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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.
Normalisation control in deep inference via atomic flows
, 2008
"... Abstract. We introduce ‘atomic flows’: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a n ..."
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Cited by 23 (11 self)
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Abstract. We introduce ‘atomic flows’: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax. 1.
A folk model structure on omegacat
, 2009
"... The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generat ..."
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Cited by 1 (0 self)
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The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on ncategories for arbitrary n ∈ N, as specialisations of the ωcategorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.
Polygraphic resolutions and homology of monoids
, 2007
"... We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1 ..."
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Cited by 1 (1 self)
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We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1
◮ Atomic rules:
, 2009
"... a ∨ a ai↓ t faw↓ aac↓ ∨ a ai ↓ a ∨ ā aw ↓ a ac ↓ a a ∨ identity ā weakening ..."
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a ∨ a ai↓ t faw↓ aac↓ ∨ a ai ↓ a ∨ ā aw ↓ a ac ↓ a a ∨ identity ā weakening
Complexity Classes
, 2010
"... ◮ N P = class of problems that are verifiable in polynomial time. ◮ SAT = ‘Is a propositional formula satisfiable? ’ (Yes: here is a satisfying assignment.) ◮ coN P = class of problems that are disqualifiable in polynomial time. ◮ VAL = ‘Is a propositional formula valid? ’ (No: here is a falsifying ..."
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◮ N P = class of problems that are verifiable in polynomial time. ◮ SAT = ‘Is a propositional formula satisfiable? ’ (Yes: here is a satisfying assignment.) ◮ coN P = class of problems that are disqualifiable in polynomial time. ◮ VAL = ‘Is a propositional formula valid? ’ (No: here is a falsifying assignment.) ◮ P = class of problems that can be solved in polynomial time. ◮ N P ̸ = coN P implies P ̸ = N P.Proof Systems ◮ Proof complexity = proof size. ◮ Proof system = algorithm that verifies proofs in polynomial time on their size. ◮ Important question: What is the relation between size of tautologies and size of minimal proofs? inference rules. There are several equivalent ways of presenting such a system, the 3.2following Let HTis be onethe of formal the simplest, systemgiving whosesyntax axiomto schemes propositional are: Example classical logic. of Proof System: Frege