There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces. We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame. This yields completeness theorems over X for a number of languages: (i) the modal mu-calculus with the closure operator ◇; (ii) ◇ and the tangled closure operators 〈t〉 (in fact 〈t〉 can express ◇); (iii) ◇,∀; (iv) ◇,∀,〈t〉; (v) the derivative operator 〈d〉; (vi) 〈d〉 and the associated tangled closure operators 〈dt〉; (vii) 〈d〉,∀; (viii) 〈d〉,∀,〈dt〉. Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with 〈d〉, X validates the well-known axiom G1. For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connected.

We consider the temporal language with the Priorean operators G and H expressing that a formula is true at all future times and all past times, plus an operator □ expressing that a formula is true throughout some open interval containing the evaluation time (i.e. it is true ‘around now’). We show that the logic of time based on the real numbers in this language is finitely axiomatizable, answering an implicit question of Shehtman (1993). We also show that the logic has PSPACE-complete complexity, but is not Kripke complete and has no strongly complete axiomatization.

We show that for finite n⩾3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.

Abstract We show that the variety of weakly representable relation algebras is neither canonical nor closed under Monk completions.

Abstract It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RAn and wRRA contains the other.

This paper is talking about temporal logics with semantics based on intervals of time. Interval operators allow wide choice of temporal operators which includes unary operators with semantics and binary operators are considered.

Abstract A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ 3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ, we construct an n-dimensional atom structure , and prove, for infinite Γ, that is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs Γk(k < ω) with infinite chromatic number, but having a non-principal ultraproduct ΠDΓk whose chromatic number is just two. It follows that is strongly representable (each k < ω) but is not.

Abstract We consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.

In this paper we prove decidability of two kinds of branching time temporal logics. First we show that the non-local version of propositional PCTL*, in which truth values of atoms may depend on the branch of evaluation, is decidable. Then we use this result to establish decidability of various fragments of quantified PCTL*, where the next-time operator can be applied only to formulas with at most one free variable, all other temporal operators and path quantifiers are applicable only to sentences, and the first-order constructs follow the pattern of any of several decidable fragments of first-order logic.

We study the complexity of some fragments of first-order temporal logic over natural numbers time. The one-variable fragment of linear first-order temporal logic even with sole temporal operator /spl square/ is EXPSPACE-complete (this solves an open problem of J. Halpern and M. Vardi (1989)). So are the one-variable, two-variable and monadic monodic fragments with Until and Since. If we add the operators O/sup n/, with n given in binary, the fragment becomes 2EXPSPACE-complete. The packed monodic fragment has the same complexity as its pure first-order part - 2EXPTIME-complete. Over any class of flows of time containing one with an infinite ascending sequence - e.g., rationals and real numbers time, and arbitrary strict linear orders - we obtain EXPSPACE lower bounds (which solves an open problem of M. Reynolds (1997)). Our results continue to hold if we restrict to models with finite first-order domains.

In this paper we analyze the decision problem for fragments of first-order extensions of branching time temporal logics such as computational tree logics CTL and CTL* or Prior's Ockhamist logic of historical necessity. On the one hand, we show that the one-variable fragments of logics like first-order CTL*-such as the product of propositional CTL* with simple propositional modal logic S5, or even the one-variable bundled first-order temporal logic with sole temporal operator 'some time in the future'-are undecidable. On the other hand, it is proved that by restricting applications of first-order quantifiers to state (i.e., path-independent) formulas, and applications of temporal operators and path quantifiers to formulas with at most one free variable, we can obtain decidable fragments. The positive decidability results can serve as a unifying framework for devising expressive and effective time-dependent knowledge representation formalisms, e.g., temporal description or spatio-temporal logics.

We prove decidability of 'non local' propositional CTL*, where truth values of atoms may depend on the branch of evaluation. This result is then used to show decidability of the 'weak' one-variable fragment of first-order (local) CTL*, in which all temporal operators and path quantifiers except 'tomorrow' are applicable only to sentences. Various spatio-temporal logics based on combinations of CTL* and RCC-8 can be embedded into this fragment, and so are decidable.

Abstract We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras.

Abstract We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.

Abstract Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the ‘finite base property’ and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

Abstract We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method.

Abstract A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

A Hilbert style axiomatization of the temporal logic with connectives Until and Since for the real numbers is presented. We prove independence of the axioms, and completeness for this semantics with respect to single formulas.

Abstract A countably infinite relational structure M is called absolutely ubiquitous if the following holds: whenever N is a countably infinite structure, and M and N have the same isomorphism types of finite induced substructures, there is an isomorphism from M to N. Here a characterisation is given of absolutely ubiquitous structures over languages with finitely many relation symbols. A corresponding result is proved for uncountable structures.