NP?=EXP

We only point out that the work of algorithmic algebra community is not enough, at least so far.

On Algebras of Algorithms and Specifications over Uninterpreted Data

Summary This paper continues formalization in Mizar [2, 1] of basic notions of the composition-nominative approach to program semantics [13] which was started in [8, 11]. The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a mathematical systems theory [4, 6, 7, 5, 3]. In the paper we introduce a definition of the notion of a binominative function over a set D understood as a partial function which maps elements of D to D. The sets of binominative functions and nominative predicates [11] over given sets form the carrier of the generalized Glushkov algorithmic algebra [14]. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties. We formalize the operations of this algebra (also called compositions) which are valid over uninterpretated data and which include among others the sequential composition, the prediction composition, the branching composition, the monotone Floyd-Hoare composition, and the cycle composition. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in [10, 12, 9].

On an Algorithmic Algebra over Simple-Named Complex-Valued Nominative Data

Summary This paper continues formalization in the Mizar system [2, 1] of basic notions of the composition-nominative approach to program semantics [14] which was started in [8, 12, 10]. The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. In particular, data in computer systems are modeled as nominative data [15]. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a mathematical systems theory [4, 6, 7, 5, 3]. In the paper we give a formal definition of the notions of a binominative function over given sets of names and values (i.e. a partial function which maps simple-named complex-valued nominative data to such data) and a nominative predicate (a partial predicate on simple-named complex-valued nominative data). The sets of such binominative functions and nominative predicates form the carrier of the generalized Glushkov algorithmic algebra for simple-named complex-valued nominative data [15]. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties. In particular, we formalize the operations of this algebra which require a specification of a data domain and which include the existential quantifier, the assignment composition, the composition of superposition into a predicate, the composition of superposition into a binominative function, the name checking predicate. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in [11, 13, 9].

EFFICIENT ALGORITHMS FOR HIGHLY COMPRESSED DATA: THE WORD PROBLEM IN HIGMAN'S GROUP IS IN P

Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is in P. Before that the best known upper bound was non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic algebra and algorithmic group theory: (1) We define a modified reduction procedure on power circuits which runs in quadratic time, thereby improving the known cubic time complexity. The improvement is crucial for our other results. (2) We improve the complexity of the Word Problem for the Baumslag group to cubic time, thereby providing the first practical algorithm for that problem. (The algorithm has been implemented and is available in the CRAG library.) (3) The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of power circuits.