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18
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.
On transforming intuitionistic matrix proofs into standardsequent proofs
 TABLEAUX–95, LNAI 918
, 1995
"... We present a procedure transforming intuitionistic matrix proofs into proofs within the intuitionistic standard sequent calculus. The transformation is based on L. Wallen’s proof justifying his matrix characterization for the validity of intuitionistic formulae. Since this proof makes use of Fitting ..."
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We present a procedure transforming intuitionistic matrix proofs into proofs within the intuitionistic standard sequent calculus. The transformation is based on L. Wallen’s proof justifying his matrix characterization for the validity of intuitionistic formulae. Since this proof makes use of Fitting‘s nonstandard sequent calculus our procedure consists of two steps. First a nonstandard sequent proof will be extracted from a given matrix proof. Secondly we transform each nonstandard proof into a standard proof in a structure preserving way. To simplify the latter step we introduce an extended standard calculus which is shown to be sound and complete.
A Constructive Formalization of the Catch and Throw Mechanism
 Conf. Rec. IEEE Symposium on Logic in Computer Science
, 1992
"... The catch/throw mechanism is a programming construct for nonlocal exit. In the practical programming, this mechanism plays an important role when programmers handle exceptional situations. In this paper we give a constructive formalization which captures the mechanism in the proofsasprograms noti ..."
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Cited by 16 (1 self)
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The catch/throw mechanism is a programming construct for nonlocal exit. In the practical programming, this mechanism plays an important role when programmers handle exceptional situations. In this paper we give a constructive formalization which captures the mechanism in the proofsasprograms notion. We introduce a modified version of LJ equipped with inference rules corresponding to the operations of catch and throw. Then we show that we can actually extract programs which make use of the catch/throw mechanism from proofs under a certain realizability interpretation. Although the catch/throw mechanism provides only a restricted access to the current continuation, the formulation remains constructive in contrast to the works due to Griffin and Murthy on more powerful facilities such as call/cc (callwithcurrentcontinuation) of Scheme.
Logical Structures of the Catch and Throw Mechanism
, 1995
"... The catch and throw mechanism is a programming construct for nonlocal exit. In practical programming, this mechanism plays an important role when programmers handle exceptional situations. In this thesis we give typing systems which capture the mechanism in the proofsasprograms notion. The typing ..."
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Cited by 11 (0 self)
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The catch and throw mechanism is a programming construct for nonlocal exit. In practical programming, this mechanism plays an important role when programmers handle exceptional situations. In this thesis we give typing systems which capture the mechanism in the proofsasprograms notion. The typing systems can be regarded as a constructive logic with facilities for exception handling, which includes inference rules corresponding to the operations of catch and throw. We show that we can actually regard their proofs as programs which make use of the catch and throw mechanism by a natural interpretation. On one hand the catch and throw mechanism provides only a restricted access to the current continuation, on the other hand its logic is still constructive, in contrast to the works due to Griffin and Murthy on more powerful facilities such as call/cc (callwithcurrentcontinuation) of Scheme. We also capture the nondeterminism introduced by the catch and throw mechanism in a consistent way. Acknowledgements I would like to thank Susumu Hayashi for a great number of helpful suggestions and invaluable encouragement on this work. This work has been deeply influenced by him in a number of aspects. He pointed out the connection between this work and the literature concerning variants of LJ and the treatment of goto in Hoare's logic. I would like to thank Satoru Takasu, who enlightened me on the fundamental notions of constructive programming. I would like to thank Masami Hagiya for his valuable comments and helpful support on this thesis. I would also thank Takayasu Ito, Masahiko Sato, Makoto Tatsuta, Satoshi Kobayashi, and many other people who contributed directly or indirectly to the development of this thesis. Finally I wish to thank Hiromi Nakano, my wife, for her pa...
Efficiently Deciding Intuitionistic Propositional Logic via Translation into Classical Logic
, 1996
"... We present a technique that efficiently translates propositional intuitionistic formulas into propositional classical formulas. This technique allows the use of arbitrary classical theorem provers for deciding the intuitionistic validity of a given propositional formula. The translation is based on ..."
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Cited by 6 (3 self)
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We present a technique that efficiently translates propositional intuitionistic formulas into propositional classical formulas. This technique allows the use of arbitrary classical theorem provers for deciding the intuitionistic validity of a given propositional formula. The translation is based on the constructive description of a finite countermodel for any intuitionistic nontheorem. This enables us to replace universal quantification over all accessible worlds by a conjunction over the constructed finite set of these worlds within the encoding of a refuting Kripkeframe. This way, no additional theory handling by the theorem prover is required.
Timing Analysis of Combinational Circuits in Intuitionistic Propositional Logic
 Formal Methods in System Design
, 1999
"... Classical logic has so far been the logic of choice in formal hardware verification. This paper proposes the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes. The modeltheoretic properties are exploited to handle the s ..."
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Cited by 5 (1 self)
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Classical logic has so far been the logic of choice in formal hardware verification. This paper proposes the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes. The modeltheoretic properties are exploited to handle the secondorder nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way. We present a natural Kripkestyle semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic [15], in which validity is validity up to stabilisation, and implication oe comes out as "boundedly gives rise to." We show that this semantics is equivalently characterised by a notion of realisability with stabilisation bounds as realisers...
On the modal logic K plus theories
 Proc. CSL'95
, 1996
"... . K + T is the propositional modal logic K with the elements of the nite set T as additional axioms. We develop a sequent calculus that is suited for proof search in K + T and discuss methods to improve the eciency. An implementation of the resulting decision procedure is part of the Logics Workben ..."
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Cited by 3 (1 self)
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. K + T is the propositional modal logic K with the elements of the nite set T as additional axioms. We develop a sequent calculus that is suited for proof search in K + T and discuss methods to improve the eciency. An implementation of the resulting decision procedure is part of the Logics Workbench LWB. Then we show that { in contrast to K, KT, S4 { there are theories T and formulas A where a countermodel must have a superpolynomial diameter in the size of T plus A. In the last part we construct an embedding of S4 in K + T . 1 Introduction A Hilbertstyle calculus for (propositional) K + T is obtained from the usual Hilbertstyle calculus for the modal logic K by adding the formulas of T as additional axioms. There is also a natural characterization of K + T from the point of view of Kripke structures: We restrict ourselves to Kripke structures whose worlds satisfy T . One application of K+T is the proof of the equivalence of propositional normal modal logics ([5]). Another app...
A firstorder temporal logic for actions
 The Computing Research Repository (CoRR
"... We present a multimodal action logic with firstorder modalities, which contain terms which can be unified with the terms inside the subsequent formulas and which can be quantified. This makes it possible to handle simultaneously time and states. We discuss applications of this language to action t ..."
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We present a multimodal action logic with firstorder modalities, which contain terms which can be unified with the terms inside the subsequent formulas and which can be quantified. This makes it possible to handle simultaneously time and states. We discuss applications of this language to action theory where it is possible to express many temporal aspects of actions, as for example, beginning, end, time points, delayed preconditions and results, duration and many others. We present tableaux rules for a decidable fragment of this logic. 1
Modal Logic
, 1995
"... If is a formula then we denote by x [c] the formula obtained from by replacing every free occurrence of x by c. If c is the name of an element of a set O then x [c] is called Oinstance of . In order to omit parentheses, we assume that the oneplace operators bind closer that the two place oper ..."
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If is a formula then we denote by x [c] the formula obtained from by replacing every free occurrence of x by c. If c is the name of an element of a set O then x [c] is called Oinstance of . In order to omit parentheses, we assume that the oneplace operators bind closer that the two place operators and ^, _ bind closer than !, (i.e. ^ ! _ is ( ^ ) ! ( _ ), and a sequence of ! is parenthesized to the right (i.e. ! ! ! i
Bicomplete calculi for intuitionistical propositional logic
"... . We define a general notion of refutability for logical calculi, similar to finite failure for logic programs, and we call a sequent calculus K for a logic L bicomplete, iff every nonderivable sequent of L is refutable by K. Thus for a bicomplete sequent calculus every sequent is either derivab ..."
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. We define a general notion of refutability for logical calculi, similar to finite failure for logic programs, and we call a sequent calculus K for a logic L bicomplete, iff every nonderivable sequent of L is refutable by K. Thus for a bicomplete sequent calculus every sequent is either derivable or refutable. Now we show that from any bicomplete calculus for a logic L we may define a canonical semantics for L. For the semantics obtained in this way the corresponding bicomplete calculus provides a solution to the problem of constructing semantical counterexamples for nonprovable formulae of L. In particular for intuitionistic logic we are going to give three such calculi and we are thus obtaining three different types of semantics for it. One of them is the familiar Kripkean semantics. Thus the bicomplete calculus generating this semantics gives at the same time a very perspicuous solution to the notorious problem of finding Kripkean counterexamples for intuitionistically...