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16
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.
Canonical sequent proofs via multifocusing
 Fifth IFIP International Conference on Theoretical Computer Science, volume 273 of IFIP International Federation for Information Processing
, 2008
"... Abstract The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calc ..."
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Cited by 18 (10 self)
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Abstract The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calculus, for they identify proofs more abstractly and geometrically. In this paper we recover permutative canonicity directly in the cutfree sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multifocused proofs. We validate this definition by proving a bijection to the wellknown proofnets for the unitfree multiplicative linear logic, and discuss the possibility of a similar correspondence for larger fragments. 1
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 ..."
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Cited by 17 (5 self)
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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
On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
"... ..."
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.
System BV is NPcomplete
, 2005
"... System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a selfdual, noncommutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of MLL. Due to the operator seq, syste ..."
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Cited by 9 (4 self)
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System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a selfdual, noncommutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of MLL. Due to the operator seq, system BV extends the applications of MLL to those where sequential composition is crucial, e.g., concurrency theory. System FBV is an extension of MLL with the rules mix and nullary mix. In this paper, by relying on the fact that system BV is a conservative extension of system FBV, I show that system BV is NPcomplete by encoding the 3Partition problem in FBV. I provide a simple completeness proof of this encoding by resorting to a novel proof theoretical method for reducing the nondeterminism in proof search, which is also of independent interest.
Expanding the realm of systematic proof theory
"... Abstract. This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics. Generalizing our previous work on intuitionisticsubstructural axioms and singleconclusion (hyper)sequent calculi, we define a hierarchy on Hilbert axioms in the language ..."
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Cited by 4 (2 self)
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Abstract. This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics. Generalizing our previous work on intuitionisticsubstructural axioms and singleconclusion (hyper)sequent calculi, we define a hierarchy on Hilbert axioms in the language of classical linear logic without exponentials. We then give a systematic procedure to transform axioms up to the level P ′ 3 of the hierarchy into inference rules in multipleconclusion (hyper)sequent calculi, which enjoy cutelimination under a certain condition. This allows a systematic treatment of logics which could not be dealt with in the previous approach. Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P ′ 3. The case study of Abelian and ̷Lukasiewicz logic is outlined. 1
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
Binding bigraphs as symmetric monoidal closed theories
, 810
"... Abstract. Milner’s bigraphs [1] are a general framework for reasoning about distributed and concurrent programming languages. Notably, it has been designed to encompass both the πcalculus [2] and the Ambient calculus [3]. This paper is only concerned with bigraphical syntax: given what we here call ..."
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Cited by 1 (1 self)
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Abstract. Milner’s bigraphs [1] are a general framework for reasoning about distributed and concurrent programming languages. Notably, it has been designed to encompass both the πcalculus [2] and the Ambient calculus [3]. This paper is only concerned with bigraphical syntax: given what we here call a bigraphical signature K, Milner constructs a (pre) category of bigraphs Bbg(K), whose main features are (1) the presence of relative pushouts (RPOs), which makes them wellbehaved w.r.t. bisimulations, and that (2) the socalled structural equations become equalities. Examples of the latter are, e.g., in π and Ambients, renaming of bound variables, associativity and commutativity of parallel composition, or scope extrusion for νbound names. Also, bigraphs follow a scoping discipline ensuring that, roughly, bound variables never escape their scope. Here, we reconstruct bigraphs using a standard categorical tool: symmetric monoidal closed (smc) theories. Our theory enforces the same scoping discipline as bigraphs, as a direct property of smc structure. Furthermore, it elucidates the slightly mysterious status of socalled edges in