On varieties of cylindric algebras with applications to logic

AbstractMnα, Mgα, and Bgα denote the classes of minimal, monadic-generated, and binary-generated cylindric algebras of dimension α respectively, and EqK denotes the equational theory of the class K of algebras. In Theorem 2, we characterize those classes K ⊆ Mgα, α > 2, for which EqK is recursively enumerable (r.e.). As a corollary we obtain that EqMnα is not r.e. iff α ⩾ ω, EqMgα is not r.e. iff α > 2, EqBgα is r.e. for α ⩾ ω and EqMnα = EqMgα iff (α = 0 or α ⩾ ω). These results solve Problems 4.11, 4.12 and the problem in item (1) on p. (ii) of the introduction of Part II of Henkin-Monk-Tarski [11] and continue the investigations started in Monk [22]. We discuss at length the logical meaning and consequences in the introduction and in Section 2. The proofs of the above results rely on the fact that the set of satisfiable Diophantine equations is not decidable. We also show that the equational theory of monadic-generated relation algebras is not r.e. Some further results can be found in Theorems 5 and 6: in Theorem 5 we give a single equation that characterizes being of characteristic 0 in Mgω, in Theorem 6 we investigate how big Mgα is. We do some investigations concerning the lattice of varieties of CAα's, α ⩾ ω

On Tarski's axiomatic foundations of the calculus of relations

It is shown that Tarski's set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski's axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski's other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski's axiom system

Using Isabelle/HOL to verify first-order relativity theory

Logicians at the Rényi Mathematical Institute in Budapest have spent several years developing versions of relativity theory (special, general, and other variants) based wholly on first-order logic, and have argued in favour of the physical decidability, via exploitation of cosmological phenomena, of formally unsolvable questions such as the Halting Problem and the consistency of set theory. As part of a joint project, researchers at Sheffield have recently started generating rigorous machine-verified versions of the Hungarian proofs, so as to demonstrate the soundness of their work. In this paper, we explain the background to the project and demonstrate a first-order proof in Isabelle/HOL of the theorem “no inertial observer can travel faster than light”. This approach to physical theories and physical computability has several pay-offs, because the precision with which physical theories need to be formalised within automated proof systems forces us to recognise subtly hidden assumptions

A unifying theorem for algebraic semantics and dynamic logics

AbstractA unified single proof is given which implies theorems in such diverse fields as continuous algebras of algebraic semantics, dynamic algebras of logics of programs, and program verification methods for total correctness. The proof concerns ultraproducts and diagonalization

Varieties generated by completions

We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the completions of representable relation algebras does not contain all relation algebras. This answers Problem 1.1(1) from Maddux's 2018 Algebra Universalis paper in the negative. At the same time, we confirm the suggestion in that paper that the finite maximal relation algebras constructed in M. Frias and R. Maddux's 1997 Algebra Universalis paper are not in the variety generated by the completions of representable relation algebras. We prove that there are continuum many varieties between the variety generated by the completions of representable relation algebras and the variety of relation algebras

Testing definitional equivalence of theories via automorphism groups

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, Glymour and Halvorson