Results 1  10
of
32
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 ..."
Abstract

Cited by 162 (4 self)
 Add to MetaCart
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.
Program extraction from classical proofs
 Annals of Pure and Applied Logic
, 1994
"... 1 Introduction It is well known that it is undecidable in general whether a given program meets its specification. In contrast, it can be checked easily by a machine whether a formal proof is correct, and from a constructive proof one can automatically extract a corresponding program, which by its v ..."
Abstract

Cited by 54 (9 self)
 Add to MetaCart
1 Introduction It is well known that it is undecidable in general whether a given program meets its specification. In contrast, it can be checked easily by a machine whether a formal proof is correct, and from a constructive proof one can automatically extract a corresponding program, which by its very construction is correct as well. This at least in principle opens a way to produce correct software, e.g. for safetycritical applications. Moreover, programs obtained from proofs are "commented " in a rather extreme sense. Therefore it is easy to maintain them, and also to adapt them to particular situations. We will concentrate on the question of classical versus constructive proofs. It is known that any classical proof of a specification of the form 8x9yB with B quantifierfree can be transformed into a constructive proof of the same formula. However, when it comes to extraction of a program from a proof obtained in this way, one easily ends up with a mess. Therefore, some refinements of the standard transformation are necessary.
Categorical Reconstruction of a Reduction Free Normalization Proof
, 1995
"... Introduction We present a categorical proof of the normalization theorem for simply typed calculus, i.e. we derive a computable function nf which assigns to every typed term a normal form, s.t. M ' N nf(M ) = nf(N ) nf(M ) ' M where ' is fij equality. Both the function nf and its correctness ..."
Abstract

Cited by 23 (5 self)
 Add to MetaCart
Introduction We present a categorical proof of the normalization theorem for simply typed calculus, i.e. we derive a computable function nf which assigns to every typed term a normal form, s.t. M ' N nf(M ) = nf(N ) nf(M ) ' M where ' is fij equality. Both the function nf and its correctness properties can be deduced from the categorical construction. To substantiate this, we present an ML program in the appendix which can be extracted from our argument. We emphasize that this presentation of normalization is reduction free, i.e. we do not mention term rewriting or use properties of term rewriting systems such as the ChurchRosser property. An immediate consequence of normalization is the decidability of ' but there are other useful corollaries; for instance we can show that
Turchin's Supercompiler Revisited  An operational theory of positive information propagation
, 1996
"... Turchin`s supercompiler is a program transformer that includes both partial evaluation and deforestation. Although known in the West since 1979, the essence of its techniques, its more precise relations to other transformers, and the properties of the programs that it produces are only now becoming ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
Turchin`s supercompiler is a program transformer that includes both partial evaluation and deforestation. Although known in the West since 1979, the essence of its techniques, its more precise relations to other transformers, and the properties of the programs that it produces are only now becoming apparent in the Western functional programming community. This thesis gives a new formulation of the supercompiler in familiar terms; we study the essence of it, how it achieves its effects, and its relations to related transformers; and we develop results dealing with the problems of preserving semantics, assessing the efficiency of transformed programs, and ensuring termination.
A Compositional Semantics for Logic Programs and Deductive Databases
, 1996
"... Considering integrity constraints and program composition, it is argued that a semantics for logic programs and deductive databases should not accommodate inconsistencies globally like in classical logic, but locally. It is shown that minimal logic, a weakening of classical logic which precludes ref ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Considering integrity constraints and program composition, it is argued that a semantics for logic programs and deductive databases should not accommodate inconsistencies globally like in classical logic, but locally. It is shown that minimal logic, a weakening of classical logic which precludes refutation proofs, is sufficient to provide for a proof theory for generalized programs corresponding to deductive databases and disjunctive logic programs. A (nonclassical) model theory is proposed for these programs, which allows local inconsistencies. The proposed semantics naturally extends the minimal model and completion semantics of positive programs and is compositional. Arguably, it appropriately conveys a practician's intuition. 1 Introduction The semantics of positive and definite logic programs does not easily extend to nonpositive programs, i.e. programs containing clauses with negative body literals. Although each approach to specifying the semantics of positive programs can be ...
Collapsing Partial Combinatory Algebras
 HigherOrder Algebra, Logic, and Term Rewriting
, 1996
"... Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We establish several facts concerning final collapses (maximal identification of elements). `En passant' we find another example of a pca that cannot be extended to a total one. 1
On model theory for intuitionistic bounded arithmetic with applications to independence results
 Feasible Mathematics
, 1990
"... Abstract IPV+ is IPV (which is essentially IS12) with polynomialinduction on \Sigma b+1formulas disjoined with arbitrary formulas in which the induction variable does not occur. This paper proves that IPV+ is sound and complete with respect to Kripke structures in which every world is a model of C ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Abstract IPV+ is IPV (which is essentially IS12) with polynomialinduction on \Sigma b+1formulas disjoined with arbitrary formulas in which the induction variable does not occur. This paper proves that IPV+ is sound and complete with respect to Kripke structures in which every world is a model of CPV (essentially S12). Thus IPV is sound with respect to such structures. In this setting, this is a strengthening of the usual completeness and soundness theorems for firstorder intuitionistic theories. Using Kripke structures a conservation result is proved for PV1 over IPV.
Constructive algebraic integration theory without choice. Dagstuhl proceedings
 Mathematics, Algorithms, Proofs, number 05021 in Dagstuhl Seminar Proceedings. Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2005
"... Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpret ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in MartinL type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.