2,144 research outputs found
The decision problem of modal product logics with a diagonal, and faulty counter machines
In the propositional modal (and algebraic) treatment of two-variable
first-order logic equality is modelled by a `diagonal' constant, interpreted in
square products of universal frames as the identity (also known as the
`diagonal') relation. Here we study the decision problem of products of two
arbitrary modal logics equipped with such a diagonal. As the presence or
absence of equality in two-variable first-order logic does not influence the
complexity of its satisfiability problem, one might expect that adding a
diagonal to product logics in general is similarly harmless. We show that this
is far from being the case, and there can be quite a big jump in complexity,
even from decidable to the highly undecidable. Our undecidable logics can also
be viewed as new fragments of first- order logic where adding equality changes
a decidable fragment to undecidable. We prove our results by a novel
application of counter machine problems. While our formalism apparently cannot
force reliable counter machine computations directly, the presence of a unique
diagonal in the models makes it possible to encode both lossy and
insertion-error computations, for the same sequence of instructions. We show
that, given such a pair of faulty computations, it is then possible to
reconstruct a reliable run from them
Modal Logics that Bound the Circumference of Transitive Frames
For each natural number we study the modal logic determined by the class
of transitive Kripke frames in which there are no cycles of length greater than
and no strictly ascending chains. The case is the G\"odel-L\"ob
provability logic. Each logic is axiomatised by adding a single axiom to K4,
and is shown to have the finite model property and be decidable.
We then consider a number of extensions of these logics, including
restricting to reflexive frames to obtain a corresponding sequence of
extensions of S4. When , this gives the famous logic of Grzegorczyk, known
as S4Grz, which is the strongest modal companion to intuitionistic
propositional logic. A topological semantic analysis shows that the -th
member of the sequence of extensions of S4 is the logic of hereditarily
-irresolvable spaces when the modality is interpreted as the
topological closure operation. We also study the definability of this class of
spaces under the interpretation of as the derived set (of limit
points) operation.
The variety of modal algebras validating the -th logic is shown to be
generated by the powerset algebras of the finite frames with cycle length
bounded by . Moreover each algebra in the variety is a model of the
universal theory of the finite ones, and so is embeddable into an ultraproduct
of them
Topological Semantics and Decidability
It is well-known that the basic modal logic of all topological spaces is
. However, the structure of basic modal and hybrid logics of classes of
spaces satisfying various separation axioms was until present unclear. We prove
that modal logics of , and topological spaces coincide and are
S4T_1 spaces coincide.Comment: presentation changes, results about concrete structure adde
The Relevant Logic E and Some Close Neighbours: A Reinterpretation
This paper has two aims. First, it sets out an interpretation of the relevant logic E of relevant entailment based on the theory of situated inference. Second, it uses this interpretation, together with Anderson and Belnap’s natural deduc- tion system for E, to generalise E to a range of other systems of strict relevant implication. Routley–Meyer ternary relation semantics for these systems are produced and completeness theorems are proven
Uniform interpolation and coherence
A variety V is said to be coherent if any finitely generated subalgebra of a
finitely presented member of V is finitely presented. It is shown here that V
is coherent if and only if it satisfies a restricted form of uniform deductive
interpolation: that is, any compact congruence on a finitely generated free
algebra of V restricted to a free algebra over a subset of the generators is
again compact. A general criterion is obtained for establishing failures of
coherence, and hence also of uniform deductive interpolation. This criterion is
then used in conjunction with properties of canonical extensions to prove that
coherence and uniform deductive interpolation fail for certain varieties of
Boolean algebras with operators (in particular, algebras of modal logic K and
its standard non-transitive extensions), double-Heyting algebras, residuated
lattices, and lattices
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