In one respect the present general completeness result differs from most of the others in the literature. Tests are given for determining whether a given axiom expresses an elementary condition and for determining what it is in case it does. It turns out that the compact logics are just those whose axioms express an elementary condition. It is ascertained which of the subframe logics are compact. As a consequence, every logic complete for a condition closed under subframes has the finite model property. It turns out that the subframe logics are exactly those complete for a condition that is closed under subframes (any subframe of a frame satisfying the condition also satisfies the condition). There are a continuum of subframe logics and they include many of the standard ones, such as T, S 4, S 4.3, S 5 and G. The general result is that each subframe logic has the finite model property. By a subframe logic we mean the result of adding such formulas as axioms to K 4.

With each finite transitive frame в„ we may associate a formula - B в„ which validates just those frames в„‘ in which в„ is not in a certain sense embeddable (to be exact, в„ is not the p -morphic image of any subframe of в„‘.

This paper establishes another very general completeness result for the logics within the field of K 4. The chapter then outlines some most important ideas, such as local versus global consequence, reducing multimodal consequence to monomodal consequence, interpolation theorems, and the admissibility of rules. An algebraic characterization of interpolation and ways of establishing interpolation for logics, Beth-definability, and fixed point theorems are also reviewed. The chapter the reviews results establishing that the lattices of polymodal and polyadic logics can be naturally embedded into the lattice of monomodal logics preserving and reflecting a good deal of properties. The consequence relations from an algebraic perspective the global consequence relations, and the connection between semisimple varieties of modal algebras and weak transitivity are outlined in the chapter. The notion of consequence is a fundamental logical concept and in the setting of modal logic, it can be defined in a number of different ways. It also focuses at modal consequence relations and the structure of the lattice they form and the notion of a splitting.

This chapter describes the basic concepts from universal algebra and basic logical notions such as consequence relations, rules, the deduction theorem, and interpolation. While the EXPTIME upper bound for the universal theory of normal modal algebras can be obtained in a more straightforward way, as discussed in the paper, due to its close connection to the equational theory of normal modal algebras with the universal modality operator, the technique based on the theory of partial algebras is applicable to the study of universal theories of algebras corresponding to a wide range of logics with intensional operators, where no such connection is available. It also follows that the quasi-equational theory of monotonic modal algebras is co-NP-complete. We also obtain the corresponding lower bounds, which means that the universal theory of monotonic modal algebras is co-NP-complete and the universal theory of normal modal algebras is EXPTIME-complete. We show how the theory of partial algebras can be deployed to obtain co-NP and EXPTIME upper bounds for the universal theories of, respectively, monotonic and normal modal algebras. We apply the theory of partial algebras, following the approach developed by Van Alten (Theor Comput Sci 501:82вЂ“92, 2013), to the study of the computational complexity of universal theories of monotonic and normal modal algebras.

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