An operational domain-theoretic treatment of recursive types

2013 ◽  
Vol 24 (1) ◽  
Author(s):  
WENG KIN HO
Keyword(s):  
Operational Semantics ◽  
Domain Theory ◽  
Invariance Property ◽  
Inverse Limits ◽  
Classical Domain ◽  
Contextual Equivalence

We develop an operational domain theory for treating recursive types with respect to contextual equivalence. The principal approach we take deviates from classical domain theory in that we do not produce the recursive types using the usual inverse limits constructions – we get them for free by working directly with the operational semantics. By extending type expressions to functors between some ‘syntactic’ categories, we establish algebraic compactness. To do this, we rely on an operational version of the minimal invariance property, for which we give a purely operational proof.

scholarly journals Domain Theory for Concurrency

2003 ◽  
Vol 10 (43) ◽  
Author(s):  
Mikkel Nygaard ◽  
Glynn Winskel
Keyword(s):  
Linear Logic ◽  
Operational Semantics ◽  
Denotational Semantics ◽  
Higher Order ◽  
The Other ◽  
Domain Theory ◽  
Parallel Composition ◽  
Concurrent Computation ◽  
Contextual Equivalence

A simple domain theory for concurrency is presented. Based on a categorical model of linear logic and associated comonads, it highlights the role of linearity in concurrent computation. Two choices of comonad yield two expressive metalanguages for higher-order processes, both arising from canonical constructions in the model. Their denotational semantics are fully abstract with respect to contextual equivalence. One language derives from an exponential of linear logic; it supports a straightforward operational semantics with simple proofs of soundness and adequacy. The other choice of comonad yields a model of affine-linear logic, and a process language with a tensor operation to be understood as a parallel composition of independent processes. The domain theory can be generalised to presheaf models, providing a more refined treatment of nondeterministic branching. The article concludes with a discussion of a broader programme of research, towards a fully fledged domain theory for concurrency.

Linear types and approximation

2000 ◽  
Vol 10 (6) ◽  
pp. 719-745 ◽  
Author(s):  
MICHAEL HUTH ◽  
ACHIM JUNG ◽  
KLAUS KEIMEL
Keyword(s):  
Full Subcategory ◽  
Linear Logic ◽  
Domain Theory ◽  
Classical Domain ◽  
Complete Distributivity ◽  
Linear Types ◽  
Continuous Lattices

We study continuous lattices with maps that preserve all suprema rather than only directed ones. We introduce the (full) subcategory of FS-lattices, which turns out to be *-autonomous, and in fact maximal with this property. FS-lattices are studied in the presence of distributivity and algebraicity. The theory is extremely rich with numerous connections to classical Domain Theory, complete distributivity, Topology and models of Linear Logic.

scholarly journals A call-by-need lambda calculus with locally bottom-avoiding choice: context lemma and correctness of transformations

2008 ◽  
Vol 18 (3) ◽  
pp. 501-553 ◽  
Author(s):  
DAVID SABEL ◽  
MANFRED SCHMIDT-SCHAUSS
Keyword(s):  
Operational Semantics ◽  
Lambda Calculus ◽  
Equational Theory ◽  
Order Reduction ◽  
Single Step ◽  
Program Transformations ◽  
Normal Order ◽  
Choice Context ◽  
Contextual Equivalence ◽  
Fairness Condition

We present a higher-order call-by-need lambda calculus enriched with constructors, case expressions, recursive letrec expressions, a seq operator for sequential evaluation and a non-deterministic operator amb that is locally bottom-avoiding. We use a small-step operational semantics in the form of a single-step rewriting system that defines a (non-deterministic) normal-order reduction. This strategy can be made fair by adding resources for book-keeping. As equational theory, we use contextual equivalence (that is, terms are equal if, when plugged into any program context, their termination behaviour is the same), in which we use a combination of may- and must-convergence, which is appropriate for non-deterministic computations. We show that we can drop the fairness condition for equational reasoning, since the valid equations with respect to normal-order reduction are the same as for fair normal-order reduction. We develop a number of proof tools for proving correctness of program transformations. In particular, we prove a context lemma for both may- and must- convergence that restricts the number of contexts that need to be examined for proving contextual equivalence. Combining this with so-called complete sets of commuting and forking diagrams, we show that all the deterministic reduction rules and some additional transformations preserve contextual equivalence. We also prove a standardisation theorem for fair normal-order reduction. The structure of the ordering ≤c is also analysed, and we show that Ω is not a least element and ≤c already implies contextual equivalence with respect to may-convergence.

scholarly journals A Presheaf Semantics of Value-Passing Processes

1996 ◽  
Vol 3 (44) ◽  
Author(s):  
Glynn Winskel
Keyword(s):  
Operational Semantics ◽  
Denotational Semantics ◽  
Domain Theory ◽  
Process Calculi ◽  
Traditional Definition ◽  
Open Maps

This paper investigates presheaf models for process calculi with<br />value passing. Denotational semantics in presheaf models are shown<br />to correspond to operational semantics in that bisimulation obtained<br />from open maps is proved to coincide with bisimulation as defined<br />traditionally from the operational semantics. Both "early" and "late"<br />semantics are considered, though the more interesting "late" semantics<br />is emphasised. A presheaf model and denotational semantics is proposed<br />for a language allowing process passing, though there remains<br />the problem of relating the notion of bisimulation obtained from open<br />maps to a more traditional definition from the operational semantics.<br />A tentative beginning is made of a "domain theory" supporting<br />presheaf models.

Mechanizing proofs with logical relations – Kripke-style

2018 ◽  
Vol 28 (9) ◽  
pp. 1606-1638 ◽  
Author(s):  
ANDREW CAVE ◽  
BRIGITTE PIENTKA
Keyword(s):  
Case Studies ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Equational Theory ◽  
Higher Order ◽  
Abstract Syntax ◽  
Completeness Proof ◽  
Typed Lambda Calculus ◽  
Contextual Equivalence ◽  
The Right

Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe two case studies using the proof environmentBeluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms. The development of these proofs inBelugarelies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using higher order abstract syntax (HOAS) thereby avoiding the need to manipulate and deal with binders, renaming and substitutions, (2) we take advantage ofBeluga's support for representing derivations that depend on assumptions and first-class contexts to directly state inductive properties such as logical relations and inductive proofs, (3) we exploitBeluga's rich equational theory for simultaneous substitutions; as a consequence, users do not need to establish and subsequently use substitution properties, and proofs are not cluttered with references to them. We believe these examples demonstrate thatBelugaprovides the right level of abstractions and primitives to mechanize challenging proofs using HOAS encodings. It also may serve as a valuable benchmark for other proof environments.

scholarly journals HOPLA--A Higher-Order Process Language

2002 ◽  
Vol 9 (49) ◽  
Author(s):  
Mikkel Nygaard ◽  
Glynn Winskel
Keyword(s):  
Operational Semantics ◽  
Lambda Calculus ◽  
Expressive Power ◽  
Higher Order ◽  
Domain Theory ◽  
Order Process ◽  
Encode Process ◽  
Mobile Ambients ◽  
Computation Path ◽  
Congruence Properties

A small but powerful language for higher-order nondeterministic processes is introduced. Its roots in a linear domain theory for concurrency are sketched though for the most part it lends itself to a more operational account. The language can be viewed as an extension of the lambda calculus with a ``prefixed sum'', in which types express the form of computation path of which a process is capable. Its operational semantics, bisimulation, congruence properties and expressive power are explored; in particular, it is shown how it can directly encode process languages such as CCS, CCS with process passing, and mobile ambients with public names.

scholarly journals New-HOPLA--A Higher-Order Process Language with Name Generation

2004 ◽  
Vol 11 (21) ◽  
Author(s):  
Glynn Winskel ◽  
Francesco Zappa Nardelli
Keyword(s):  
Operational Semantics ◽  
Expressive Power ◽  
Higher Order ◽  
Domain Theory ◽  
Process Algebras ◽  
Order Process ◽  
Pi Calculus ◽  
Mobile Ambients ◽  
Congruence Properties

This paper introduces new-HOPLA, a concise but powerful language for higher-order nondeterministic processes with name generation. Its origins as a metalanguage for domain theory are sketched but for the most part the paper concentrates on its operational semantics. The language is typed, the type of a process describing the shape of the computation paths it can perform. Its transition semantics, bisimulation, congruence properties and expressive power are explored. Encodings are given of well-known process algebras, including pi-calculus, Higher-Order pi-calculus and Mobile Ambients.

scholarly journals Compilation and equivalence of imperative objects

1999 ◽  
Vol 9 (4) ◽  
pp. 373-426 ◽  
Author(s):  
ANDREW D. GORDON ◽  
PAUL D. HANKIN ◽  
SØREN B. LASSEN
Keyword(s):  
Operational Semantics ◽  
Direct Proof ◽  
Object Oriented ◽  
Small Step ◽  
Abstract Machine ◽  
Program Equivalence ◽  
Contextual Equivalence ◽  
Object Oriented Languages

We adopt the untyped imperative object calculus of Abadi and Cardelli as a minimal setting in which to study problems of compilation and program equivalence that arise when compiling object-oriented languages. We present both a big-step and a small-step substitution-based operational semantics for the calculus. Our first two results are theorems asserting the equivalence of our substitution-based semantics with a closure-based semantics like that given by Abadi and Cardelli. Our third result is a direct proof of the correctness of compilation to a stack-based abstract machine via a small-step decompilation algorithm. Our fourth result is that contextual equivalence of objects coincides with a form of Mason and Talcott's CIU equivalence; the latter provides a tractable means of establishing operational equivalences. Finally, we prove correct an algorithm, used in our prototype compiler, for statically resolving method offsets. This is the first study of correctness of an object-oriented abstract machine, and of operational equivalence for the imperative object calculus.

Contextual equivalence for inductive definitions with binders in higher order typed functional programming

2013 ◽  
Vol 23 (6) ◽  
pp. 658-700
Author(s):  
MATTHEW R. LAKIN ◽  
ANDREW M. PITTS
Keyword(s):  
Functional Programming ◽  
Operational Semantics ◽  
Higher Order ◽  
European Symposium ◽  
Proof Search ◽  
Inductive Definitions ◽  
University Of Cambridge ◽  
Meta Programming ◽  
Contextual Equivalence ◽  
Phd Thesis

AbstractCorrect handling of names and binders is an important issue in meta-programming. This paper presents an embedding of constraint logic programming into the αML functional programming language, which provides a provably correct means of implementing proof search computations over inductive definitions involving names and binders modulo α-equivalence. We show that the execution of proof search in the αML operational semantics is sound and complete with regard to the model-theoretic semantics of formulae, and develop a theory of contextual equivalence for the subclass of αML expressions which correspond to inductive definitions and formulae. In particular, we prove that αML expressions, which denote inductive definitions, are contextually equivalent precisely when those inductive definitions have the same model-theoretic semantics. This paper is a revised and extended version of the conference paper (Lakin, M. R. & Pitts, A. M. (2009) Resolving inductive definitions with binders in higher-order typed functional programming. InProceedings of the 18th European Symposium on Programming (ESOP 2009), Castagna, G. (ed), Lecture Notes in Computer Science, vol. 5502. Berlin, Germany: Springer-Verlag, pp. 47–61) and draws on material from the first author's PhD thesis (Lakin, M. R. (2010)An Executable Meta-Language for Inductive Definitions with Binders. University of Cambridge, UK).

scholarly journals Denotational semantics of recursive types in synthetic guarded domain theory

2018 ◽  
Vol 29 (3) ◽  
pp. 465-510 ◽  
Author(s):  
RASMUS E. MØGELBERG ◽  
MARCO PAVIOTTI
Keyword(s):  
Programming Languages ◽  
Type Theory ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Denotational Semantics ◽  
Domain Theory ◽  
Logical Relation ◽  
Dependent Type Theory ◽  
Semantics Of Programming Languages ◽  
Dependent Type

Just like any other branch of mathematics, denotational semantics of programming languages should be formalised in type theory, but adapting traditional domain theoretic semantics, as originally formulated in classical set theory to type theory has proven challenging. This paper is part of a project on formulating denotational semantics in type theories with guarded recursion. This should have the benefit of not only giving simpler semantics and proofs of properties such as adequacy, but also hopefully in the future to scale to languages with advanced features, such as general references, outside the reach of traditional domain theoretic techniques.Working inGuarded Dependent Type Theory(GDTT), we develop denotational semantics for Fixed Point Calculus (FPC), the simply typed lambda calculus extended with recursive types, modelling the recursive types of FPC using the guarded recursive types ofGDTT. We prove soundness and computational adequacy of the model inGDTTusing a logical relation between syntax and semantics constructed also using guarded recursive types. The denotational semantics is intensional in the sense that it counts the number of unfold-fold reductions needed to compute the value of a term, but we construct a relation relating the denotations of extensionally equal terms, i.e., pairs of terms that compute the same value in a different number of steps. Finally, we show how the denotational semantics of terms can be executed inside type theory and prove that executing the denotation of a boolean term computes the same value as the operational semantics of FPC.

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