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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 165 (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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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
A Relationship among Gentzen’s ProofReduction
 KirbyParis’ Hydra Game, and Buchholz’s Hydra Game, Math. Logic Quarterly
, 1997
"... KirbyParis [9] found a certain combinatorial game called Hydra Game whose termination is true but cannot be proved in $PA $. Cichon [4] gave a new proof based on Wainer’s finite characterization of the $\mathrm{P}\mathrm{A}$provably recursive functions by the use of Hardy functions. Both KirbyP ..."
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KirbyParis [9] found a certain combinatorial game called Hydra Game whose termination is true but cannot be proved in $PA $. Cichon [4] gave a new proof based on Wainer’s finite characterization of the $\mathrm{P}\mathrm{A}$provably recursive functions by the use of Hardy functions. Both KirbyParis and Cichon’s proofs on the unprovability result were obtained by a certain investigation on the ordinals less than $\epsilon_{0} $ , the critical ordinal of $\mathrm{P}\mathrm{A} $ , with the help of the fast and slow growing hierarchies, respectively. On the other hand, Jervell [7] proposed a combinatorial game, called Gentzen Game, on finite binary labeled trees. His Game was defined by abstracting some of the proof reduction procedure of Gentzen’s consistency proof of PA [5], which directly implies Gentzen Game’s unprovability of PA via G\"odel’s incompleteness theorem. The rules of Gentzen Game look rather artificial and complicated while those of KirbyParis ’ are more natural and simpler as a combinatorial game. Moreover, the modified proof reduction Jervell considered ignores Gentzen’s notion of potential (or height), which makes the Gentzen Game less complicated but the natural termination proof requires much larger than $\epsilon_{0} $. Hence, the resulting Gentzen Game is much stronger than $\mathrm{P}\mathrm{A} $ , while KirbyParis ’ Game is considered an optimal game, in the sense that any subgame restricted to the hydras with an upper bound size turns out to be provable in $\mathrm{P}\mathrm{A} $. On the other hand, Jervell’s unprovability proof (on Gentzen Game) is direct and clear (thanks to the fact that the Game is directly connected to Gentzen’s consistency proof) while the unprovability proofs of KirbyParis Game are more involved and complicated. Hence, it is very natural to ask if one can find another way of interpreting Gentzen’s proofreduction procedure into a more natural combinatorial game such as KirbyParis’, (so that one can get a natural combinatorial game and a direct unprovabil
Classifying the provably total functions of PA
 BSL
, 2006
"... We give a selfcontained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as good as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for ..."
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We give a selfcontained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as good as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for teaching purposes and just requires basic familiarity with PA and the ordinals below ε0. (Familiarity with a cut elimination theorem for a Gentzen or Tait calculus is helpful but not presupposed). 1
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 clas ..."
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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.
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
A Prooftheoretical Investigation of Global
, 2004
"... the date of receipt and acceptance should be inserted later – c ○ SpringerVerlag ..."
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the date of receipt and acceptance should be inserted later – c ○ SpringerVerlag
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 ...