Tell category theorists about the concept of abstract syntax for a language and they may say &... more Tell category theorists about the concept of abstract syntax for a language and they may say "that's just the initial algebra for a sum-of-products functor on the category of sets". Despite what you might think, they are trying to be helpful since the initiality property is the common denominator of both definitions by structural recursion and proofs by structural induction [5, Sect. 4.4]. In recent years we have learned how to extend this initial algebra view of abstract syntax to encompass languages with statically scoped binders. In the presence of such binders one wants to abstract away from the specific names of bound variables, either by quotienting parse trees by a suitable notion of alpha-equivalence, or by replacing conventional trees with ones containing de Bruijn indices [1]. By changing from the category of sets to other well-known, but still 'set-like' categories of sheaves or presheaves, one can regain an initial algebra view of this even more than normally abstract syntax---the pay-off being new and automatically generated forms of structural recursion and induction that respect alpha-equivalence [2, 3]. One good test of these new ideas is to see if they give rise to new forms of functional programming. In fact they do. The paper [6] sketches a functional programming language for representing and manipulating syntactical structure involving binders, based on the mathematical model of variable-binding in [3, 4]. In this ML-like language there are new forms of type for names and name-binding that come along with facilities for declaring fresh names, for binding names in abstractions and for pulling apart such name-abstractions via pattern-matching. The key idea is that properly abstract uses of names, i.e. ones that do not descend below the level of alpha-conversion, can be imposed on the user by a static type system that deduces information about the freshness of names. Even though we appear to be giving users a 'gensym' facility, the type system restricts the way it can be used to the extent that we keep within effect-free functional programming, in the sense that the usual laws of pure functional programming remain valid (augmented with new laws for names and name-abstractions). In this talk I will introduce this new approach to representing languages static binders in functional programming and discuss some of the difficulties we have had verifying its semantic properties.
Abstract. Existential types have proved useful for classifying various kinds of information hidin... more Abstract. Existential types have proved useful for classifying various kinds of information hiding in programming languages, such as occurs in abstract datatypes and objects. In this paper we address the question of when two elements of an existentiM type are semanticMly ...
Abstract. For programming languages whose denotational semantics uses fixed points of domain cons... more Abstract. For programming languages whose denotational semantics uses fixed points of domain constructors of mixed variance, proofs of correspondence between operational and denotational semantics (or be-tween two different denotational semantics) often depend upon the ...
Page 1. Reasoning about Local Variables with Operationally-Based Logical Relations Andrew M. Pitt... more Page 1. Reasoning about Local Variables with Operationally-Based Logical Relations Andrew M. Pitts* Cambridge University Computer Laboratory, Pembroke Street, Cambridge CB2 3QG, UK [email protected] Abstract A parametric ...
Page 255. Operationally-Based Theories of Program Equivalence Andrew Pitts Contents 1 Introductio... more Page 255. Operationally-Based Theories of Program Equivalence Andrew Pitts Contents 1 Introduction 241 2 Contextual Equivalence 243 3 Similarity and Bisimilarity 257 4 Rational Completeness and Syntactic Continuity 271 5 Further Directions 278 A Proof of the Operational ...
Pitts, A.M., A co-induction principle for recursively defined domains, Theoretical Computer Scien... more Pitts, A.M., A co-induction principle for recursively defined domains, Theoretical Computer Science 124 (1994) 195-219. This paper establishes a new property of predomains recursively defined using the cartesian product, disjoint union, partial function space and convex powerdomain constructors. We prove that the partial order on such a recursive predomain D is the greatest fixed point of a certain monotone operator associated to D. This provides a structurally defined family of proof principles for these recursive predomains: to show that one element of D approximates another, it suffices to find a binary relation containing the two elements that is a post-fixed point for the associated monotone operator. The statement of the proof principles is independent of any of the various methods available for explicit construction of recursive predomains. Following Milner and Tofte (1991), the method of proof is called co-induction. It closely resembles the way bisimulations are used in concurrent process calculi .
Page 1. THE JOURNAL OF SYMBOLIC LOGIC Volume 57, Number 1, March 1992 ON AN INTERPRETATION OF SEC... more Page 1. THE JOURNAL OF SYMBOLIC LOGIC Volume 57, Number 1, March 1992 ON AN INTERPRETATION OF SECOND ORDER QUANTIFICATION IN FIRST ORDER INTUITIONISTIC PROPOSITIONAL LOGIC ANDREW M. PITTS Abstract. ...
Page 1. Alpha-Structural Recursion and Induction ANDREW M. PITTS Cambridge University, Cambridge,... more Page 1. Alpha-Structural Recursion and Induction ANDREW M. PITTS Cambridge University, Cambridge, UK Abstract. The nominal approach to abstract syntax deals with the issues of bound names and α-equivalence by considering ...
New tools are presented for reasoning about properties of recursivelydefined domains. We work wit... more New tools are presented for reasoning about properties of recursivelydefined domains. We work within a general, category-theoretic frameworkfor various notions of `relation" on domains and for actions of domain constructorson relations. Freyd's analysis of recursive types in terms of aproperty of mixed initiality/finality is transferred to a corresponding propertyof invariant relations. The existence of invariant relations is provedunder completeness assumptions
The FreshML project made a fundamental and rigorously- founded extension to functional programmin... more The FreshML project made a fundamental and rigorously- founded extension to functional programming languages. The extension has to do with metaprogramming, the activ- ity of creating software systems that manipulate syntactical structures (interpreters, compilers, proof checkers, proof assistants, etc). In all but the most trivial cases, these syn- tactical structures involve name binding, with associated notions of free and bound
Page 1. Relational Properties of Recursively Defined Domains Andrew M. Pitts* University of Cambr... more Page 1. Relational Properties of Recursively Defined Domains Andrew M. Pitts* University of Cambridge Computer Laboratory Cambridge CB2 3QG, UK Abstract This paper describes a mixed induction/co-in-duction property of relations on recursively defined do-mains. ...
In loc.cit., it is shown that the standard interpretation of the first order typed lambda calculu... more In loc.cit., it is shown that the standard interpretation of the first order typed lambda calculus in the category of sets cannot be extended to a model of the second order typed lambda calculus. Reynolds works with quite a general notion of what constitutes a second order ...
Abstract. This tutorial paper discusses a particular style of operational semantics that enables ... more Abstract. This tutorial paper discusses a particular style of operational semantics that enables one to give a 'syntax-directed' inductive defini-tion of termination which is very useful for reasoning about operational equivalence of programs. We restrict attention to contextual ...
We present a generalisation of first-order unification tothe practically important case of equati... more We present a generalisation of first-order unification tothe practically important case of equations between termsinvolving binding operations. A substitution of terms forvariables solves such an equation if it makes the equatedterms #-equivalent, i.e. equal up to renaming bound names.For the applications we have in mind, we must consider thesimple, textual form of substitution in which names occurringin terms may be
This paper describes a syntactic translation for a substantial fragment of the core Standard ML l... more This paper describes a syntactic translation for a substantial fragment of the core Standard ML language into a typed -calculus with recursive types and imperative features in the form of reference types. The translation compiles SML's use of declarations and pattern matching into -terms, and transforms the use of environments in the operational semantics into a simpler relation of evaluation
Tell category theorists about the concept of abstract syntax for a language and they may say &... more Tell category theorists about the concept of abstract syntax for a language and they may say "that's just the initial algebra for a sum-of-products functor on the category of sets". Despite what you might think, they are trying to be helpful since the initiality property is the common denominator of both definitions by structural recursion and proofs by structural induction [5, Sect. 4.4]. In recent years we have learned how to extend this initial algebra view of abstract syntax to encompass languages with statically scoped binders. In the presence of such binders one wants to abstract away from the specific names of bound variables, either by quotienting parse trees by a suitable notion of alpha-equivalence, or by replacing conventional trees with ones containing de Bruijn indices [1]. By changing from the category of sets to other well-known, but still 'set-like' categories of sheaves or presheaves, one can regain an initial algebra view of this even more than normally abstract syntax---the pay-off being new and automatically generated forms of structural recursion and induction that respect alpha-equivalence [2, 3]. One good test of these new ideas is to see if they give rise to new forms of functional programming. In fact they do. The paper [6] sketches a functional programming language for representing and manipulating syntactical structure involving binders, based on the mathematical model of variable-binding in [3, 4]. In this ML-like language there are new forms of type for names and name-binding that come along with facilities for declaring fresh names, for binding names in abstractions and for pulling apart such name-abstractions via pattern-matching. The key idea is that properly abstract uses of names, i.e. ones that do not descend below the level of alpha-conversion, can be imposed on the user by a static type system that deduces information about the freshness of names. Even though we appear to be giving users a 'gensym' facility, the type system restricts the way it can be used to the extent that we keep within effect-free functional programming, in the sense that the usual laws of pure functional programming remain valid (augmented with new laws for names and name-abstractions). In this talk I will introduce this new approach to representing languages static binders in functional programming and discuss some of the difficulties we have had verifying its semantic properties.
Abstract. Existential types have proved useful for classifying various kinds of information hidin... more Abstract. Existential types have proved useful for classifying various kinds of information hiding in programming languages, such as occurs in abstract datatypes and objects. In this paper we address the question of when two elements of an existentiM type are semanticMly ...
Abstract. For programming languages whose denotational semantics uses fixed points of domain cons... more Abstract. For programming languages whose denotational semantics uses fixed points of domain constructors of mixed variance, proofs of correspondence between operational and denotational semantics (or be-tween two different denotational semantics) often depend upon the ...
Page 1. Reasoning about Local Variables with Operationally-Based Logical Relations Andrew M. Pitt... more Page 1. Reasoning about Local Variables with Operationally-Based Logical Relations Andrew M. Pitts* Cambridge University Computer Laboratory, Pembroke Street, Cambridge CB2 3QG, UK [email protected] Abstract A parametric ...
Page 255. Operationally-Based Theories of Program Equivalence Andrew Pitts Contents 1 Introductio... more Page 255. Operationally-Based Theories of Program Equivalence Andrew Pitts Contents 1 Introduction 241 2 Contextual Equivalence 243 3 Similarity and Bisimilarity 257 4 Rational Completeness and Syntactic Continuity 271 5 Further Directions 278 A Proof of the Operational ...
Pitts, A.M., A co-induction principle for recursively defined domains, Theoretical Computer Scien... more Pitts, A.M., A co-induction principle for recursively defined domains, Theoretical Computer Science 124 (1994) 195-219. This paper establishes a new property of predomains recursively defined using the cartesian product, disjoint union, partial function space and convex powerdomain constructors. We prove that the partial order on such a recursive predomain D is the greatest fixed point of a certain monotone operator associated to D. This provides a structurally defined family of proof principles for these recursive predomains: to show that one element of D approximates another, it suffices to find a binary relation containing the two elements that is a post-fixed point for the associated monotone operator. The statement of the proof principles is independent of any of the various methods available for explicit construction of recursive predomains. Following Milner and Tofte (1991), the method of proof is called co-induction. It closely resembles the way bisimulations are used in concurrent process calculi .
Page 1. THE JOURNAL OF SYMBOLIC LOGIC Volume 57, Number 1, March 1992 ON AN INTERPRETATION OF SEC... more Page 1. THE JOURNAL OF SYMBOLIC LOGIC Volume 57, Number 1, March 1992 ON AN INTERPRETATION OF SECOND ORDER QUANTIFICATION IN FIRST ORDER INTUITIONISTIC PROPOSITIONAL LOGIC ANDREW M. PITTS Abstract. ...
Page 1. Alpha-Structural Recursion and Induction ANDREW M. PITTS Cambridge University, Cambridge,... more Page 1. Alpha-Structural Recursion and Induction ANDREW M. PITTS Cambridge University, Cambridge, UK Abstract. The nominal approach to abstract syntax deals with the issues of bound names and α-equivalence by considering ...
New tools are presented for reasoning about properties of recursivelydefined domains. We work wit... more New tools are presented for reasoning about properties of recursivelydefined domains. We work within a general, category-theoretic frameworkfor various notions of `relation" on domains and for actions of domain constructorson relations. Freyd's analysis of recursive types in terms of aproperty of mixed initiality/finality is transferred to a corresponding propertyof invariant relations. The existence of invariant relations is provedunder completeness assumptions
The FreshML project made a fundamental and rigorously- founded extension to functional programmin... more The FreshML project made a fundamental and rigorously- founded extension to functional programming languages. The extension has to do with metaprogramming, the activ- ity of creating software systems that manipulate syntactical structures (interpreters, compilers, proof checkers, proof assistants, etc). In all but the most trivial cases, these syn- tactical structures involve name binding, with associated notions of free and bound
Page 1. Relational Properties of Recursively Defined Domains Andrew M. Pitts* University of Cambr... more Page 1. Relational Properties of Recursively Defined Domains Andrew M. Pitts* University of Cambridge Computer Laboratory Cambridge CB2 3QG, UK Abstract This paper describes a mixed induction/co-in-duction property of relations on recursively defined do-mains. ...
In loc.cit., it is shown that the standard interpretation of the first order typed lambda calculu... more In loc.cit., it is shown that the standard interpretation of the first order typed lambda calculus in the category of sets cannot be extended to a model of the second order typed lambda calculus. Reynolds works with quite a general notion of what constitutes a second order ...
Abstract. This tutorial paper discusses a particular style of operational semantics that enables ... more Abstract. This tutorial paper discusses a particular style of operational semantics that enables one to give a 'syntax-directed' inductive defini-tion of termination which is very useful for reasoning about operational equivalence of programs. We restrict attention to contextual ...
We present a generalisation of first-order unification tothe practically important case of equati... more We present a generalisation of first-order unification tothe practically important case of equations between termsinvolving binding operations. A substitution of terms forvariables solves such an equation if it makes the equatedterms #-equivalent, i.e. equal up to renaming bound names.For the applications we have in mind, we must consider thesimple, textual form of substitution in which names occurringin terms may be
This paper describes a syntactic translation for a substantial fragment of the core Standard ML l... more This paper describes a syntactic translation for a substantial fragment of the core Standard ML language into a typed -calculus with recursive types and imperative features in the form of reference types. The translation compiles SML's use of declarations and pattern matching into -terms, and transforms the use of environments in the operational semantics into a simpler relation of evaluation
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