Results 1  10
of
12
Comparing and Implementing Calculi of Explicit Substitutions with Eta Reduction
 Annals of Pure and Applied Logic
, 2005
"... The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that e ..."
Abstract

Cited by 10 (8 self)
 Add to MetaCart
The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the etareductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of betacontraction.
The theory of calculi with explicit substitutions revisited
 CSL 2007
, 2007
"... Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we fi ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambdacalculus which enjoys fundamental properties such as simulation of onestep betareduction, confluence on metaterms, preservation of betastrong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proofnets.
Pure Type Systems with Explicit Substitution
, 2000
"... We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is established. A more complicated extension is proposed for which subject reduction does hold in general.
Absolute Explicit Unification
 in International Conference on Rewriting Techniques and Applications (RTA'2000
, 2000
"... . This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higherorder unification, and type inference for PTS using purely firstorder machinery. A novel feature of our system is that it combines substituti ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
. This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higherorder unification, and type inference for PTS using purely firstorder machinery. A novel feature of our system is that it combines substitutions and variable declarations. This allows as a sideeffect to type check letbindings. Our treatment of metavariables is also explicit, such that instantiations of metavariables is internalized in the calculus. This produces a confluent calculus with distinguished holes and explicit substitutions that is insensitive to ffconversion, and allows directly embedding the system into rewriting logic. 1 Introduction Explicit substitutions provide a convenient framework for encoding higherorder typed calculus using firstorder machinery. In particular, this allows to integrate higherorder unification with firstorder provers, rewriting logic, and to delay evaluation and resolve scoping...
Explicit Substitutions and All That
, 2000
"... Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In this paper we explore new developments on two of the most successful styles of explicit substitution calculi: the lambdasigma and lambda_secalculi.
Unification via the ...Style of Explicit Substitutions
, 2001
"... A unication method based on the se style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the sty ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
A unication method based on the se style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the style of explicit substitution. Correctness and completeness properties of the proposed seunication method are shown and its advantages, inherited from the qualities of the se calculus, are pointed out. Our method needs only one sort of objects: terms. And in contrast to the HOU approach based on the calculus, it avoids the use of substitution objects. This makes our method closer to the syntax of the calculus. Furthermore, detection of redices depends on the search for solutions of simple arithmetic constraints which makes our method more operational than the one based on the style of explicit substitution. Keywords: Higher order unication, explicit substitution, lambdacalculi. 1
Use and Meaning of Open Terms in Interactive Formal Problem Solving
"... By formal problem solving, we mean problem solving where the problem and the possible solutions are stated in a precise way, and so is the notion of correctness, stating whether a solution solves a problem. Formal problem solving occurs in many places in computer science, like a program editor (wher ..."
Abstract
 Add to MetaCart
By formal problem solving, we mean problem solving where the problem and the possible solutions are stated in a precise way, and so is the notion of correctness, stating whether a solution solves a problem. Formal problem solving occurs in many places in computer science, like a program editor (where a program has to meet a specification), a structure editor (where syntactically correct expressions have to be built), verification (of protocols), testing, theorem proving and data bases (where information is extracted using queries). An important aspect of computer systems for problem solving is that they are interactive. The machine will in general not be able to solve our problems by itself; solving problems requires humanmachine interaction. Ideally, one would let the computer construct a part of the solution, returning a `solutionwithholes', and then the user provides information on how to proceed to refine the holes. Such a hole in a `solutionwithholes' is also known as an ope...
Higher Order Unification via ...Style of Explicit Substitution
"... A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach ..."
Abstract
 Add to MetaCart
A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.