Complexity theoretic algebra I - vector spaces over finite fields

This is the first of a series of papers on complexity-theoretic algebra. We carne to the subject as follows. In 1975 Metakides and Nerode [15] initiated the systematic study of recursion-theoretic algebra. The motivation was to establish the recursive content of mathematical constructions. The novelty was the use of the finite injury priority method from recursion theory as a uniform tool to meet algebraic requirements. Prior to that time the priority method had been limited primarily to internai applications within recursion theory in the theory of recursively enumerable sets and in the theory of degrees of unsolvability and their generalizations. Recursion theoretic algebra has been developed since, in depth, by many authors in such subjects as commutative fields, vector spaces, orderings, and boolean algebras (see [6] for references and a cross-section of results before 1980). Recursion theoretic algebra yielded as a byproduct a theory óf re-cursively enumerable substructures (see the survey article [18] fòr references). Simultaneously in computer science there was a vast development of P and NP complexity theory. This subject started out as a tool for measuring the relative difficulties of classes of computational problems (see Cobham [4], Cook [5]. Hartmains and Stearns [9], Karp [13]). Many papers in this area have dealt with coding a given problem M into a calibrated problem to find an upper bound on the complexity of M , and coding a calibrated problem into a given problem M to find a lower bound on the complexity of M (se Hopcroft and Ullman [11] and Garey and Johnson