Sparse language

Abstract: Two complexity measures for query languages are proposed. Data complexity is the complexity of evaluating a query in the language as a function of the size of the database, and expression complexity is the complexity of evaluating a query in the language as a function of the size of the expression defining the query. We study the data and expression complexity of logical languages - relational calculus and its extensions by transitive closure, fixpoint and second order existential quantification - and algebraic languages - relational algebra and its extensions by bounded and unbounded looping. The pattern which will be shown is that the expression complexity of the investigated languages is one exponential higher then their data complexity, and for both types of complexity we show completeness in some complexity class.

Abstract: @ PTAPE. We show that all NP complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets. Thus showing that, inspite of the radically different origins and attempted simplification of these sets, all the known NP complete sets are identical but for polynomially time bounded permutations.Furthermore, if all NP complete sets are p-isomorphic then they all must have similar densities and, for example, no language over a single letter alphabet can be NP complete, nor can any sparse language over an arbitrary alphabet be NP complete. We show that complete sets in EXPTIME and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet. Similarly, we show that the hardest context-sensitive languages cannot be sparse. We also relate the existence of sparse complete sets to the existence of simple combinatorial circuits for the corresponding truncated recognition problem of these languages.

Abstract: AaSTancr. A recurslve padding technique is used to obtain conditions sufficient for separation of nondetermlmsttc multltape Turlng machine time complexity classes If T2 is a running time and Tl(n + 1) grows more slowly than T~(n), then there is a language which can be accepted nondetermmlstlcally within time bound T~ but which cannot be accepted nondetermlnlStlcally within time bound T1. If even T~(n + f(n)) grows more slowly than Tz(n), where f is the very slowly growing "rounded reverse" of some real-time countable function, then there is such a language over a single-letter alphabet. The strongest known dmgonalization results for both deterministic and nondetermlmstlc time complexity classes are reviewed and orgamzed for comparison with the results of the new padding technique