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Fast and Loose Reasoning is Morally Correct
, 2006
"... Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to nontotal (partial) languages. We justify such reasoning. ..."
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Cited by 28 (1 self)
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Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to nontotal (partial) languages. We justify such reasoning.
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 for ..."
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Cited by 23 (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.
An Illative Theory of Relations
, 1990
"... this paper we present a nonstandard logic for our structures. It is a typefree intensional logic, and is also in the tradition of Curry's illative logic [HS86]; see also [AczN, FM87, Smi84, MA88]. The logic has two judgments: that an object is a fact and that an object is a stateofa#airs (c ..."
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Cited by 15 (2 self)
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this paper we present a nonstandard logic for our structures. It is a typefree intensional logic, and is also in the tradition of Curry's illative logic [HS86]; see also [AczN, FM87, Smi84, MA88]. The logic has two judgments: that an object is a fact and that an object is a stateofa#airs (cf. truth and proposition). Objects are given using a variant of the traditional situation theory notation which is more standard, logically speaking, with explicit negation and quantification (see also [Bar87]). No metalinguistic apparatus is employed
Constructing type systems over an operational semantics
 Journal of Symbolic Computation
, 1992
"... Type theories in the sense of MartinLof and the NuPRL system are based on taking as primitive a typefree programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general f ..."
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Cited by 8 (0 self)
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Type theories in the sense of MartinLof and the NuPRL system are based on taking as primitive a typefree programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general form of inductive definition that may either be taken as acceptable in its own right, or further explicated in terms of other patterns of induction. One such account, based on a general theory of inductivelydefined relations, was given by Allen. An alternative account, based on an essentially settheoretic argument, is presented. 1
Optimized Encodings of Fragments of Type Theory in First Order Logic
 JLC: Journal of Logic and Computation
, 1994
"... The paper presents sound and complete translations of several fragments of MartinLof's monomorphic type theory to first order predicate calculus. The translations are optimised for the purpose of automated theorem proving in the mentioned fragments. The implementation of the theorem prover ..."
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Cited by 7 (4 self)
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The paper presents sound and complete translations of several fragments of MartinLof's monomorphic type theory to first order predicate calculus. The translations are optimised for the purpose of automated theorem proving in the mentioned fragments. The implementation of the theorem prover Gandalf and several experimental results are described. 1 Introduction The subject of this paper is the problem of automated theorem proving in MartinLof's monomorphic type theory [19, 8], which is the underlying logic of the interactive proof development system ALF [2, 14]. In the scope of our paper the task of automated theorem proving in type theory is understood as demonstrating that a certain type is inhabited by constructing a term of that type. The problem of inhabitedness of a type A is understood in the following way: given a set of judgements \Gamma (these may be constant declarations, explicit definitions and defining equalities), find a term a such that a2A is derivable from \Gam...
A Framework for the Hyperintensional Semantics of Natural Language with Two Implementations
 Logical Aspects of Computational Linguistics, Springer Lecture Notes in Artificial Intelligence
, 2001
"... In this paper we present a framework for constructing hyperintensional semantics for natural language. On this approach, the axiom of extensionality is discarded from the axiom base of a logic. Weaker conditions are specified for the connection between equivalence and identity which prevent the redu ..."
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Cited by 4 (1 self)
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In this paper we present a framework for constructing hyperintensional semantics for natural language. On this approach, the axiom of extensionality is discarded from the axiom base of a logic. Weaker conditions are specified for the connection between equivalence and identity which prevent the reduction of the former relation to the latter. In addition, by axiomatising an intensional number theory we can provide an internal account of proportional cardinality quantifiers, like most. We use a (pre)lattice defined in terms of a (pre)order that models the entailment relation. Possible worlds/situations/indices are then prime filters of propositions in the (pre)lattice. Truth in a world/situation is then reducible to membership of a prime filter. We show how this approach can be implemented within (i) an intensional higherorder type theory, and (ii) firstorder property theory.
Towards a formal theory of program construction
 REVUE D'INTELLIGENCE ARTIFICIELLE
, 1990
"... A unified framework for formal reasoning about programs and deductive mechanisms involved in programming is developed. Within it principal approaches to program synthesis are formally investigated. We will show that a high degree of abstraction opens a way to combine their strengths, simplifies form ..."
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Cited by 4 (2 self)
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A unified framework for formal reasoning about programs and deductive mechanisms involved in programming is developed. Within it principal approaches to program synthesis are formally investigated. We will show that a high degree of abstraction opens a way to combine their strengths, simplifies formal proofs, and leads to clearer insights into the metamathematics of program construction. All definitions and theorems are presented completely formal which allows to straightforwardly implement them with a proof system for the underlying calculus and derive verified implementations of programming methods from them.
An expressive firstorder logic with flexible typing for natural language semantics
 Logic Journal of the Interest Group in Pure ans Applied Logics 12(2):135–168
, 2003
"... We present Property Theory with Curry Typing (PTCT), an intensional firstorder logic for natural language semantics. PTCT permits finegrained specifications of meaning. It also supports polymorphic types and separation types. 1 We develop an intensional number theory within PTCT in order to repres ..."
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Cited by 3 (0 self)
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We present Property Theory with Curry Typing (PTCT), an intensional firstorder logic for natural language semantics. PTCT permits finegrained specifications of meaning. It also supports polymorphic types and separation types. 1 We develop an intensional number theory within PTCT in order to represent proportional generalized quantifiers like most. We use the type system and our treatment of generalized quantifiers in natural language to construct a typetheoretic approach to pronominal anaphora that avoids some of the difficulties that undermine previous typetheoretic analyses of this phenomenon. 1
Propositional Functions and Families of Types
 In Workshop on Programming Logic
, 1989
"... Introduction In order to capture some of the programmers errors, several computer languages, like Pascal and ML, are equipped with a type system. Using the CurryHoward interpretation of propositions as types [3, 8], or as we shall say here, propositions as sets, a type system can be made strong en ..."
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Cited by 1 (0 self)
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Introduction In order to capture some of the programmers errors, several computer languages, like Pascal and ML, are equipped with a type system. Using the CurryHoward interpretation of propositions as types [3, 8], or as we shall say here, propositions as sets, a type system can be made strong enough to be used to specify the task a program is supposed to do. This is one of the basis for MartinLof's suggestion in [11] to use his formulation of type theory for programming; his ideas are exploited in [14] and there are several computer implementations of type theory [4, 16]. Similar ideas are also behind Coquand and Huet's calculus of constructions [2]. The idea of propositions as sets is closely related to the intuitionistic explanations of the logical constants given by Heyting [7]. In MartinLof's type theory, the interpretation of propositions as sets is fundamental since the notions of proposition and set are identical. So a logical constant is definitionally equal to th
Precise Reasoning About Nonstrict Functional Programs
"... This thesis consists of two parts. Both concern reasoning about nonstrict functional programming languages with partial and infinite values and lifted types, including lifted function spaces. The first part is a case study in program verification: We have written a simple parser and a corresponding ..."
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This thesis consists of two parts. Both concern reasoning about nonstrict functional programming languages with partial and infinite values and lifted types, including lifted function spaces. The first part is a case study in program verification: We have written a simple parser and a corresponding prettyprinter in Haskell. A natural aim is to prove that the programs are, in some sense, each other’s inverses. The presence of partial and infinite values makes this exercise interesting. We have tackled the problem in different ways, and report on the merits of those approaches. More specifically, first a method for testing properties of programs in the presence of partial and infinite values is described. By testing before proving we avoid wasting time trying to prove statements that are not valid. Then it is proved that the programs we have written are in fact (more or less) inverses using first fixpoint induction and then the approximation lemma.