Results 21  30
of
263
Natural Semantics and Some of its MetaTheory in Elf
 PROCEEDINGS OF THE SECOND INTERNATIONAL WORKSHOP ON EXTENSIONS OF LOGIC PROGRAMMING
, 1991
"... Operational semantics provide a simple, highlevel and elegant means of specifying interpreters for programming languages. In natural semantics, a form of operational semantics, programs are traditionally represented as firstorder tree structures and reasoned about using natural deductionlike meth ..."
Abstract

Cited by 44 (14 self)
 Add to MetaCart
Operational semantics provide a simple, highlevel and elegant means of specifying interpreters for programming languages. In natural semantics, a form of operational semantics, programs are traditionally represented as firstorder tree structures and reasoned about using natural deductionlike methods. Hannan and Miller combined these methods with higherorder representations using Prolog. In this paper we go one step further and investigate the use of the logic programming language Elf to represent natural semantics. Because Elf is based on the LF Logical Framework with dependent types, it is possible to write programs that reason about their own partial correctness. We illustrate these techniques by giving type checking rules and operational semantics for MiniML, a small functional language based on a simply typed calculus with polymorphism, constants, products, conditionals, and recursive function definitions. We also partially internalize proofs for some metatheoretic properti...
A finite axiomatization of inductiverecursive definitions
 Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science
, 1999
"... Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an in ..."
Abstract

Cited by 43 (14 self)
 Add to MetaCart
Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductiverecursive definitions. We prove consistency by constructing a settheoretic model which makes use of one Mahlo cardinal. 1
Indexed InductionRecursion
, 2001
"... We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in ..."
Abstract

Cited by 43 (15 self)
 Add to MetaCart
We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductiverecursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.
Set Theory for Verification: II  Induction and Recursion
 Journal of Automated Reasoning
, 2000
"... A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. ..."
Abstract

Cited by 42 (20 self)
 Add to MetaCart
A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.
TypeBased Termination of Recursive Definitions
, 2002
"... This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the techn ..."
Abstract

Cited by 41 (3 self)
 Add to MetaCart
This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the technical machinery behind our system puts a slightly different emphasis on the interpretation of types. More precisely, we formalize the notion of typebased termination using a restricted form of type dependency (a.k.a. indexed types), as popularized by (Xi and Pfenning 1998; Xi and Pfenning 1999). This leads to a simple and intuitive system which is robust under several extensions, such as mutually inductive datatypes and mutually recursive function definitions; however, such extensions are not treated in the paper
A Predicative Analysis of Structural Recursion
, 1999
"... We introduce a language based upon lambda calculus with products, coproducts and strictly positive inductive types that allows the definition of recursive terms. We present the implementation (foetus) of a syntactical check that ensures that all such terms are structurally recursive, i.e., recursive ..."
Abstract

Cited by 41 (20 self)
 Add to MetaCart
We introduce a language based upon lambda calculus with products, coproducts and strictly positive inductive types that allows the definition of recursive terms. We present the implementation (foetus) of a syntactical check that ensures that all such terms are structurally recursive, i.e., recursive calls appear only with arguments structurally smaller than the input parameters of terms considered. To ensure the correctness of the termination checker, we show that all structurally recursive terms are normalizing with respect to a given operational semantics. To this end, we define a semantics on all types and a structural ordering on the values in this semantics and prove that all values are accessible with regard to this ordering. Finally, we point out how to do this proof predicatively using set based operators.
Syntax and Semantics of Dependent Types
 Semantics and Logics of Computation
, 1997
"... ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe ..."
Abstract

Cited by 40 (4 self)
 Add to MetaCart
ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The Eloperator is omitted. For example the \Pitype is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...
On the Interpretation of Type Theory in Locally Cartesian Closed Categories
 Proceedings of Computer Science Logic, Lecture Notes in Computer Science
, 1994
"... . We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of exten ..."
Abstract

Cited by 39 (1 self)
 Add to MetaCart
. We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory. 1 Introduction and Motivation Interpreting dependent type theory in locally cartesian closed categories (lcccs) and more generally in (non split) fibrational models like the ones described in [7] is an intricate problem. The reason is that in order to interpret terms associated with substitution like pairing for \Sigma types or application for \Pitypes one needs a semantical equivalent to syntactic substitution. To clarify the issue let us have a look at the "naive" approach described in Seely's seminal paper [14] which contains a subtle inaccuracy. Assume some dependently typed calculus like the one defined in [10] and an lccc C (a category ...
Modelling General Recursion in Type Theory
 Mathematical Structures in Computer Science
, 2002
"... Constructive type theory is an expressive programming language where both algorithms and proofs can be represented. However, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that satisfy no syntactic condition guaranteeing termination. ..."
Abstract

Cited by 38 (6 self)
 Add to MetaCart
Constructive type theory is an expressive programming language where both algorithms and proofs can be represented. However, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that satisfy no syntactic condition guaranteeing termination.