P. Martin-Lof. Constructive Mathematics and Computer Programming. In Logic, Methodology and Philosophy of Science, volume VI, pages 153-175. North Holland, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an extension of Martin L of type theory where the inductive recursive nature of this and other de nitions is formalized. In this way we hope to help clarify the reason why it is an acceptable notion from the point of view of intuitionistic meaning explanations in the sense of Martin L of [14, 16, 15]. First recall that for the case of the simply typed lambda calculus the Taitcomputability predicates A are predicates on terms of type A which are de ned by recursion on the structure of A. We read A (a) as a is a computable term of type A . To match Martin L of s de nition [17] we consider ....

P. Martin-Lof. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science, VI, 1979, pages 153-175. North-Holland, 1982.

....its treatment of equality. In Coq s Calculus of Constructions, for example, there is a single global equality relation which is not the desired one for many types (e.g. function types) The desired equalities have to be handled explicitly, which is quite burdensome. As in Martin L of type theory [17] (of which NuPRL type theory is an extension) in NuPRL each type comes with its own equality relation (the extensional one in the case of functions) and the typing rules guarantee that well typed terms respect these equalities. Semantically, a quotient in NuPRL is trivial to de ne: it is simply ....

Per Martin-Lof. Constructive mathematics and computer programming. In Proceedings of the Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153-175, Amsterdam, 1982. North Holland.

....types are taken to be predicates, which therefore denote sets (or some variant thereof, such as domains) However, many advocates of this view are more proof theoretically inclined, and hence might resist such denotations. Perhaps the best known work along this line is Martin Lof s type theory [44], which also provides dependent types, as implemented in Pebble [5] and other languages. Note that type theory is not a general theory of types, but rather a specific intuitionistic logic which provides one specific notion of type) Although much effort has been put into the types as sets ....

Per Martin-Lof. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science VI, pages 153--175. North-Holand, 1982.

....there is considerable precedent for such augmentations of the type system. For example, Lucassen and Gifford introduce state into functions using the type system to declare whether functions are free of side effects [65] Martin Lf introduces dependent types, in which types are indexed by terms [70]. Xi uses dependent types to augment the type system to include array sizes, and uses type resolution to annotate programs that do not need dynamic array bounds checking [88] The technique uses singleton types instead of general terms [37] to help avoid undecidability. While much of the ....

P. Martin-Lf, "Constructive Mathematics and Computer Programming," in Logic, Methodology, and Philosophy of Science VI, pp. 153-175, North-Holland, 1980.

....is used to determine the admissibility of inductive de nitions, using positive type pointers that is, terms that specify the positive occurrence of parameters in recursive de nitions. In Section 4 we represent inductive types using Martin L of s type constructor for wellorderings (W types) see [14, 15] and chapter 15 of [18] extending the work by Dybjer [10] This solution has the disadvantage that structurally equal elements of a W type are not always convertible, thus making the W type representation only extensionally isomorphic to the desired inductive type. Alternatively, we can ....

.... Wellorderings (also called W types) are types of trees speci ed by a type of nodes A and, for every element a of A, a type of branches (B a) This means that every node labelled with the element a has as many branches as the elements of (B a) Wellorderings were introduced by Martin L of [14, 15] and used by Dybjer [10] to encode all inductive types obtained from strictly positive operators. Here we extend Dybjer s construction to strongly positive operators. De nition 3. Let A : and B : A . The type W(A;B) is de ned by the rules formation W(A;B) introduction a : A f : B a) ....

Per Martin-Lof. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science, VI,

....Although Coq is based on the Extended Calculus of Constructions (see [15] everything could be formalized in a weaker system. Any Pure Type System that is at least as expressive as P (see [2] endowed with inductive types (see [22] or Martin L of s Type Theory with at least two universes (see [16], 17] or [19] is enough. We assume that we have two universes of types s for sets and p for propositions (Set and Prop in the syntax of Coq) and that they both belong to the higher universe 2 (Type in Coq) The product type x : A:B is written using Coq notation (x : A)B. If B : p ....

....will be the argument of a future paper. Here we adopt a solution that represents every inductive type by a type of trees. Solution using Wellorderings. W types are a type theoretic implementation or the notion of well orderings as well founded trees. They were introduced by Per Martin L of in [16] (see also [17] and [19] chapter 15) Suppose that we want to de ne a type of trees such that the nodes of the trees are labeled by elements of the type B, and for each node labeled by an element b : B, the branches stemming from the node are labeled by the elements of a set (C b) i.e. the ....

Per Martin-Lof. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science, VI,

....a given type system, if there exists, for some type oe, a derivation of Gamma e : oe, in this type system. In this work, we present some examples of type systems that support the definition of constructs used in modern programming languages. Some recent research topics, like dependent types [ML78, ML84, Mac86, C 86, CH88] intersection types [Jim96] impredicative polymorphism [Gir71, Gir89, Rey74] and works related with module systems [HL94, Mac85, MTH90, Mac86, MMM91] will not be treated here. It should be noted that formal systems can be used to define not only types, but also ....

....for a typing formula Gamma p : oe, for some oe, where Gamma is a context containing typings for predefined terms of the language. Some type systems of programming languages are undecidable, such as those of Quest [CL91] and Cayenne [Aug99] The type system of Cayenne uses dependent types [ML78, ML84, Mac86, C 86, CH88] that is, the type of an expression depends not only on the types of other expressions, but also on these expressions themselves. For example, the type of a function result can depend on the type of its argument as well as on the value of this argument; the type of a ....

Per Martin-Lof. Constructive Mathematics and Computer Programming. In Proc. of the 6th International Congress for Logic, Methodology, and Phylosophy of Science, pages 153--175. North-Holland, Amsterdam, 1978.

....in Martin Lof s Type Theory [86] by Nordstrom, Petersson and Smith. A recent book by Ranta, Type theoretical Grammar [93] has a good general account of type theory. Martin Lof type theory is presented in his Intuitionistic Type Theory [80] and Constructive Mathematics and Computer Programming [79]. Section 2 Subtypes and Set Types The notion of subtype is not very thoroughly presented in the literature. There is the article by Constable and Hickey [39] which cites the basic literature. Another key paper is by Pierce and Turner [91] The PhD theses from Cornell and Edinburgh deal with ....

....is the heart of the untyped and typed lambda calculus. See Barendregt [7, 8] Stenlund [102] and Church [29] The treatment of relations goes back to Frege and Russell and is covered well in Church [30] Section 6 Universes, Powers and Openness The account of universes is from Per Martin Lof [79] and is informed by Allen [4] and Palmgren [88] Insights about power sets can be found in Fraenkel et al. 50] Beeson [11] and Troelstra [105] Section 7 Families Families are important in set theory, and accounts such as Bourbaki [18] inform Martin Lof s approach [79] Section 8 Lists ....

[Article contains additional citation context not shown here]

Per Martin-Lof. Constructive mathematics and computer programming. In Proceedings of the Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153--175, Amsterdam, 1982. NorthHolland. REFERENCES 45

....of type theory are formulated as tactics which makes a top down construction of programs possible. These ideas are illustrated by a formal derivation of a program for a partitioning problem. Keywords: type, specification, program derivation, goal, tactic. 1 Introduction Martin Lof s type theory [4, 5] was originally developed as a foundation of constructive mathematics. One of the main reasons for using type theory for programming is that it is a theory both for writing specifications and constructing programs. In type theory a specification is expressed by a type and an element of that type ....

Per Martin-Lof. Constructive Mathematics and Computer Programming. In Logic, Methodology and Philosophy of Science, VI,

....argument. For example, append would have a spec list(int) list(int) list(int) Every program is associated with a corresponding specification and synthesis proof. Our proofs are written in the proof editor Oyster [HS90] which is based on a constructive logic known as Martin Lof s Type Theory [ML79]. The synthesis proof essentially guarantees the correctness of the program extracted from it. The more detailed the specification, the more we can guarantee about the program. Our simple specifications prove that C Y NTHIA programs are syntactically correct, well typed, well defined and ....

Per Martin-Lof. Constructive mathematics and computer programming. In 6th International Congress for Logic, Methodology and Philosophy of Science, pages 153--175, Hanover, August

....every term. There are many ways to approach this mathematically. We can give a denotational semantics, or a reduction (rewrite) semantics, or a structural operational semantics or something else. We choose to base the account on inductively defined relations and partial functions in the style of [13, 35, 39, 43], sometimes called natural semantics . We use a lazy evaluator [1, 13, 35] We define first a relation evaluates to, t evals to t 0 . Then we define val as a map val : fx : term j 9t : term x evals to tg term: We give some cases of the evaluation relation to illustrate the method. ffl ....

....denotational semantics, or a reduction (rewrite) semantics, or a structural operational semantics or something else. We choose to base the account on inductively defined relations and partial functions in the style of [13, 35, 39, 43] sometimes called natural semantics . We use a lazy evaluator [1, 13, 35]. We define first a relation evaluates to, t evals to t 0 . Then we define val as a map val : fx : term j 9t : term x evals to tg term: We give some cases of the evaluation relation to illustrate the method. ffl f evals to (x.b) b[a=x] evals to c ap(f;a) evals to c ffl (x.b) evals to ....

[Article contains additional citation context not shown here]

P. Martin-Lof. Constructive mathematics and computer programming. In Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153--75. North-Holland, Amsterdam, 1982.

....issues. There are several approaches to constructivism (see [4,5,19] for an overview) We are especially interested in the constructive recursive mathematics (CRM) approach developed by Markov [12,13] and in constructive type theories (especially those that are based on Martin L of type theory [14]) since we believe them to be highly relevant to Computer Science. In this paper we demonstrate how to apply the ideas of CRM to a constructive type theory thus creating a more powerful type theory that combines the strengths of both approaches to constructive mathematics. According to Markov s ....

Per Martin-Lof. Constructive mathematics and computer programming. In Proceedings of the Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153-175, Amsterdam, 1982. North Holland.

.... original motivation for this work was to justify certain useful kinds of type free inference for the type theory of the Nuprl proof development system [8, 3, 2] The type theories of Nuprl and Martin This research was supported in part by NSF grant CCR8616552 and ONR grant N00014 88 K 0409 Lof [10] are based on an untyped computation system. A type system is constructed by selecting certain terms of the computation system to denote types and by specifying for each type what terms are its members and when two members of the type are to be considered equal. A major practical problem ....

....We take the largest R satisfying R ae [R] and then prove it is a congruence. We call this a maximal congruence. We start by defining a lazy computation system or lcs. The class of lazy computation systems is a direct generalization of the computation system of MartinL of s type theory [10]. An lcs consists of a term language, where the operators of the language may have variable binding structure, together with an evaluation relation which can be any relation on closed terms that is the identity function on those terms that are taken to be values. For an lcs S we define a ....

[Article contains additional citation context not shown here]

P. Martin-Lof. Constructive mathematics and computer programming. In Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153--175, Amsterdam, 1982. North Holland.

.... Constructive Synthesis The Deductive Tableau is a classical example of the deductive synthesis approach to program development, which is sometimes contrasted in with constructive synthesis methods (Deville and Lau, 1993) The latter are typically based on constructive type theories, such as (Martin L of, 1982; Coquand and Huet, 1988) where a proof of a formula 8 x: 9 y: Q(x; y) also yields a function f for which Q(x; f(x) holds. From the viewpoint of proof by higher order resolution, many approaches to constructive synthesis can also be understood as deductive synthesis. For example, as mentioned ....

Martin-Lof, P. (1982). Constructive mathematics and computer programming. In Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153-175, Amsterdam. North Holland.

....of formal semantics for natural language of the 1970s (Montague, 1973) In exible Montague grammar (Hendriks, 1993) simple type theory was replaced by more exible typing schemes. In a somewhat di erent direction, there have been proposals to use Martin L of style type theory (Martin L of, 1984; Martin L of, 1982) as a basis for NL processing systems. A meta mathematical investigation of Martin L of type theory is carried out in (Aczel, 1980) Unfortunately, the perspectives of model theoretic semantics and of Martin L of type theory are at odds. Martin L of type theory is inspired by proof theory. Type ....

P. Martin-Lof. 1982. Constructive mathematics and computer programming. In Cohen, Los, Pfeier, and Podewski, editors, Logic, Methodology and Philosophy of Science VI, pages 153-179. North Holland.

....by Richard Montague in his program of formal semantics for natural language of the 1970s [22] In flexible Montague grammar [12] simple type theory was replaced by more flexible typing schemes. In a somewhat different direction, there have been proposals to use Martin Lof style type theory [21, 20] as a basis for NL processing systems. A meta mathematical investigation of Martin Lof type theory is carried out in [1] Unfortunately, the perspectives of model theoretic semantics and of Martin Lof type theory are at odds. Martin Lof type theory is inspired by proof theory. Type theorists in ....

P. Martin-Lof. Constructive mathematics and computer programming. In Cohen, Los, Pfeiffer, and Podewski, editors, Logic, Methodology and Philosophy of Science VI, pages 153--179. North Holland, 1982.

....of natural language. To some, this may seem a minor bureaucratic matter easily resolved by appropriate regulations. Another view, however, is that the question is an instance of a deep conceptual problem, also manifest in the analysis of proofs as functional objects in proof theory and programming [16, 19, 27, 44], in the question of parameters in situation theory [15] and in treatments within discourse representation theory of phenomena, such as VP ellipsis [25] whose natural analysis requires abstraction. While it would be presumptuous to claim that the present analysis will solve fully the di#cult ....

P. Martin-Lof. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science, VI, pages 153--175. NorthHolland, 1982.

No context found.

P. Martin-Lof. Constructive mathematics and computer programming. In Sixth International Congress for Logic, Methodology and Philosophy of Science, 153{ 175, Amsterdam, 1982. North-Holland.

....are used for many purposes in computer science, one important example is syntax trees for representing phrases in languages. It is therefore vital that such a type constructor, together with its proof rules, should be available in a programming logic such as Martin Lof s intuitionistic type theory [ML82, ML84]. In this paper, we will define a set constructor that one could use for defining many inductive data types. The rest of the paper is organized as follows: We first explain the wellorder set constructor W introduced by Martin Lof in [ML82] This set constructor is the least solution of a ....

....such as Martin Lof s intuitionistic type theory [ML82, ML84] In this paper, we will define a set constructor that one could use for defining many inductive data types. The rest of the paper is organized as follows: We first explain the wellorder set constructor W introduced by Martin Lof in [ML82]. This set constructor is the least solution of a particular parameterized fixed point set equation. And since the parameters of this equation are closely related to the different parts of a single ML datatype definition [Mil84] we continue to discuss the correspondence between the wellorder ....

Per Martin-Lof. Constructive Mathematics and Computer Programming. In Logic, Methodology and Philosophy of Science, VI,

No context found.

P. Martin-Lof. Constructive Mathematics and Computer Programming. In Logic, Methodology and Philosophy of Science, volume VI, pages 153-175. North Holland, 1979.

No context found.

P. Martin-Lof. Constructive Mathematics and Computer Programming. In Logic, Methodology and Philosophy of Science, volume VI, pages 153{ 175. North Holland, 1979.

No context found.

P. Martin-Lof. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science, VI, 1979.

No context found.

Per Martin-Lof. Constructive mathematics and computer programming. In Proceedings of the Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153-175, Amsterdam, 1982. North Holland.

No context found.

P. Martin-Lo"f, Constructive mathematics and computer programming, in Logic, Methodology and Philosophy of Science, VI, North-Holland, Amsterdam, 1982, pp. 153-175.

No context found.

P. Martin-Lof. Constructive mathematics and computer programming. In L.J. Cohen, J. Los, H. Pfei#er, and K.-P. Podewski, editors, Logic, Methodology and Philosophy of Science VI, pages 153--175, Amsterdam, 1982. North-Holland.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC