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Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
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Cited by 162 (4 self)
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Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. For constructive (intuitionistic)
arithmetic, Kleene’s realizability interpretation is given; this provides an example
of the possibility of a constructive mathematical practice which diverges from classical
mathematics. The crucial notion in intuitionistic analysis, choice sequence, is
briefly described and some principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
Isabelle Tutorial and User's Manual
, 1990
"... This manual describes how to use the theorem prover Isabelle. For beginners, it explains how to perform simple singlestep proofs in the builtin logics. These include firstorder logic, a classical sequent calculus, zf set theory, Constructive Type Theory, and higherorder logic. Each of these logi ..."
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Cited by 26 (2 self)
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This manual describes how to use the theorem prover Isabelle. For beginners, it explains how to perform simple singlestep proofs in the builtin logics. These include firstorder logic, a classical sequent calculus, zf set theory, Constructive Type Theory, and higherorder logic. Each of these logics is described. The manual then explains how to develop advanced tactics and tacticals and how to derive rules. Finally, it describes how to define new logics within Isabelle. Acknowledgements. Isabelle uses Dave Matthews's Standard ml compiler, Poly/ml. Philippe de Groote wrote the first version of the logic lk. Funding and equipment were provided by SERC/Alvey grant GR/E0355.7 and ESPRIT BRA grant 3245. Thanks also to Philippe Noel, Brian Monahan, Martin Coen, and Annette Schumann. Contents 1 Basic Features of Isabelle 5 1.1 Overview of Isabelle : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.1.1 The representation of logics : : : : : : : : : : : : : : : : : : : 6 1.1.2 The...
Modular cutelimination: Finding proofs or counterexamples
 Proceedings of the 13th International Conference of Logic for Programming AI and Reasoning (LPAR06), LNAI 4246
, 2006
"... Abstract. Modular cutelimination is a particular notion of ”cutelimination in the presence of nonlogical axioms ” that is preserved under the addition of suitable rules. We introduce syntactic necessary and sufficient conditions for modular cutelimination for standard calculi, a wide class of (p ..."
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Abstract. Modular cutelimination is a particular notion of ”cutelimination in the presence of nonlogical axioms ” that is preserved under the addition of suitable rules. We introduce syntactic necessary and sufficient conditions for modular cutelimination for standard calculi, a wide class of (possibly) multipleconclusion sequent calculi with generalized quantifiers. We provide a ”universal” modular cutelimination procedure that works uniformly for any standard calculus satisfying our conditions. The failure of these conditions generates counterexamples for modular cutelimination and, in certain cases, for cutelimination. 1
Streams and strings of formal proofs
 Theoretical Computer Science
, 2000
"... www.elsevier.com/locate/tcs ..."
Complexity of the Intuitionistic Sequent Calculus
, 1998
"... . We initiate the study of proof complexity for intuitionistic propositional proof systems: It is known that the set of intuitionistic tautologies is PSPACEcomplete (as opposed to coNPcomplete in the classical case). We show that formulas derived from the "clique tautologies " used in classical pr ..."
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. We initiate the study of proof complexity for intuitionistic propositional proof systems: It is known that the set of intuitionistic tautologies is PSPACEcomplete (as opposed to coNPcomplete in the classical case). We show that formulas derived from the "clique tautologies " used in classical proof complexity have only exponentialsize proofs in intuitionistic sequent calculi or Frege systems (without substitution). This is in contrast to the classical case where the complexity of Frege systems is still open. Introduction The theory of classical propositional proof systems is motivated by the conjecture NP 6= coNP. Moreover, recently there is increased interest in the impact of this theory on questions of a more algorithmic nature,for example [Bo et al.97],[Be et al. 97]. As the set of classical tautologies is coNP complete and propositional proof systems are just nondeterministic algorithms having at least 1 accepting computation for each tautology, there should be tautologies...
Finitary reductions for local predicativity, I: recursively regular ordinals
"... We define notation system for infinitary derivations arising from cutelimination for a theory T 1 \Sigma of recursively regular ordinals by the method of local predicativity. Using these notations, we derive finitary cutelimination steps together with corresponding ordinal assignments. Introductio ..."
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We define notation system for infinitary derivations arising from cutelimination for a theory T 1 \Sigma of recursively regular ordinals by the method of local predicativity. Using these notations, we derive finitary cutelimination steps together with corresponding ordinal assignments. Introduction There is an extensive literature connecting infinitary "Schuttestyle" and finitary "GentzenTakeutistyle" sides of proof theory. For example, in papers [Mi75, Mi75a, Mi79, Bu91, Bu97a] this was done for systems not exceeding in strength Peano Arithmetic. But most recently, there has been an interest to what one can get on the side of finitary proof theory from the methods which are used for prooftheoretical analysis of impredicative theories (see [Wei96, Bu97]). Especially we want to mention paper [Bu97], where it was shown that Takeuti's reduction steps for \Pi 1 1 \Gamma CA+ BI [Tak87, x27] can be derived from Buchholz' method of\Omega +1 rule ([BFPS, Ch. IVV], [BS88]). Here we ...
an der
"... i Acknowledgements I would like to thank my advisor Agata Ciabattoni for introducing me to the field of systematic proof theory, for her patience and her continuous support. I really appreciate the time she spent on explanations when a proof didn’t work out as planned. I also want to thank Gernot Sa ..."
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i Acknowledgements I would like to thank my advisor Agata Ciabattoni for introducing me to the field of systematic proof theory, for her patience and her continuous support. I really appreciate the time she spent on explanations when a proof didn’t work out as planned. I also want to thank Gernot Salzer who was of great help when I was struggling with PROLOG. I am really grateful to my parents, Patrizia and Ingmar, my brother Dominik and my family for supporting me in every possible way. I also want to thank Christoph for always being there. Lara Nonclassical logics are logics different from classical, boolean logic. They encompass, amongst others, the family of intermediate, fuzzy, and substructural logics. During the past few years, these logics have gained importance, especially in many fields of computer science, artificial intelligence, and knowledge representation. By now, there are many useful and interesting nonclassical logics and practitioners in various fields
Hypersequent and Labelled Calculi for Intermediate Logics ⋆
"... Abstract. Hypersequent and labelled calculi are often viewed as antagonist formalisms to define cutfree calculi for nonclassical logics. We focus on the class of intermediate logics to investigate the methods of turning Hilbert axioms into hypersequent rules and frame conditions into labelled rule ..."
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Abstract. Hypersequent and labelled calculi are often viewed as antagonist formalisms to define cutfree calculi for nonclassical logics. We focus on the class of intermediate logics to investigate the methods of turning Hilbert axioms into hypersequent rules and frame conditions into labelled rules. We show that these methods are closely related and we extend them to capture larger classes of intermediate logics. 1