Sensible λ-theories are equational extensions of the untyped lambda calculus that equate all the unsolvable λ-terms and are closed under derivation. A longstanding open problem in lambda calculus is whether there exists a nonsyntactic model whose equational theory is the least sensible λ-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of models, there exist a minimal and maximal sensible λ-theory represented by it. In this paper we give a positive answer to this question for the semantics of lambda calculus given in terms of graph models. We conjecture that the least sensible graph theory, where “graph theory ” means “λ-theory of a graph model”, is equal to H, while in the main result of the paper we characterize 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 fact that all the equations between solvable λ-terms, which have different Böhm trees, fail in every sensible graph model. Further results of the paper are: (i) the existence of a continuum of different sensible graph theories strictly included in B (this result positively answers Question 2 in [7, Section 6.3]); (ii) the non-existence of a graph model whose equational theory is exactly the minimal lambda theory λβ (this result negatively answers Question 1 in [7, Section 6.2] for the restricted class of graph models).

A longstanding open problem in lambda-calculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambda-calculus whose theory is exactly the beta-theory or the beta-eta-theory. A related question, raised recently by C.Berline, is whether, given a class of lambda-models, there is a minimal equational theory represented by it.

In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory T, there is a continuum of lambda theories including T which are omitted by the indecomposable semantics. 1

Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λ-calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim-Skolem theorem.

Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriting has also

In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λ-theories, the equational incompleteness of lambda calculus semantics, and the λ-theories induced by graph models of lambda calculus.

A longstanding open problem in lambda calculus is whether there exists a non-syntactical model of the untyped lambda calculus whose theory is exactly the least λ-theory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a model can be recursively enumerable (r.e. for brevity). We introduce the notion of an effective model of lambda calculus and prove the following results: (i) The equational theory of an effective model cannot be λβ, λβη; (ii) The order theory of an effective model cannot be r.e.; (iii) No effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove the following, where “graph theory ” is a shortcut for “theory of a graph model”: (iv) There exists a minimum order graph theory (for equational graph theories this was proved in [9, 10]). (v) The minimum equational/order graph theory is the theory of an effective graph model. (vi) No order graph theory can be r.e. (vii) Every equational/order graph theory is the theory of a graph model having a countable web. This last result proves that the class of graph models enjoys a kind of (downwards) Löwenheim-Skolem theorem, and it answers positively Question 3 in [4, Section 6.3] for the class of graph models. 1.

Abstract. We generalize to universal algebra concepts originating from λ-calculus and programming to prove a new result on the lattice λT of λ-theories, and a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. In this paper we introduce the class of Church algebras (which includes all Boolean algebras, combinatory algebras, rings with unit and the term algebras of all λ-theories) to model the “if-then-else ” instruction of programming. The interest of Church algebras is that each has a Boolean algebra of central elements, which play the role of the idempotent elements in rings. Central elements are the key tool to represent any Church algebra as a weak Boolean product of indecomposable Church algebras and to prove the meta representation theorem mentioned above. We generalize the notion of easy λ-term of lambda calculus to prove that any Church algebra with an “easy set ” of cardinality n admits (at the top) a lattice interval of congruences isomorphic to the free Boolean algebra with n generators. This theorem has the following consequence for the lattice of λ-theories: for every recursively enumerable λ-theory φ and each n, there is a λ-theory φn ⊇ φ such that {ψ: ψ ⊇ φn} “is ” the Boolean lattice with 2 n elements.

Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as first-order logic, the lambda-calculus, or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitely-supported permutation action); previous work proved soundness and completeness. The HSP theorem characterises the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras, and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to so-called freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this paper we give the constructions which show that, after all, a ‘nominal ’ version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products, and an atoms-abstraction construction specific to nominal-style semantics. Keywords: universal algebra, equational logic, nominal algebra, HSP or Birkhoff’s theorem, nominal sets, nominal terms 1

A longstanding open problem is whether there exists a model of the untyped λ-calculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λ-theory λβ or the least extensional λ-theory λβη. In this paper we analyze the class of reflexive Scott domains, the models of λ-calculus living in the category of Scott domains (a full subcategory of Cpo). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the λβη-calculus, i.e., there are equations not in λβη which hold in all extensional reflexive Scott domains. (ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott’s information systems, called iweb-models. The class of iweb-models includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in Cpo. Based on a fine-grained study of an “effective” version of Scott’s information systems, we have shown that there are equations not in λβ (resp. λβη) which hold in all (extensional) iweb-models.