21 research outputs found
Lambda theories of effective lambda models
A longstanding open problem is whether there exists a non-syntactical model
of untyped lambda-calculus whose theory is exactly the least equational
lambda-theory (=Lb). In this paper we make use of the Visser topology for
investigating the more general question of whether the equational (resp. order)
theory of a non syntactical model M, say Eq(M) (resp. Ord(M)) can be
recursively enumerable (= r.e. below). We conjecture that no such model exists
and prove the conjecture for several large classes of models. In particular we
introduce a notion of effective lambda-model and show that for all effective
models M, Eq(M) is different from Lb, and Ord(M) is not r.e. If moreover M
belongs to the stable or strongly stable semantics, then Eq(M) is not r.e.
Concerning Scott's continuous semantics we explore the class of (all) graph
models, show that it satisfies Lowenheim Skolem theorem, that there exists a
minimum order/equational graph theory, and that both are the order/equ theories
of an effective graph model. We deduce that no graph model can have an r.e.
order theory, and also show that for some large subclasses, the same is true
for Eq(M).Comment: 15 pages, accepted CSL'0
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus 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 l-beta or l-beta-eta. 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 minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34
Graph Lambda Theories
to appear in MSCSInternational audienceA longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly lambda-beta or the the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question, raised recently by C. Berline, is whether, given a class of lambda models, there are a minimal lambda-theory and a minimal sensible lambda-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-Engeler. In particular, we build two graph models whose theories are respectively the set of equations satisfied in any graph model and in any sensible graph model. We conjecture that the least sensible graph theory, where ''graph theory" means ''lambda-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 beta-theory. Another related question is whether, given a class of lambda models, there is a maximal sensible lambdatheory represented by it. In the main result of the paper we characterize the greatest sensible graph theory as the lambda-theory B generated by equating lambda-terms with the same Boehm tree. This result is a consequence of the main technical theorem of the paper: all the equations between solvable lambda-terms, which have different Boehm 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
On Noncommutative Generalisations of Boolean Algebras
Skew Boolean algebras (SBA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed SBAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use SBAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed SBAs in terms of nBAs of n-partitions
Exploring New Topologies for the Theory of Clones
Clones of operations of arity (referred to as -operations)
have been employed by Neumann to represent varieties of infinitary algebras
defined by operations of at most arity . More recently, clone algebras
have been introduced to study clones of functions, including
-operations, within the framework of one-sorted universal algebra.
Additionally, polymorphisms of arity , which are -operations
preserving the relations of a given first-order structure, have recently been
used to establish model theory results with applications in the field of
complexity of CSP problems.
In this paper, we undertake a topological and algebraic study of
polymorphisms of arity and their corresponding invariant relations.
Given a Boolean ideal on the set , we propose a method to endow
the set of -operations on with a topology, which we refer to as
-topology. Notably, the topology of pointwise convergence can be retrieved
as a special case of this approach. Polymorphisms and invariant relations are
then defined parametrically, with respect to the -topology. We characterise
the -closed clones of -operations in terms of
- and present a method to relate
- to the classical (finitary) -
Sequent calculi of finite dimension
In recent work, the authors introduced the notion of n-dimensional Boolean
algebra and the corresponding propositional logic nCL. In this paper, we
introduce a sequent calculus for nCL and we show its soundness and
completeness.Comment: arXiv admin note: text overlap with arXiv:1806.0653
Classical logic with n truth values as a symmetric many-valued logic
We introduce Boolean-like algebras of dimension n (nBA s) having n constants e1, … , en, and an (n+ 1) -ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The nBA s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, nCL , and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in nCL , and, via the embeddings, in all the finite tabular logics
Factor Varieties
The universal algebraic literature is rife with generalisations of discriminator varieties, whereby several investigators have tried to preserve in more general settings as much as possible of their structure theory. Here, we modify the definition of discriminator algebra by having the switching function project onto its third coordinate in case the ordered pair of its first two coordinates belongs
to a designated relation (not necessarily the diagonal relation). We call these algebras factor algebras and the varieties they generate factor varieties. Among other things, we provide an equational description of these varieties and match equational conditions involving the factor term with properties of the associated factor relation. Factor varieties include, apart from discriminator varieties, several varieties of algebras from quantum and fuzzy logics