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Constructing Recursion Operators in Intuitionistic Type Theory
 Journal of Symbolic Computation
, 1984
"... MartinLöf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of wellfounded relations is presented. Using primitive recursion over higher types, induction and recursion are formally ..."
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Cited by 22 (5 self)
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MartinLöf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of wellfounded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of wellfounded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.
Information Loss in the Programming Logic TK
 Programming Concepts and Methods
, 1990
"... this paper we investigate the topic of information loss in the constructive and intensional theory for programming development TK. The term information loss arose during the investigation of MartinLf's Type Theory [Mar 82] (MLTT) as a programming logic and it refers to techniques for removing compu ..."
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Cited by 3 (2 self)
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this paper we investigate the topic of information loss in the constructive and intensional theory for programming development TK. The term information loss arose during the investigation of MartinLf's Type Theory [Mar 82] (MLTT) as a programming logic and it refers to techniques for removing computationally redundant data from programs which are obtained by formal derivation from specifications. Earlier papers [Hen 89a] [Hen 89b] contain details of the theory TK and [HeT 88] presents a theory of which TK is a restriction. We have taken the opportunity in this paper of describing TK in its entirety and this appears as an appendix. We will devote the rest of this introduction to a motivation for the current work and explain how it is related to similar research which has used MLTT as a basis for a programming logic [Abb 87] [Con 86] [Kha 86] [Bac 89]. The reasons for investigating and using systems like TK and MLTT are, by now, quite well known: program specifications are assertions (in MLTT qua type) and it is possible to prove them within the system. Such proofs show that they are, in principle, satisfiable specifications and it is possible to extract programs that meet them from such proofs. Thus the enterprises of program derivation and specification are unified and one inherits a basic methodology for program derivation from the logical structure governing programs and types. Like MLTT, TK is a constructive theory of sets (sets in TK are types or kinds) but it differs from it in a number of respects, the most important of which, for the purposes of this paper, is that the language of TK separates the assertions or formulae from the types. MLTT, in contrast, makes use of the propositions as types identification [How 80] and so does not make this separation. We have ...
Safe Positive Induction in the Programming Logic TK
 in: Logic Programming (ed. Voronkov, A.), LNCS 592
, 1992
"... We describe an alternative schema of induction for the programming logic TK based on safe positive induction. This replaces the original schema based on the well founded part of a relation. We show how the new schema can be included into the realizability definition and how the soundness of realizab ..."
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Cited by 1 (1 self)
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We describe an alternative schema of induction for the programming logic TK based on safe positive induction. This replaces the original schema based on the well founded part of a relation. We show how the new schema can be included into the realizability definition and how the soundness of realizability can be extended to allow for the derivation of recursive programs from proofs of specifications which use the new schema. We further show how systems of mutual induction can be handled naturally with the new schema. In particular we show how useful systems of mutually recursive combinators can be derived which realize the principles of mutual induction. 2 Introduction The apparatus which a programming logic provides for inductive types is, perhaps, its most important component. This is because we rely on inductive types for the definition of many recursive types ubiquitous in programming languages: natural numbers, lists, trees, and so on. Moreover, it is from proofs which involve in...
Semantics and Parsing in Intuitionistic Categorial Grammar
, 1995
"... The present work is a contribution to the formal approach to natural language semantics founded on the antirealistic philosophical school. This school claims that the proper job of a semantic theory is not just to state the truthconditions of a sentence, but to provide an effective procedure for rec ..."
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The present work is a contribution to the formal approach to natural language semantics founded on the antirealistic philosophical school. This school claims that the proper job of a semantic theory is not just to state the truthconditions of a sentence, but to provide an effective procedure for recognising when such conditions hold in the world; this is called a verificationist theory of meaning. The system has been developed under the framework of MartinL of's Type Theory (MLTT) [ML84], which consists of a constructive theory proposed to serve as a foundation for constructive mathematics. Application of the type theory to the analysis of natural language had already been achieved by Ranta [Ran91] and Sundholm [Sun89], among others. The main contribution of the present document is in parsing; to provide an interpretation of a given English sentence as an expression in MLTT. I show that it is possible to recast English sentences in MLTT in a compositional fashion, by defining a set of rules in the Type Theory that can be effectively computed. Following the tradition of Montague's PTQ paper [Mon74b], the system consists of a set of syntactic and semantic rules that define a fragment of English. The semantic rules are so defined as to be in a onetoone correspondence with the sysntactic rules. This provides a compositional semantic analysis of the infinitely many expressions generated by the grammar. Attention has been given to the problem of discourse representation and crosssentencial anaphora. Rules have been defined to provide an interpretation for these natural language phenomena and a procedure has been developed to support a process looking for the resolution of anaphoric referents. A discussion of the problem of intensional contexts is included, and a formal f...