RPO, Second-order Contexts, and Lambda-calculus

📝 Abstract

First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda-calculus in Combinatory Logic.

💡 Analysis

First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda-calculus in Combinatory Logic.

📄 Content

Logical Methods in Computer Science Vol. 5 (3:6) 2009, pp. 1–35 www.lmcs-online.org Submitted Aug. 13, 2008 Published Aug. 6, 2009 RPO, SECOND-ORDER CONTEXTS, AND λ-CALCULUS ∗ PIETRO DI GIANANTONIO, FURIO HONSELL, AND MARINA LENISA Dip. di Matematica e Informatica, Universit`a di Udine, via delle Scienze 206, 33100 Udine, Italy e-mail address: {digianantonio,honsell,lenisa}@dimi.uniud.it Abstract. First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labeled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda- calculus in Combinatory Logic.

1. Introduction Recently, much attention has been devoted to derive labeled transition systems and bisimilarity congruences from reactive systems, in the context of process languages and graph rewriting, [Sew02, LM00, SS03, GM05, BGK06, BKM06, EK06]. In the theory of process algebras, the operational semantics of CCS was originally given via a labeled transi- tion system (lts), while more recent process calculi have been presented via reactive systems plus structural rules. Reactive systems naturally induce behavioral equivalences which are congruences w.r.t. contexts, while lts’s naturally induce bisimilarity equivalences with coin- ductive characterizations. However, such equivalences are not congruences in general, or else it is an heavy, ad-hoc task to prove that they are congruences. 1998 ACM Subject Classification: F.3.2, F.4.1. Key words and phrases: λ-calculus, reactive system, labeled transition system, weak bisimilarity, RPO technique. ∗Work supported by ART PRIN Project prot. 2005015824 and by FIRB Project RBIN04M8S8 (both funded by MIUR). LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.2168/LMCS-5 (3:6) 2009 c ⃝ P. Di Gianantonio, F. Honsell, and M. Lenisa CC ⃝ Creative Commons 2 P. DI GIANANTONIO, F. HONSELL, AND M. LENISA Generalizing [Sew02], Leifer and Milner [LM00] presented a general categorical method for deriving a transition system from a reactive system, in such a way that the induced bisimilarity is a congruence. The labels in Leifer-Milner’s transition system are those con- texts which are minimal for a given reaction to fire. Minimal contexts are identified via the categorical notion of relative pushout (RPO). Leifer-Milner’s central result guarantees that, under a suitable categorical condition, the induced bisimilarity is a congruence w.r.t. all contexts. In the literature, some case studies have been carried out, especially in the setting of process calculi, for testing the expressivity of Leifer-Milner’s approach. Some difficulties have arisen in applying the approach directly to such languages, viewed as Lawvere theo- ries, because of structural rules. To overcome this problem, two different approaches have been considered. The first approach consists in using more complex categorical construc- tions, where structural rules are accounted for explicitly, [Lei01, SS03, SS05]. In the second approach, intermediate encodings have been considered in graph theory, for which the ap- proach of “borrowed contexts” has been developed [EK06], and in Milner’s bigraph theory. Here structural rules are avoided, since structurally equivalent terms are equated in the target language. Moreover, the following further issues have arisen in applying Leifer-Milner’s technique. (i) Leifer-Milner’s bisimilarity is still redundant, and many labels have to be eliminated a posteriori, by an ad-hoc reasoning. Thus general results are called for, in order to reduce the complexity of the bisimilarity a priori. (ii) In some cases it is useful to consider weak variants of Leifer-Milner technique. However, for the weak bisimilarity we only have a partial congruence result, stating that such bisimilarity is a congruence w.r.t. a certain class of contexts. However, in many concrete cases, the weak bisimilarity t

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