We present the first (polynomial-time) algorithm for reducing a given deterministic finite state automaton (DFA) into a hyperminimized DFA, which may have fewer states than the classically minimized DFA. The price we pay is that the language recognized by the new machine can differ from the original on a finite number of inputs. These hyper-minimized automata are optimal, in the sense that every DFA with fewer states must disagree on infinitely many inputs. With small modifications, the construction works also for finite state transducers producing outputs. Within a class of finitely differing languages, the hyper-minimized automaton is not necessarily unique. There may exist several non-isomorphic machines using the minimum number of states, each accepting a separate language finitely-different from the original one. We will show that there are large structural similarities among all these smallest automata.

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Two formal languages are f-equivalent if their symmetric difference L 1 △ L 2 is a finite set -that is, if they differ on only finitely many words. The study of f-equivalent languages, and particularly the DFAs that accept them, was recently introduced [3]. First, we restate the fundamental results in this new area of research. Second, we introduce our main result, which is a faster algorithm for the natural minimization problem: given a starting DFA D, find the smallest (by number of states) DFA D ′ such that L(D) and L(D ′ ) are f-equivalent. Finally, we suggest a technique that combines this hyper-minimization with the well-studied notion of a deterministic finite cover automaton [2, 4, 5], or DFCA, thereby extending the applicability of DFCAs from finite to infinite regular languages.a There, f-equivalence is called either "almost equivalence" or "finite difference". We use the new term here because it is shorter, and cannot be misunderstood as excluding total equivalence. 735 Int. J. Found. Comput. Sci. 2009.20:735-746. Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 02/05/15. For personal use only.

The aim of this article is to derive several Pólya and Steffensen type integral inequalities for trigonometrically ρ-convex functions.

The aim of this article is to introduce the definition of the conjugate of sub $E$-functions, which plays an important role in linking the concept of duality among sub $E$-functions. Furthermore, some properties for this class are established.