Polyadic Algebras

This chapter surveys the development in the theory of polyadic algebras in the last decades

Axiomatizability of reducts of algebras of relations

Submitted versio

On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results

Fix a finite ordinal $n\geq 3$ and let $\alpha$ be an arbitrary ordinal. Let $\mathsf{CA}_n$ denote the class of cylindric algebras of dimension $n$ and $\sf RA$ denote the class of relation algebras. Let $\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})$ stand for the class of polyadic (equality) algebras of dimension $\alpha$. We reprove that the class $\mathsf{CRCA}_n$ of completely representable $\mathsf{CA}_n$s, and the class $\sf CRRA$ of completely representable $\mathsf{RA}$s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety $\sf V$ between polyadic algebras of dimension $n$ and diagonal free $\mathsf{CA}_n$s. We show that that the class of completely and strongly representable algebras in $\sf V$ is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class $\sf CRRA$ is not closed under $\equiv_{\infty,\omega}$. In contrast, we show that given $\alpha\geq \omega$, and an atomic $\mathfrak{A}\in \mathsf{PEA}_{\alpha}$, then for any \(n/p

Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems

Algebras of partial functions

This thesis collects together four sets of results, produced by investigating modifications, in four distinct directions, of the following. Some set-theoretic operations on partial functions are chosen—composition and intersection are examples—and the class of algebras isomorphic to a collection of partial functions, equipped with those operations, is studied. Typical questions asked are whether the class is axiomatisable, or indeed finitely axiomatisable, in any fragment of first-order logic, what computational complexity classes its equational/quasiequational/first-order theories lie in, and whether it is decidable if a finite algebra is in the class. The first modification to the basic picture asks that the isomorphisms turn any existing suprema into unions and/or infima into intersections, and examines the class so obtained. For composition, intersection, and antidomain together, we show that the suprema and infima conditions are equivalent. We show the resulting class is axiomatisable by a universal-existential-universal sentence, but not axiomatisable by any existential-universal-existential theory. The second contribution concerns what happens when we demand partial functions on some finite base set. The finite representation property is essentially the assertion that this restriction that the base set be finite does not restrict the algebras themselves. For composition, intersection, domain, and range, plus many supersignatures, we prove the finite representation property. It follows that it is decidable whether a finite algebra is a member of the relevant class. The third set of results generalises from unary to ‘multiplace’ functions. For the signatures investigated, finite equational or quasiequational axiomatisations are obtained; similarly when the functions are constrained to be injective. The finite representation property follows. The equational theories are shown to be coNP-complete. In the last section we consider operations that may only be partial. For most signatures the relevant class is found to be recursively, but not finitely, axiomatisable. For others, finite axiomatisations are provided