STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES

The Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.

What is a categorical model of the differential and the resource λ-calculi?

The differential λ-calculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows the application of a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows us to write the full Taylor expansion of a program. Through this expansion, every program can be decomposed into an infinite sum (representing non-deterministic choice) of ‘simpler’ programs that are strictly linear.The aim of this paper is to develop an abstract ‘model theory’ for the untyped differential λ-calculus. In particular, we investigate what form a general categorical definition of a denotational model for this calculus should take. Starting from the work of Blute, Cockett and Seely on differential categories, we develop the notion of a Cartesian closed differential category and prove that linear reflexive objects living in such categories constitute sound and complete models of the untyped differential λ-calculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This requires that every model living in such categories equates all programs having the same full Taylor expansion.We then provide a concrete example of a Cartesian closed differential category modelling the Taylor expansion, namely the category MRel of sets and relations from finite multisets to sets. We prove that the extensional model of λ-calculus we have recently built in MRel is linear, and is thus also an extensional model of the untyped differential λ-calculus. In the same category, we build a non-extensional model and prove that it is, nevertheless, extensional on its differential part.Finally, we study the relationship between the differential λ-calculus and the resource calculus, which is a functional programming language combining the ideas behind the differential λ-calculus with those behind Boudol's λ-calculus with multiplicities. We define two translation maps between these two calculi and study the properties of these translations. In particular, this analysis shows that the two calculi share the same notion of a model, and thus that the resource calculus can be interpreted by translation into every linear reflexive object living in a Cartesian closed differential category.

Abstract Beth definability in institutions

This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. We generalise the concept of definability to arbitrary logics, formalised as institutions, and we develop three general definability results. One generalises the classical Beth theorem by relying on the interpolation properties of the institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for many new actual definability results, including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for ‘borrowing’ definability properties from another institution via an ‘adequate’ encoding between institutions.The power of our general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-)varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalised as institutions from computing science and logic.

Barwise: Abstract Model Theory and Generalized Quantifiers

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties areκ-compactness. Any set of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model. Löwenheim-Skolem Theorem down to κ. If a sentence of the logic has a model, it has a model of cardinality at most κ. Interpolation Property. If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languages Lκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory.