Results 1  10
of
30
Noncommutativity and MELL in the Calculus of Structures
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2001
"... We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a selfdual noncommutative operator inspired by CCS, that seems not to be expressible in ..."
Abstract

Cited by 55 (22 self)
 Add to MetaCart
We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a selfdual noncommutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.
On the Proof Complexity of Deep Inference
, 2000
"... We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential ..."
Abstract

Cited by 31 (13 self)
 Add to MetaCart
We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential speedup over analytic Gentzen proof systems that they polynomially simulate.
L.: Constructing free Boolean categories
, 2005
"... By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category and not in a weakly distributive one, which simplifies issues like the Mix rule. An important axiom, which is introduced later, is a “graphical ” condition, which is closely related to denotational semantics and the Geometry of Interaction. Then we show that a previously
Reducing Nondeterminism in the Calculus of Structures
, 2005
"... The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: in contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than all other formalisms supporting a ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: in contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than all other formalisms supporting analytical proofs. However, deep applicability of inference rules causes greater nondeterminism than in the sequent calculus regarding proof search. In this paper, we introduce a new technique which reduces nondeterminism without breaking proof theoretical properties, and provides a more immediate access to shorter proofs. We present our technique on system BV, the smallest technically nontrivial system in the calculus of structures, extending multiplicative linear logic with the rules mix, nullary mix and a self dual, noncommutative logical operator. Since our technique exploits a scheme common to all the systems in the calculus of structures, we argue that it generalizes to these systems for classical logic, linear logic and modal logics.
A system of interaction and structure II: the need for deep inference
 Logical Methods in Computer Science
, 2006
"... Vol. 2 (2:4) 2006, pp. 1–24 ..."
Classical Modal Display Logic . . .
, 2007
"... We begin by showing how to faithfully encode the Classical Modal Display Logic (CMDL) of Wansing into the Calculus of Structures (CoS) of Guglielmi. Since every CMDL calculus enjoys cutelimination, we obtain a cutelimination theorem for all corresponding CoS calculi. We then show how our result le ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
We begin by showing how to faithfully encode the Classical Modal Display Logic (CMDL) of Wansing into the Calculus of Structures (CoS) of Guglielmi. Since every CMDL calculus enjoys cutelimination, we obtain a cutelimination theorem for all corresponding CoS calculi. We then show how our result leads to a minimal cutfree CoS calculus for modal logic S5. No other existing CoS calculi for S5 enjoy both these properties simultaneously.
A local system for intuitionistic logic
 of Lecture Notes in Artificial Intelligence
, 2006
"... Abstract. This paper presents systems for firstorder intuitionistic logic and several of its extensions in which all the propositional rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The ma ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract. This paper presents systems for firstorder intuitionistic logic and several of its extensions in which all the propositional rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The main source of nonlocality is the contraction rules. We show that the contraction rules can be restricted to the atomic ones, provided we employ deepinference, i.e., to allow rules to apply anywhere inside logical expressions. We further show that the use of deep inference allows for modular extensions of intuitionistic logic to Dummett’s intermediate logic LC, Gödel logic and classical logic. We present the systems in the calculus of structures, a proof theoretic formalism which supports deepinference. Cut elimination for these systems are proved indirectly by simulating the cutfree sequent systems, or the hypersequent systems in the cases of Dummett’s LC and Gödel logic, in the cut free systems in the calculus of structures.
Breaking Paths in Atomic Flows for Classic Logic
, 2010
"... This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make cruci ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make crucial use of the path breaker, an atomicflow construction that avoids some nasty termination problems, and that can be used in any proof system with sufficient symmetry. This paper contains an original 2dimensionaldiagram exposition of atomic flows, which helps us to connect atomic flows with other known formalisms.
What is the Problem with Proof Nets for Classical Logic?
"... Abstract. This paper is an informal (and nonexhaustive) overview over some existing notions of proof nets for classical logic, and gives some hints why they might be considered to be unsatisfactory. 1 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. This paper is an informal (and nonexhaustive) overview over some existing notions of proof nets for classical logic, and gives some hints why they might be considered to be unsatisfactory. 1