Unsolvable Terms Research Articles

The theory of call-by-value solvability

The semantics of the untyped (call-by-name) lambda-calculus is a well developed field built around the concept of solvable terms, which are elegantly characterized in many different ways. In particular, unsolvable terms provide a consistent notion of meaningless term. The semantics of the untyped call-by-value lambda-calculus (CbV) is instead still in its infancy, because of some inherent difficulties but also because CbV solvable terms are less studied and understood than in call-by-name. On the one hand, we show that a carefully crafted presentation of CbV allows us to recover many of the properties that solvability has in call-by-name, in particular qualitative and quantitative characterizations via multi types. On the other hand, we stress that, in CbV, solvability plays a different role: identifying unsolvable terms as meaningless induces an inconsistent theory.

No solvable lambda-value term left behind

In the lambda calculus a term is solvable iff it is operationally relevant. Solvable terms are a superset of the terms that convert to a final result called normal form. Unsolvable terms are operationally irrelevant and can be equated without loss of consistency. There is a definition of solvability for the lambda-value calculus, called v-solvability, but it is not synonymous with operational relevance, some lambda-value normal forms are unsolvable, and unsolvables cannot be consistently equated. We provide a definition of solvability for the lambda-value calculus that does capture operational relevance and such that a consistent proof-theory can be constructed where unsolvables are equated attending to the number of arguments they take (their "order" in the jargon). The intuition is that in lambda-value the different sequentialisations of a computation can be distinguished operationally. We prove a version of the Genericity Lemma stating that unsolvable terms are generic and can be replaced by arbitrary terms of equal or greater order.

Graph easy sets of mute lambda terms

Among the unsolvable terms of the lambda calculus, the mute ones are those having the highest degree of undefinedness. In this paper, we define for each natural number n, an infinite and recursive set Mn of mute terms, and show that it is graph-easy: for any closed term t of the lambda calculus there exists a graph model equating all the terms of Mn to t. Alongside, we provide a brief survey of the notion of undefinedness in the lambda calculus.

Minimal lambda-theories by ultraproducts

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the least lambda-theory lambda-beta or the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there is a minimal lambda-theory represented by it. In this paper, we give a general tool to answer positively to this question and we apply it to a wide class of webbed models: the i-models. The method then applies also to graph models, Krivine models, coherent models and filter models. In particular, we build an i-model whose theory is the set of equations satisfied in all i-models.

Solution to the Range Problem for Combinatory Logic

The λ-theory H is obtained from β-conversion by identifying all closed unsolvable terms (or, equivalently, terms without head normal form). The range problem for the theory H asks whether a closed term has always (up to equality in H) either an infinite range or a singleton range (that is, it is a constant function). Here we give a solution to a natural version of this problem, giving a positive answer for the theory H restricted to Combinatory Logic. The method of proof applies also to the Lazy λ-Calculus.

Graph lambda theories

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the λβ or the least sensible λ-theory ℋ (which is generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there are a minimal λ-theory and a minimal sensible λ-theory represented by it. In this paper, we give a positive answer to this question for the class of graph models à la Plotkin, Scott and Engeler. In particular, we build two graph models whose theories are the set of equations satisfied in, respectively, any graph model and any sensible graph model. We conjecture that the least sensible graph theory, where ‘graph theory’ means ‘λ-theory of a graph model’, is equal to ℋ, while in one of the main results of the paper we show the non-existence of a graph model whose equational theory is exactly the λβ theory.Another related question is whether, given a class of lambda models, there is a maximal sensible λ-theory represented by it. In the main result of the paper, we characterise the greatest sensible graph theory as the λ-theory ℬ generated by equating λ-terms with the same Böhm tree. This result is a consequence of the main technical theorem of the paper, which says that all the equations between solvable λ-terms that have different Böhm trees fail in every sensible graph model. A further result of the paper is the existence of a continuum of different sensible graph theories strictly included in ℬ.

Towards the range property for the lambda theory [formula omitted

A sketch of the proof is given for an open problem, the range property for H : the range of a closed λ -term in the closed term model modulo β -conversion and equating unsolvable terms is either a singleton or infinite. The proof depends on one unresolved technical conjecture.

A domain-theoretic semantics of lax generic functions

The semantic structure of a polymorphic calculus λ m is studied. λ m is defined over a hierarchical type structure, and a function in this calculus, called a generic function, can be composed from more than one lambda expression and the ways it behaves on each type are weakly related in that it lax commutes with the coercion functions defined from the subtypes to the supertypes. Since laxness is intermediate between ad hocness (behaviors on each type are not related) and coherency (commuting with the coercion functions), λ m has syntactic properties lying between those of calculi with ad hoc generic functions and coherent generic functions studied in Tsuiki (Math. Struct. Comput. Sci. 8 (1998) 321). That is, although λ m allows self application and thus is not normalizing, it does not have any unsolvable terms. For this reason, all the semantic domains are connected by mutually recursive equations and, at the same time, they do not have the least elements.We solve them by considering fibrations and expressing the equations as a recursive equation about fibrations. We also show the adequacy theorem for λ m following the construction of Pitts and use it to derive some syntactic properties.

Full Intersection Types and Topologies in Lambda Calculus

Topologies are introduced on the set of lambda terms by their typeability in the full intersection type assignment system. These topologies give rise to simple proofs of some fundamental results of the lambda calculus such as the continuity theorem and the genericity lemma. We show that application is continuous, unsolvable terms are bottoms, and normal forms are isolated points with respect to these topologies. The restriction of all these topologies to the set of closed lambda terms appears to be unique. We compare the introduced topology with the filter topology on the set of (closed) lambda terms and show that they coincide.

Infinite λ-calculus and types

Recent work on infinitary versions of the lambda calculus has shown that the infinite lambda calculus can be a useful tool to study the unsolvable terms of the classical lambda calculus. Working in the framework of the intersection type disciplines, we devise a type assignment system such that two terms are equal in the infinite lambda calculus iff they can be assigned the same types in any basis. A novel feature of the system is the presence of a type constant to denote the set of all terms of order zero, and the possibility of applying a type to another type. We prove a completeness and an approximation theorem for our system. Our results can be considered as a first step towards the goal of giving a denotational semantics for the lambda calculus which is suited for the study of the unsolvable terms. However, some noncontinuity phenomena of the infinite lambda calculus make a full realization of this idea (namely the construction of a filter model) a quite difficult task.

Every countable poset is embeddable in the poset of unsolvable terms

In this paper we shall show that the poset of unsolvable terms is universal.