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21
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.
Power domains and iterated function systems
 Information and Computation
, 1996
"... We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniquene ..."
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Cited by 30 (10 self)
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We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.] 1996 Academic Press, Inc. 1.
Topology and Iterates in Computational Logic
 Proceedings of the 12th Summer Conference on Topology and its Applications: Special Session on Topology in Computer Science
, 1997
"... We consider the problem of finding models for logic programs P via fixed points of immediate consequence operators, T P . Certain extensions of syntax invalidate the classical approach, adopted in the case of definite programs, using iterates of T P and the KnasterTarski theorem. We discuss alterna ..."
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Cited by 18 (13 self)
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We consider the problem of finding models for logic programs P via fixed points of immediate consequence operators, T P . Certain extensions of syntax invalidate the classical approach, adopted in the case of definite programs, using iterates of T P and the KnasterTarski theorem. We discuss alternatives to the use of this theorem based on elementary notions from topological dynamics. This leads us to consider simple syntactic conditions on P , employing level mappings taking values in a countable ordinal fl, which ensure convergence (to models and fixed points) of the requisite sequences of iterates. We obtain, as a result, a constructive approach to the perfect model semantics of Przymusinski for locally stratified programs, somewhat along the lines of the approach adopted by Apt, Blair and Walker for stratified programs. In particular, when certain inequalities are sharp, we show the existence of unique supported models, which improves Przymusinski's results for perfect models. This...
Lambda theories of effective lambda models
 In 16th EACSL Annual Conference on Computer Science and Logic (CSL’07), LNCS
, 2007
"... Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recu ..."
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Cited by 9 (5 self)
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Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λcalculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards LöwenheimSkolem theorem.
Density and Choice for Total Continuous Functionals
 About and Around Georg Kreisel
, 1996
"... this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are rel ..."
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Cited by 8 (3 self)
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this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are relatively short and hopefully perspicious, and may contribute to redirect attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Section 3 gives a characterization of the continuous functions between domains, in terms of approximable mappings. In section 4 cartesian products and function spaces of domains and information systems are introduced. In section 5 the partial and total continuous functionals are defined. Section 6 finally contains the proofs of the two theorems above; it will be clear that the same proofs also yield effective versions of these theorems.
Some Issues Concerning Fixed Points in Computational Logic: QuasiMetrics, Multivalued Mappings and the KnasterTarski Theorem
, 2000
"... Many questions concerning the semantics of disjunctive databases and of logic programming systems depend on the fixed points of various multivalued mappings and operators determined by the database or program. We discuss known versions, for multivalued mappings, of the KnasterTarski theorem and of ..."
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Cited by 8 (7 self)
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Many questions concerning the semantics of disjunctive databases and of logic programming systems depend on the fixed points of various multivalued mappings and operators determined by the database or program. We discuss known versions, for multivalued mappings, of the KnasterTarski theorem and of the Banach contraction mapping theorem, and formulate a version of the classical fixedpoint theorem (sometimes attributed to Kleene) which is new. All these results have applications to the semantics of disjunctive logic programs, and we will describe a class of programs to which the new theorem can be applied. We also show that a unification of the latter two theorems is possible, using quasimetrics, which parallels the wellknown unification of Rutten and Smyth in the case of conventional programming language semantics.
Recursion on the partial continuous functionals
 Logic Colloquium ’05
, 2006
"... We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the wellknown abbreviation ..."
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Cited by 7 (5 self)
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We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the wellknown abbreviation
On sequential functionals of type 3
 Math. Structures Comput. Sci
, 2006
"... We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1 ..."
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Cited by 6 (0 self)
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We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1
Classifying Recursive Functions
, 1997
"... computability. With the preparations done it is now rather straightforward to define computability in our iterated function spaces C ae (based on N viewed as a flat information system). The tokens and finite sets of tokens are encodable by integers using sequencecoding. It is easy to see that the ..."
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Cited by 4 (1 self)
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computability. With the preparations done it is now rather straightforward to define computability in our iterated function spaces C ae (based on N viewed as a flat information system). The tokens and finite sets of tokens are encodable by integers using sequencecoding. It is easy to see that the notions X 2 Con ae of consistency and X ` ae a of entailment correspond to recursive (in fact elementary) relations. Definition. A partial continuous functional ' of type ae is said to be computable if  when viewed as a set of (codes of) tokens  it is \Sigma 0 1 definable. 5. Computability in higher types 27 Lemma. For all types ae; oe; ø the functionals eval ae;oe : (C ae ! C oe ) \Theta C ae ! C oe curry ae;oe;ø : (C ae \Theta C oe ! C ø ) ! (C ae ! (C oe ! C ø )) are computable. Proof. The tokens of eval ae;oe are of the form (W; X; a) with W 2 Con ae!oe , X 2 Con ae and a 2 C oe , and we have (W; X; a) 2 eval j a 2 W (X) j a 2 WX j WX ` a: Here WX is the application of fini...
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.