Jeffrey B. Remmel

Shuffles of permutations and the Kronecker product

The Kronecker product of two homogeneous symmetric polynomialsP1,P2 is defined by means of the Frobenius map by the formulaP1oP2=F(F−1P1)(F−1P2). WhenP1 andP2 are the Schur functionsSI,SJ then the resulting productSI oSJ is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofSI oSJ with a third Schur functionsSK gives the so called Kronecker coefficientcI,J,K=. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thecI,J,K in terms of some classes of permutations. In Lascouxs workI andJ are restricted to be hooks and in Gessels both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of forSI a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessels and Lascouxs results are related and show how they can be derived from a special case of our result.

Hybrid Systems as Finsler Manifolds: Finite State Control as Approximation to Connections

Hybrid systems are networks of interacting digital devices and continuous plants reacting to a changing environment. Our multiple agent hybrid control architecture ([KN93b], [KN93c]) is based on the notion of hybrid system state. The latter incorporates evolution models using differential or difference equations, logic constraints, and geometric constraints. The set of hybrid states of a hybrid system can be construed in a variety of ways as a differentiable (or a C∞) manifold which we have called the carrier manifold ([KNRG95]). We have suggested that for control problems the coordinates of points of the carrier manifolds should be selected to incorporate all information about system state, control state, and environment needed to choose new values of control parameters.

On the Continuity of Gelfond-Lifschitz Operator and Other Applications of Proof-Theory in ASP

Using a characterization of stable models of logic programs P assatisfying valuations of a suitably chosen propositional theory,called the set of reduced defining equations rΦ P , we showthat the finitary character of that theory rΦ P is equivalentto a certain continuity property of the Gelfond-Lifschitz operator

Complexity of recursive normal default logic

{mathit{GL}}_P

The complexity of recursive constraint satisfaction problems

associated with the program P. The introductionof the formula rΦ P leads to a double-backtracking algorithmfor computation of stable models by reduction to satisfiability ofsuitably chosen propositional theories. This algorithm does not usethe reduction via loop-formulas as proposed in [1] or its extensionproposed in [2]. Finally, we discuss possible extensions oftechniques proposed in this paper to the context of cardinalityconstraints.

On the Foundations of Answer Set Programming

Normal default logic, the fragment of default logic obtained by restricting defaults to rules of the form α:Mβ/β, is the most important and widely studied part of default logic. In [20], we proved a basis theorem for extensions of recursive propositional logic normal default theories and hence for finite predicate logic normal default theories. That is, we proved that every recursive propositional normal default theory possesses an extension which is r.e. in 0′. Here we show that this bound is tight. Specifically, we show that for every r.e. set A and every set B r.e. in A there is a recursive normal default theory 〈D, W〉 with a unique extension which is Turing-equivalent to A t B. A similar result holds for finite predicate logic normal default theories.

San Diego Convention Center, San Diego, CA January 8–9, 2008

Abstract We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem.