic complexity results. We also explore the expressivity of such systems with the cut rule in terms of axioms in a Hilbert-style system by exhibiting a corresponding syntactically defined class of axioms along with automatic translations between axioms and rules. This enables us to show a number of limitative results concerning amongst others the modal logic S5. As a step towards a generic construction of cut free and tractable sequent calculi we then introduce the notion of cut trees as representations of rules constructed by absorbing cuts. With certain limitations this allows the automatic construction of a cut free and tractable sequent system from a finite number of rules. For cases where such a system is to be constructed by hand we introduce a graphical representation of rules with context restrictions which simplifies this process. Finally, we apply the developed tools and techniques and construct new cut free sequent systems for a number of Lewis' conditional logics extending the logic V. The systems yield purely syntactic decision procedures of optimal complexity and proofs of the Craig interpolation property for the logics at hand. 5

1998

Abstract In [N] the contraction-free and cut-free sequent calculus G3ip for intuitionistic p) ropositiona. l logic was extended by rules for theories of apartness an (l order. The logical content. of the axioms of these theories is expressed by the geometry of sequent calculus rules, which have only atomic formulas as active and principal. In this way also such extensions are contraction-free and cutfree. Cut. elimination permits structural proof analysis, and syntactic proofs of conservativity results.