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23
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 ..."
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Cited by 31 (13 self)
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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.
A system of interaction and structure IV: The exponentials
 IN THE SECOND ROUND OF REVISION FOR MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2007
"... We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. T ..."
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Cited by 11 (6 self)
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We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. The interest of NEL resides in: 1) its being Turing complete, while the same for MELL is not known, and is widely conjectured not to be the case; 2) its inclusion of a selfdual, noncommutative logical operator that, despite its simplicity, cannot be axiomatised in any analytic sequent calculus system; 3) its ability to model the sequential composition of processes. We present several decomposition results for NEL and, as a consequence of those and via a splitting theorem, cut elimination. We use, for the first time, an induction measure based on flow graphs associated to the exponentials, which captures their rather complex behaviour in the normalisation process. The results are presented in the calculus of structures, which is the first, developed formalism in deep inference.
A PROOF CALCULUS WHICH REDUCES SYNTACTIC BUREAUCRACY
"... In usual proof systems, like the sequent calculus, only a very limited way of combining proofs is available through the tree structure. We present in this paper a logicindependent proof calculus, where proofs can be freely composed by connectives, and prove its basic properties. The main advantage ..."
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Cited by 9 (4 self)
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In usual proof systems, like the sequent calculus, only a very limited way of combining proofs is available through the tree structure. We present in this paper a logicindependent proof calculus, where proofs can be freely composed by connectives, and prove its basic properties. The main advantage of this proof calculus is that it allows to avoid certain types of syntactic bureaucracy inherent to all usual proof systems, in particular the sequent calculus. Proofs in this system closely reflect their atomic flow, which traces the behaviour of atoms through structural rules. The general definition is illustrated by the standard deepinference system for propositional logic, for which there are known rewriting techniques that achieve cut elimination based only on the information in atomic flows.
Quasipolynomial normalisation in deep inference via atomic flows and threshold formulae
, 2009
"... ABSTRACT. Jeˇrábek showed that analytic propositionallogic deepinference proofs can be constructed in quasipolynomial time from nonanalytic proofs. In this work, we improve on that as follows: 1) we significantly simplify the technique; 2) our normalisation procedure is direct, i.e., it is interna ..."
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Cited by 8 (4 self)
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ABSTRACT. Jeˇrábek showed that analytic propositionallogic deepinference proofs can be constructed in quasipolynomial time from nonanalytic proofs. In this work, we improve on that as follows: 1) we significantly simplify the technique; 2) our normalisation procedure is direct, i.e., it is internal to deep inference. The paper is selfcontained, and provides a starting point and a good deal of information for tackling the problem of whether a polynomialtime normalisation procedure exists. 1.
A Quasipolynomial CutElimination Procedure in Deep Inference via Atomic Flows and Threshold Formulae
"... Jerábek showed in 2008 that cuts in propositionallogic deepinference proofs can be eliminated in quasipolynomial time. The proof is an indirect one relying on a result of Atserias, Galesi and Pudlák about monotone sequent calculus and a correspondence between this system and cutfree deepinference ..."
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Cited by 7 (4 self)
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Jerábek showed in 2008 that cuts in propositionallogic deepinference proofs can be eliminated in quasipolynomial time. The proof is an indirect one relying on a result of Atserias, Galesi and Pudlák about monotone sequent calculus and a correspondence between this system and cutfree deepinference proofs. In this paper we give a direct proof of Jeˇrábek’s result: we give a quasipolynomialtime cutelimination procedure in propositionallogic deep inference. The main new ingredient is the use of a computational trace of deepinference proofs called atomic flows, which are both very simple (they trace only structural rules and forget logical rules) and strong enough to faithfully represent the cutelimination procedure.
A system of interaction and structure V: The exponentials and splitting
, 2009
"... System NEL is the mixed commutative/noncommutative linear logic BV augmented with linear logic’s exponentials, or, equivalently, it is MELL augmented with the noncommutative selfdual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential compositio ..."
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Cited by 4 (3 self)
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System NEL is the mixed commutative/noncommutative linear logic BV augmented with linear logic’s exponentials, or, equivalently, it is MELL augmented with the noncommutative selfdual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential composition and it faithfully models causal quantum evolution. In this paper, we show cut elimination for NEL, based on a property that we call splitting. NEL is presented in the calculus of structures, which is a deepinference formalism, because no Gentzen formalism can express it analytically. The splitting theorem shows how and to what extent we can recover a sequentlike structure in NEL proofs. Together with the decomposition theorem, proved in the previous paper of the series, this immediately leads to a cutelimination theorem for NEL. 1
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 ..."
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Cited by 4 (2 self)
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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.
Interaction and Depth against Nondeterminism in Deep Inference Proof Search
, 2007
"... Deep inference is a proof theoretical methodology that generalises the traditional notion of inference of the sequent calculus. Deep inference provides more freedom in design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of expone ..."
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Cited by 3 (1 self)
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Deep inference is a proof theoretical methodology that generalises the traditional notion of inference of the sequent calculus. Deep inference provides more freedom in design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, but with the cost of a greater nondeterminism than in the sequent calculus. In this paper, we extend our previous work on proof search with deep inference deductive systems. We argue that, by exploiting an interaction and depth scheme in the logical expressions, the nondeterminism in proof search can be reduced without losing the shorter proofs and without sacrificing from proof theoretical cleanliness.
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 ..."
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Cited by 3 (0 self)
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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
Complexity of deep inference via atomic flows
, 2012
"... Abstract. We consider the fragment of deep inference free of compression mechanisms and compare its proof complexity to other systems, utilising ‘atomic flows’ to examine size of proofs. Results include a simulation of Resolution and daglike cutfree Gentzen, as well as a separation from boundedde ..."
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Cited by 3 (3 self)
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Abstract. We consider the fragment of deep inference free of compression mechanisms and compare its proof complexity to other systems, utilising ‘atomic flows’ to examine size of proofs. Results include a simulation of Resolution and daglike cutfree Gentzen, as well as a separation from boundeddepth Frege. 1