Abstract not found

We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames. The constructions use the result of Erd os that there are finite graphs with arbitrarily large chromatic number and girth.

Abstract not found

The area of intersection between temporal logic, automata on finite and infinite objects, and classical first- or restricted second-order logics is one of considerable richness. All these areas have long and venerable traditions in mathematics, logic, and theoretical computer science, and their intimate relationships have been realised for quite some time. Indeed, automata of infinite objects were invented for the purpose of just answering decidability issues in classical first- or restricted second-order logics. However, the area has remained open to new points-of-view and insights, and very fundamental questions have yet to be both asked and answered. The point of departure here is that of programs, the computations that programs give rise

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems.

The Kuratowski Closure-Complement Theorem 1.1. [29] If (X, T) is a topological space and A ⊆ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained.

This is the first part of the whole work which will consist of two parts and intends to obtain a clear picture of the lattice (Df 2 ) of all subvarieties of the variety Df 2 of the two-dimensional diagonal-free cylindric algebras. Here we show that every proper subvariety of Df 2 is locally finite, and hence Df 2 is hereditarily finitely approximable. Moreover, we prove that there exist exactly six critical varieties in (Df 2 ), and characterize finite subvarieties of Df 2 . It is also shown that a variety V 2 (Df 2 ) is representable by its square algebras i either V = Df 2 or V is a finite variety, and give a necessary and sufficient condition for a finite variety to be representable. Representable varieties by their rectangular algebras are also described. The complexity of (Df 2 ) will be investigated in Part II.

Algebraic Logic. In preparation. Manuscript.

Some systems of modal logic, such as S5, which are often used as epistemic logics with the ‘necessity ’ operator read as ‘the agent knows that’, are problematic as general epistemic logics for agents whose computational capacity does not exceed that of a Turing machine because they impose unwarranted constraints on the agent’s theory of non-epistemic aspects of the world, for example by requiring the theory to be decidable rather than merely recursively axiomatizable. To generalize this idea, two constraints on an epistemic logic are formulated: r.e. conservativeness, that any recursively enumerable theory R in the sublanguage without the epistemic operator is conservatively extended by some recursively enumerable theory in the language with the epistemic operator which is permitted by the logic to be the agent’s overall theory; the weaker requirement of r.e. quasi-conservativeness is similar except for applying only when R is consistent. The logic S5 is not even r.e. quasiconservative; this result is generalized to many other modal logics. However, it is also proved that the modal logics S4, Grz and KDE are r.e. quasi-conservative and that K4, KE and the provability logic GLS are r.e. conservative. Finally, r.e. conservativeness and r.e. quasiconservativeness are compared with related non-computational constraints.