We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, and works over arbitrary commutative rings. It also lends itself to efficient parallel implementations. It has been known for some time now that the complexity class GapL characterizes the complexity of computing the determinant of matrices over the integers. We present a direct proof of this characterization.

to use journal articles in a variety of ways, limited only as required to insure fair attribution to authors and the journal, and to prohibit use in a competing commercial product. See the journal's World Wide Web site for further details. The Chicago Journal of Theoretical Computer Science is a peer-reviewed scholarly journal in theoretical computer science. The journal is committed to providing a forum for signiicant results on theoretical aspects of all topics in computer science. Abstract This paper introduces weakly growing context-sensitive grammars. Abstract-1 Such grammars generalize the class of growing context-sensitive grammars (studied by several authors), in that these grammars have rules that groww according to a position valuation. If a position valuation coincides with the initial part of an expo-Abstract-2 nential function, it is called a steady position valuation. All others are called unsteady. The complexity of the language generated by a grammar depends crucially on whether the position valuation is steady or not. More precisely, for every unsteady position valuation, the class of languages generated by WGCSGs with this valuation coincides with the class CSL of context-sensitive languages. On the other hand, for every steady position valuation, the class of languages generated corresponds to a level of the hierarchy of exponential time-bounded languages in CSL. We show that the following three conditions are equivalent: The hierarchy of exponential time-bounded languages in CSL collapses. There exists a class deened by an unsteady position valuation such that there is also a normal form of order 2 (e.g., Cremers or Kuroda normal form) for that class. There exists a class deened by a steady position valuation that is closed under inverse homomorphisms.

to use journal articles in a variety of ways, limited only as required to insure fair attribution to authors and the journal, and to prohibit use in a competing commercial product. See the journal's World Wide Web site for further details. The Chicago Journal of Theoretical Computer Science is a peer-reviewed scholarly journal in theoretical computer science. The journal is committed to providing a forum for signiicant results on theoretical aspects of all topics in computer science. Shlomi Dolev and Jennifer Welch. The article presents a technique for proving stabilization of algorithms for systems with a rooted tree communication graph. Proving correctness of a distributed algorithm is sometimes a subtle task. Proving correctness of self-stabilizing distributed algorithms is even a more subtle task, since self-stabilizing algorithms can be started in any possible state. Therefore, techniques like the one presented in this article may serve the algorithm designer in proving the stabilization property. Abstract We describe a simple tree-correction theorem that states that any Abstract-1 locally checkable protocol that works on a tree can be eeciently stabilized in time proportional to the height of the tree. We show how new protocols can be designed, and how existing work can be easily understood using this theorem.

to use journal articles in a variety of ways, limited only as required to insure fair attribution to authors and the journal, and to prohibit use in a competing commercial product. See the journal's World Wide Web site for further details. The Chicago Journal of Theoretical Computer Science is a peer-reviewed scholarly journal in theoretical computer science. The journal is committed to providing a forum for signiicant results on theoretical aspects of all topics in computer science. Abstract We initiate an investigation of probabilistically checkable debate Abstract-1 systems (PCDS), a natural generalization of probabilistically check-able proof systems (PCPS). A PCDS for a language L consists of a probabilistic polynomial-time veriier V and a debate between player 1, who claims that the input x is in L, and player 0, who claims that the input x is not in L. We show that there is a PCDS for L in which V ips O(log n) random coins and reads O(1) bits of the debate if and only if L is in PSPACE. This characterization of PSPACE is used to show that certain PSPACE-hard functions are as hard to approximate closely as they are to compute exactly.Hard Functionss CFLS93b].