In this paper it is shown that Heyting and Co-Heyting mereological systems provide a convenient conceptual framework for spatial reasoning, in which spatial concepts such as connectedness, interior parts, (exterior) contact, and boundary can be defined in a natural and intuitively appealing way. This fact refutes the wide-spread contention that mereology cannot deal with the more advanced aspec...

It is well-known that any substitution in intuitionistic propositional logic can be seen as an endomorphism between free Heyting algebras. For a given substitution σ : F (n)→ F (m), (where F (n) is the n-generated free Heyting algebra, n ∈ ω), the theory of σ is the filter σ−1(>). The theory of σ is finitely axiomatizable if σ−1(>) is a principal filter. The formula which generates this filter ...

Contract-based design is an expressive paradigm for a modular and compositional specification of programs. It is in turn becoming a fundamental concept in mainstream industrial computer-aided design tools for embedded system design. In this paper, we elaborate new foundations for contract-based embedded system design by proposing a general-purpose algebra of assume/guarantee contracts based on ...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...

The unit interval in a partially ordered abelian group with order unit forms an interval effect algebra (IEA) and can be regarded as an algebraic model for the semantics of a formal deductive logic. There is a categorical equivalence between the category of IEA’s and the category of unigroups. In this article, we study the IEA-unigroup connection, focusing on the cases in which the IEA is a Boo...

We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing theMcKinsey-Tarski theorem.As a consequence,weobtain that each intermediate logicwith thefinitemodel property is the logic of a subalgebra of the Heyting algebra of all o...

In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space Rn with n > 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the frame of all open sets of a Euclidean spac...

We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC. As basic tool we use the connection between intuitionistic logic and Heyting algebra, and investigate its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain NC-completeness for the model checking problem.

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the→-free reducts of Heyting algebras while the variety of implicative semilattices by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that ge...