0-1 Laws and Decision Problems for Fragments of Second-Order Logic

Fragments of existential second-order logic are investigated in which the patterns of first order quantifiers are restricted. The focus is on the class Sigma /sub 1//sup 1/ (Ackermann) of existential second-order sentences in which the first-order part belongs to the Ackermann class, i.e. it contains at most one universal first-order quantifier. All properties expressible by Sigma /sub 1//sup 1/ (Ackermann) sentences are NP-computable, and there are natural NP-complete properties, such as satisfiability, that are expressible by such sentences. It is established that the 0-1 law holds for the class Sigma /sub 1//sup 1/ (Ackermann), and it is shown that the associated decision problem is NEXPTIME-complete. It is also shown that the 0-1 law fails for other fragments of existential second-order logic in which first-order part belongs to certain prefix classes with an unsolvable decision problem. >

[1]  Warren D. Goldfarb.  On the Godel Class with Identity , 1981, J. Symb. Log..

[2]  Harry R. Lewis,et al.  Unsolvable classes of quantificational formulas , 1979 .

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

[4]  Yu. V. Glebskii,et al.  Range and degree of realizability of formulas in the restricted predicate calculus , 1969 .

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

[6]  Y. Gurevich.  The decision problem for the logic of predicates and of operations , 1969 .

[7]  Ronald Fagin,et al.  Probabilities on finite models , 1976, Journal of Symbolic Logic.

[8]  Etienne Grandjean,et al.  Complexity of the First-Order Theory of Almost All Finite Structures , 1983, Inf. Control..

[9]  K. Compton.  Laws in Logic and Combinatorics , 1989 .

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

[11]  Neil Immerman,et al.  Sparse Sets in NP-P: EXPTIME versus NEXPTIME , 1985, Inf. Control..

[12]  Yuri Gurevich,et al.  Toward logic tailored for computational complexity , 1984 .

[13]  Saharon Shelah,et al.  Random Models and the Godel Case of the Decision Problem , 1983, J. Symb. Log..

[14]  Neil Immerman.  Languages which capture complexity classes , 1983, STOC '83.

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

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

[17]  Saharon Shelah,et al.  On random models of finite power and monadic logic , 1985, Discrete Mathematics.

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

[19]  Harry R. Lewis,et al.  Complexity Results for Classes of Quantificational Formulas , 1980, J. Comput. Syst. Sci..

[20]  Andreas Blass,et al.  A Zero-One Law for Logic with a Fixed-Point Operator , 1986, Inf. Control..

[21]  Béla Bollobás,et al.  Random Graphs , 1985 .

[22]  Yuri Gurevich.  The Decision Problem for Standard Classes , 1976, J. Symb. Log..

[23]  B. Dreben,et al.  The decision problem: Solvable classes of quantificational formulas , 1979 .

[24]  L. Pacholski,et al.  The 0-1 law fails for the class of existential second order Godel sentences with equality , 1989, 30th Annual Symposium on Foundations of Computer Science.

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

[26]  L. Pósa,et al.  Hamiltonian circuits in random graphs , 1976, Discret. Math..