Term Graph Representations for Cyclic Lambda-Terms

Abstract

We study various representations for cyclic lambda-terms as higher-order or as first-order term graphs. We focus on the relation between 'lambda-higher-order term graphs' (lambda-ho-term-graphs), which are first-order term graphs endowed with a well-behaved scope function, and their representations as 'lambda-term-graphs', which are plain first-order term graphs with scope-delimiter vertices that meet certain scoping requirements. Specifically we tackle the question: Which class of first-order term graphs admits a faithful embedding of lambda-ho-term-graphs in the sense that (i) the homomorphism-based sharing-order on lambda-ho-term-graphs is preserved and reflected, and (ii) the image of the embedding corresponds closely to a natural class (of lambda-term-graphs) that is closed under homomorphism? We systematically examine whether a number of classes of lambda-term-graphs have this property, and we find a particular class of lambda-term-graphs that satisfies this criterion. Term graphs of this class are built from application, abstraction, variable, and scope-delimiter vertices, and have the characteristic feature that the latter two kinds of vertices have back-links to the corresponding abstraction. This result puts a handle on the concept of subterm sharing for higher-order term graphs, both theoretically and algorithmically: We obtain an easily implementable method for obtaining the maximally shared form of lambda-ho-term-graphs. Furthermore, we open up the possibility to pull back properties from first-order term graphs to lambda-ho-term-graphs, properties such as the complete lattice structure of bisimulation equivalence classes with respect to the sharing order

Maximal sharing in the Lambda calculus with letrec

“…Whereas in the paper [12] we exclusively focused on the graph formalisms, and investigated them in their own right, here we connect the results obtained there to the language λ letrec for expressing sharing and cyclicity. Since the methods presented here are based on the graph formalisms, and rely on their properties for correctness, we recapitulate the concepts and the relevant results in Sec.…”

“…Alternatively scopes can be represented by a scope function (as in [4]) that assigns to every abstraction vertex the set of vertices in its scope. In the article [12] we show that higherorder term graphs with scope functions correspond bijectively to those with abstraction prefix functions.…”

“…The converse is not the case, yet it holds up to S-sharing by item (ii). For the proof, we refer to our article [12].…”

“…As a stepping stone for the methods we develop here, we use concepts and results that we described in an earlier paper [12]. There, for interpreting λ letrec -terms, a modification of Blom's higher-order term graphs (the λ-ho-term-graphsof the class H) in which scopes are represented by means of 'abstraction prefix functions'.…”

Maximal sharing in the Lambda calculus with letrec

“…For obtaining T H(G) for a G ∈ T , retain the abstractionprefix function, and remove every delimiter vertex from G, thereby connecting its incoming edge with its outgoing edge. For formal definitions and well-definedness of T H and HT , see [12]. Note that a λ-ho-term-graph may have multiple corresponding λ-term-graphs that differ only with respect to their 'degree' of S-sharing (the extent to which S-vertices occur shared).…”