Easiness in graph models

C. Berline, A. Salibra
2006 Theoretical Computer Science  
We generalize Baeten and Boerboom's method of forcing to show that there is a fixed sequence (u k ) k∈ of closed (untyped) -terms satisfying the following properties: (a) For any countable sequence (g k ) k∈ of Scott continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that ( x.xx)( x.xx)u k represents g k in the model. (b) For any countable sequence (t k ) k∈ of closed -terms there is a graph model that satisfies ( x.xx)( x.xx)u
more » ... = t k for all k. We apply these two results, which are corollaries of a unique theorem, to prove the existence of (1) a finitely axiomatized -theory L such that the interval lattice constituted by the -theories extending L is distributive; (2) a continuum of pairwise inconsistent graph theories ( = -theories that can be realized as theories of graph models); (3) a congruence distributive variety of combinatory algebras (lambda abstraction algebras, respectively). 5 their interest on a limited number of -theories, very little is known about the structure and equational theory of T (see [38, 45] ). Since syntactic techniques are usually difficult to use in the study of -theories, then semantical methods have been extensively investigated. Topology is at the center of the known approaches to giving models of the untyped lambda calculus; in particular, the first non-syntactic model was found by Scott in 1969 in the category of complete lattices and Scott continuous functions. After Scott, a large number of mathematical models for lambda calculus, arising from syntax-free constructions, have been introduced in various categories of domains and were classified into semantics according to the nature of their representable functions, see e.g. [1, 6, 14, 43] . Scott's continuous semantics [48] is given in the category whose objects are complete partial orders and morphisms are Scott continuous functions. The stable semantics introduced by Berry [15] and the strongly stable semantics introduced by Bucciarelli-Ehrhard [16] are a strengthening of the continuous semantics, introduced to capture the sequential features of lambda calculus. All these semantics are structurally and equationally rich in the sense that each of them is able to represent 2 distinct -theories [31, 33, 36] , where a semantics (or a class of models) represents a -theory T if it contains a model M whose equational theory is exactly T. Nevertheless, each of the above denotational semantics is equationally incomplete, in the sense that it is possible to produce -theories which are not represented in it. The problem of the equational incompleteness was positively solved by Honsell and Ronchi della Rocca [25] for the continuous semantics (who even produced a -theory induced by an operational semantics as a counter-example), by Bastonero and Gouy [24, 10, 11] for the stable semantics, and by Salibra [46, 47] for the strongly stable semantics. As for T , results on the structure of the set of -theories induced by a semantics are still rare, and there exist several longstanding very basic open questions (see [14] for a survey). In particular it is still open to know whether , the least -theory, could be the theory of a non-syntactic model in Scott's continuous semantics. In this paper we concentrate on the semantics G of lambda calculus given in terms of graph models, graph semantics for short. These models, isolated in the seventies by Plotkin et al. [37] within the continuous semantics, have proved useful for giving proofs of consistency of extensions of lambda calculus and for studying operational features of lambda calculus (see [14] ). For example, the simplest graph model, namely the Engeler and Plokin's model, has been used by Berline [14] to give concise proofs of the head-normalization theorem and of the left-normalization theorem of lambda calculus. Bucciarelli and Salibra [17, 18] have recently proved that the set GT , consisting of all the graph theories (= -theories that can be represented as theories of graph models), admits a least element, which is strictly greater than ; in particular cannot be the theory of a graph model. These authors have also proved in [18] results about the "smaller" class G s T of all sensible graph theories (a theory is sensible if all the unsolvable (or non-headnormalizable) terms are congruent). Smaller here only means that G s T is strictly included in GT , since from Kerth [32, 34] and David [21] it follows that G s T also contains 2 -theories (however, the result is much harder to prove than for GT ). Graph models are "webbed models" in the sense of [14] . Roughly speaking, a model of lambda calculus is a webbed model if it can be generated from a simpler structure, called its web. The web has a carrier set D and -terms are interpreted as (possibly special) subsets of D. The reasons to concentrate on G are the following. First, G is, by far, the simplest class of models, in the sense that the webs of graph models are the simplest existing webs. Second, GT nevertheless contains a continuum of elements [31] , so it is a rich class, in the sense that its cardinality is the maximal possible one, but it contains no extensional theories. Third, it is quite clear that the techniques and results for G and GT can often be transferred to other classes of webbed models, whether more general ones or belonging to other semantics. It is a well known result by Jacopini [27] that can be consistently equated to any closed term t of the (untyped) -calculus, where is the paradigmatic unsolvable term ( x.xx) x.xx (this is called the easiness of ). Baeten and Boerboom gave in [5] the first semantical proof of this result by showing that for all closed terms t one can build a graph model satisfying the equation = t. This semantical result extends to other classes of models and to some other terms which share with enough of its good will (cf. [14] for a survey of such results). We recall that a graph model is, by definition, a reflexive Scott domain, which is generated by a web of the form (D, p) , where D is an infinite set and p : D * × D → D is a total injection, D * being the set of finite subsets of D (see Section 2.2). For brevity, we shall confuse graph models and their webs, but one should keep present in mind that the underlying domain of the model (D, p) is the full powerset P(D) ordered by inclusion, which is therefore independent of p. Starting from the set D = N of natural numbers, Baeten and Boerboom build p by a method of "forcing", which, although much simpler than the forcing techniques used in set theory, is somewhat in the same spirit. In the Baeten
doi:10.1016/j.tcs.2005.11.005 fatcat:367kbdjdh5cerfzoybtblycmvi