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11
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.
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 ..."
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Cited by 16 (5 self)
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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.
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.
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 ..."
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Cited by 7 (5 self)
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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.
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.
A Deductive Compositional Approach to Petri Nets for Systems Biology
"... We introduce the language CP, a compositional language for place transition petri nets for the purpose of modelling signalling pathways in complex biological systems. We give the operational semantics of the language CP by means of a proof theoretical deductive system which extends multiplicative e ..."
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Cited by 1 (1 self)
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We introduce the language CP, a compositional language for place transition petri nets for the purpose of modelling signalling pathways in complex biological systems. We give the operational semantics of the language CP by means of a proof theoretical deductive system which extends multiplicative exponential linear logic with a selfdual noncommutative logical operator. This allows to express parallel and sequential composition of processes at the same syntactic level as in process algebra, and perform logical reasoning on these processes. We demonstrate the use of the language on a model of a signaling pathway for Fc receptormediated phagocytosis.
Ingredients of a Deep Inference Theorem Prover
"... Deep inference deductive systems for classical logic provide exponentially shorter proofs than the sequent calculus systems, however with the cost of higher nondeterminism and larger search space in proof search. We report on our ongoing work on proof search with deep inference deductive systems. We ..."
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Cited by 1 (0 self)
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Deep inference deductive systems for classical logic provide exponentially shorter proofs than the sequent calculus systems, however with the cost of higher nondeterminism and larger search space in proof search. We report on our ongoing work on proof search with deep inference deductive systems. We present systems for classical logic where nondeterminism in proof search is reduced by constraining the context management rule of these systems. We argue that a deep inference system for classical logic can outperform sequent calculus deductive systems in proof search when nondeterminism and the application of the contraction rule are controlled by means of invertible rules.
On Linear Logic Planning and Concurrency
"... We present an approach to linear logic planning where an explicit correspondence between partial order plans and multiplicative exponential linear logic proofs is established. This is performed by extracting partial order plans from sound and complete encodings of planning problems in multiplicativ ..."
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Cited by 1 (1 self)
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We present an approach to linear logic planning where an explicit correspondence between partial order plans and multiplicative exponential linear logic proofs is established. This is performed by extracting partial order plans from sound and complete encodings of planning problems in multiplicative exponential linear logic in a way that exhibits a noninterleaving behavioral concurrency semantics. Relying on this fact, we argue that this work is a crucial step for establishing a common language for concurrency and planning that will allow to carry techniques and methods between these two fields.
September 12, 2008 — Submitted to Trends in Logic VI The Logic BV and Quantum Causality
"... We describe how a logic with commutative and noncommutative connectives can be used for capturing the essence of discrete quantum causal propagation. 1 Causal graphs and locative slices In this note we describe how the kinematics of quantum causal evolution can be captured by the logic BV [2]. The s ..."
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We describe how a logic with commutative and noncommutative connectives can be used for capturing the essence of discrete quantum causal propagation. 1 Causal graphs and locative slices In this note we describe how the kinematics of quantum causal evolution can be captured by the logic BV [2]. The setting is discrete quantum mechanics. We imagine a finite “web ” of spacetime points. They are viewed as vertices in a directed acyclic graph (DAG); the edges of the DAG represent causal links mediated by the propagation of matter [1]. The fact that the graph is acyclic captures a basic causality requirement: there are no closed causal trajectories. The DAG represents a discrete approximation to the spacetime on which a quantum system evolves. The graph
grau de Mestre em Lógica Computacional supervision:
, 2007
"... Schematic systems is a metamathematical framework used to describe deductive systems. In this Master Thesis we describe sufficient conditions for a general class of sequent calculi, presented by schematic systems, to enjoy the Craig Interpolation Theorem. These conditions are a refinement of the one ..."
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Schematic systems is a metamathematical framework used to describe deductive systems. In this Master Thesis we describe sufficient conditions for a general class of sequent calculi, presented by schematic systems, to enjoy the Craig Interpolation Theorem. These conditions are a refinement of the ones in [9] already known to be weaker than symmetricity. The proof of interpolation is based on Maehara’s technique and on the idea of tracing the flow of the formulas in a deduction. Sequent systems for classical, intuitionistic and minimal first order logic are presented by schematic systems and their properties of cutelimination and interpolation are proved. Moreover, we study the combinatorial nature of interpolation by means of structured sets. Finally, we present an analysis of the interpolant complexity (number of symbols of the interpolant) for propositional deductive systems and show its application to some fundamental questions in complexity theory. 2Contents