A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable. Supported by NSF Grant #CCR88-136...

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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of genes on corresponding chromosomes, and a reversal takes an arbitrary substring of elements and reverses their order. For this problem we develop two algorithms: a greedy approximation algorithm that finds a solution provably close to optimal in O(n 2 ) time and O(n) space for an n element permutation, and a branch and bound exact algorithm that finds an optimal solution in O(mL(n;n)) time and O(n 2 ) space, where m is the size of the branch and bound search tree and L(n; n) is the time to solve a linear program of n variables and n constraints. The greedy algorithm is the first to come within a constant factor of the optimum; it guarantees a solution that uses no more than twice the min...

Given a set r of permutations of an n-set, let G be the group of permutations generated by f. If p is a prime, a Sylow p-subgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow p-subgroup exists, and that for any two Sylow p-subgroups PI, P, of G there is an element go G such that Pz = g-‘PI g. We present polynomial-time algorithms that find (generators for) a Sylow p-subgroup of G, and that find ge G such that P, = g-‘P, g whenever (generators for) two Sylow p-subgroups PI, Pz are given. These algorithms involve the classification of all tinite simple groups. 0 1985 Academic Press. Inc. PART I 1.

Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.

The problem of memory management in totally distributed computing systems leads to the following movers' problem on graphs: Let G be a graph with n vertices with k < n pebbles numbered 1, ... ,k on distinct vertices. A move consists of transferring a pebble to an adjacent unoccupied vertex. The problem is to decide whether one· arrangement of the pebbles is reachable from another, and to find the shortest sequence of moves to find the rearrangement when it is possible. In the case that G is biconnected and k=n-1, Wilson (1974) gave an efficient decision procedure. However, he did not determine whether solutions require at most polynomially many moves. We generalize by giving a P-time decision procedure for all graphs and any number of pebbles. Further, we prove matching O(n^3) upper and lower bounds on the number. of moves required, and show how to efficiently plan solutions. It is hoped that the algebraic methods introduced for the graph puzzle will be applicable to special cases of the general geometric movers' problem, which is PSPACE-hard (Reif (1979)). We consider the related question of permutation group diameter. Driscoll and Furst (1983) obtained a polynomial upper bound on the diameter of permutation groups generated by cycles of bounded length. By making effective some standard results in permutation group theory, we obtain the following partial extension of their result to unbounded vcycles: If G (on n letters) is generated by cycles, one of which has prime length p < 2n/3, and G is primitive, then G = A_n or S_n and has diameter less than 2^((p^1/6)+4) n^8. This is a moderately exponential bound.

has been introduced as a generalization of a class of grammar formalisms known as mildly context-sensitive. The recognition problem for linear context-free rewriting languages is studied at length here, presenting evidence that, even in some restricted cases, it cannot be solved efficiently. This entails the existence of a gap between, for example, tree adjoining languages and the subclass of linear context-free rewriting languages that generalizes the former class; such a gap is attributed to "crossing configurations". A few other interesting consequences of the main result are discussed, that concern the recognition problem for linear context-free rewriting languages.

In the late 1970's, S. Magliveras invented a private-key cryptographic system called Permutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space Z jGj , for a given finite permutation group G. In this paper we prove several algebraic properties of PGM. We show that the set of PGM transformations TG is not closed under functional composition and hence not a group. This set is 2-transitive on Z jGj if the underlying group G is not hamiltonian, and not abelian. Moreover, if the order of G is not a power of 2, then the set of transformations contains an odd permutation. An important consequence of these results is that the group generated by the set of transformations is nearly always the symmetric group S jGj . Thus, allowing multiple encryption, any permutation of the message space is attainable. This property is ...

A symmetric key cryptosystem based on logarithmic signatures for nite permutation groups was described by the rst author in [6], and its algebraic properties were studied in [7]. In this paper we describe two possible approaches to the construction of new public key cryptosystems with message space a large nite group G, using logarithmic signatures and their generalizations. The rst approach relies on the fact that permutations of the message space G induced by transversal logarithmic signatures almost always generate the full symmetric group SG on the message space. The second approach could potentially lead to new ElGamal - like systems based on trap-door, one-way functions induced Research supported in part by National Science Foundation grant CCR-9610138 y Research supported in part by NSERC grants IRC #216431-96 and RGPIN # 203114-98. 1 by logarithmic signature-like objects we call meshes, which are uniform covers for G. Key words. Trap-door one-way functions...