Proof Relevant Corecursive Resolution

Resolution lies at the foundation of both logic programming and type class context reduction in functional languages. Terminating derivations by resolution have well-defined inductive meaning, whereas some non-terminating derivations can be understood coinductively. Cycle detection is a popular method to capture a small subset of such derivations. We show that in fact cycle detection is a restricted form of coinductive proof, in which the atomic formula forming the cycle plays the role of coinductive hypothesis. This paper introduces a heuristic method for obtaining richer coinductive hypotheses in the form of Horn formulas. Our approach subsumes cycle detection and gives coinductive meaning to a larger class of derivations. For this purpose we extend resolution with Horn formula resolvents and corecursive evidence generation. We illustrate our method on non-terminating type class resolution problems.Comment: 23 pages, with appendices in FLOPS 201

Bidifferential calculus, matrix SIT and sine-Gordon equations

We express a matrix version of the self-induced transparency (SIT) equations in the bidifferential calculus framework. An infinite family of exact solutions is then obtained by application of a general result that generates exact solutions from solutions of a linear system of arbitrary matrix size. A side result is a solution formula for the sine-Gordon equation.Comment: 7 pages, 2 figures, 19th International Colloquium on Integrable Systems and Quantum Symmetries (ISQS19), Prague, Czech Republic, June 201

Temporal Data Modeling and Reasoning for Information Systems

Temporal knowledge representation and reasoning is a major research field in Artificial Intelligence, in Database Systems, and in Web and Semantic Web research. The ability to model and process time and calendar data is essential for many applications like appointment scheduling, planning, Web services, temporal and active database systems, adaptive Web applications, and mobile computing applications. This article aims at three complementary goals. First, to provide with a general background in temporal data modeling and reasoning approaches. Second, to serve as an orientation guide for further specific reading. Third, to point to new application fields and research perspectives on temporal knowledge representation and reasoning in the Web and Semantic Web

Knot concordance and homology cobordism

We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for K and its (p,1)-cables.Comment: 15 pages, 8 figure

The recursive nature of cominuscule Schubert calculus

The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities.Comment: 41 pages, revisions to improve clarity of expositio

Variations on a Theme: A Bibliography on Approaches to Theorem Proving Inspired From Satchmo

This articles is a structured bibliography on theorem provers, approaches to theorem proving, and theorem proving applications inspired from Satchmo, the model generation theorem prover developed in the mid 80es of the 20th century at ECRC, the European Computer- Industry Research Centre. Note that the bibliography given in this article is not exhaustive

Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover "negative flows", leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation