Reachability is harder for directed than for undirected finite graphs

It is shown that for directed graphs, reachability can not be expressed by an existential monadic second-order sentence. The proof makes use of Ehrenfeucht-Fraisse games, along with probabilistic. However, it is shown that for directed graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence. One reason for the interest in the main result is that while there is considerable empirical evidence (in terms of the efficiency of algorithms that have been discovered) that reachability in directed graphs is 'harder' than reachability in undirected graphs, this is the first proof in a precise technical sense that this is so.<<ETX>>

[1]  Neil Immerman,et al.  Descriptive and Computational Complexity , 1989, FCT.

[2]  Allan Borodin,et al.  Two Applications of Inductive Counting for Complementation Problems , 1989, SIAM J. Comput..

[3]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[4]  Alok Aggarwal,et al.  A random NC algorithm for depth first search , 1987, Comb..

[5]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[6]  Jeffrey D. Ullman,et al.  Database theory—past and future , 1987, PODS.

[7]  Catriel Beeri,et al.  Bounds on the propagation of selection into logic programs , 1987, J. Comput. Syst. Sci..

[8]  Phokion G. Kolaitis,et al.  The decision problem for the probabilities of higher-order properties , 1987, STOC.

[9]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[10]  Daniel Leivant.  Characterization of complexity classes in higher-order logic , 1987, Computational Complexity Conference.

[11]  Michel de Rougemont.  Second-order and Inductive Definability on Finite Structures , 1987, Math. Log. Q..

[12]  Richard Cole,et al.  Approximate and exact parallel scheduling with applications to list, tree and graph problems , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[13]  Neil Immerman,et al.  Relational Queries Computable in Polynomial Time , 1986, Inf. Control..

[14]  Victor Y. Pan,et al.  Efficient parallel solution of linear systems , 1985, STOC '85.

[15]  László Lovász,et al.  Computing ears and branchings in parallel , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[16]  Peter H. Hochschild,et al.  Parallel graph algorithms , 1984 .

[17]  Piotr Berman,et al.  Lower bounds on graph threading by probabilistic machines , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[18]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[19]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[20]  Neil Immerman,et al.  Upper and lower bounds for first order expressibility , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[21]  David Harel,et al.  Structure and complexity of relational queries , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[22]  Stephen A. Cook,et al.  Space Lower Bounds for Maze Threadability on Restricted Machines , 1980, SIAM J. Comput..

[23]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[24]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[25]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[26]  Ronald Fagin,et al.  Monadic generalized spectra , 1975, Math. Log. Q..

[27]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[28]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[29]  N. S. Barnett,et al.  Private communication , 1969 .

[30]  A. Ehrenfeucht.  An application of games to the completeness problem for formalized theories , 1961 .

[31]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .