In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

Abstract. The need to integrate several versions of a program into a common one arises frequently, but it is a tedious and time consuming task to merge programs by hand. The program-integration algorithm proposed by Horwitz, Prins, and Reps provides a way to create a semantics-based tool for integrating a base program with two or more variants. The integration algorithm is based on the assumption that any change in the behavior, rather than the text, of a program variant is significant and must be incorporated in the merged program. An integration system based on this algorithm will determine whether the variants incorporate interfering changes, and, if they do not, create an integrated program that includes all changes as well as all features of the base program that are preserved in all variants. To determine this information, the algorithm employs a program representation that is similar to the program dependence graphs that have been used previously in vectorizing and parallelizing compilers. This paper studies the algebraic properties of the program-integration operation, such as whether there are laws of associativity and distributivity. (For example, in this context associativity means: “If three variants of a given base are to be integrated by a pair of two-variant integrations, the same result is produced no matter which two variants are integrated first.”) To answer such questions, we reformulate the Horwitz-Prins-Reps integration algorithm as an operation in a Brouwerian algebra constructed from sets of dependence graphs. (A Brouwerian algebra is a distributive lattice with an operation a. − b characterized by a. ¡ ¢

The ability to transform between distinct geometric representations is the key to success of multiple-representation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of classical problems of CSG $ b-rep conversions, CSG optimization, and other representation conversions suggests a natural relationship between a representation scheme and an appropriate decomposition of space. We show that a hierarchy of space decompositions corresponding to different representation schemes can be used to enhance the theory and to develop a systematic approach to maintenance of geometric representations. 1. Motivation 1.1. Modern theory of representations The modern field of solid modeling owes much of its success to the theoretical foundations laid by members of the Production Automation Project at the University of Rochester in the 1970's. The history of these development...

This article revisits the main ideas and foundations of solid modeling in engineering, summarizes recent progress and bottlenecks, and speculates on possible future directions

this paper, we present various forms of set approximations via the unifying concept of modal--style operators. Two examples indicate the usefulness of the approach

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.

Algebraic Logic. In preparation. Manuscript.

This paper studies operations for creating new variants of a program that relate, in a well-defined way, to existing variants of the program. We formalize a program modification as a (special kind of) function from programs to programs, and study the algebra of these program modifications. We make use of the algebraic structure to formalize several intuitive concepts, such as that of "compatibility among program modifications", and establish several new results concerning the problems of program merging and separating consecutive edits. We also identify a category in which the objects are programs and the morphisms are program modifications, and show how program integration relates to the pushout in this category

2.1 Syntax and semantics of intermediate logics.......... 4 2.2 Operations on Kripke frames.................. 5

Abstract: We study inference systems of weak functional dependencies in relational and complex-value databases. Functional dependencies form a very common class of database constraints. Designers and administrators proficiently utilise them in everyday database practice. Functional dependencies correspond to the linear-time decidable fragment of Horn clauses in propositional logic. Weak functional dependencies take advantage of arbitrary clauses, and therefore represent full propositional reasoning about data in databases. Moreover, they can be specified in a way that is very similar to functional dependencies. In relational databases the class of weak functional dependencies is finitely axiomatisable and the associated implication problem is coNP-complete in general. Our first main result extends this axiomatisation to databases in which complex elements can be derived from atomic ones by finitely many nestings of record, list and disjoint union constructors. In particular, we construct two nested tuples that can serve as a counterexample relation for the implication of weak functional dependencies. We further apply this construction to show an equivalence to truth assignments that serve as counterexamples for the implication of propositional clauses. Hence, we characterise the implication of weak functional dependencies in complex-value databases in completely logical terms. Consequently, state-of-the-art SAT solvers can be applied to reason about weak functional dependencies in relational and complex-value databases.