ABSTRACT
The classical zero-one law for first-order logic on random graphs says that for every first-order property φ in the theory of graphs and every p ∈ (0,1), the probability that the random graph G(n, p) satisfies φ approaches either 0 or 1 as n approaches infinity. It is well known that this law fails to hold for any formalism that can express the parity quantifier: for certain properties, the probability that G(n,p) satisfies the property need not converge, and for others the limit may be strictly between 0 and 1. In this work, we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier, FOP, over G(n,p), thus eluding the above hurdles. Specifically, we establish the following "modular convergence law": For every FOP sentence φ, there are two explicitly computable rational numbers a0, a1, such that for i ∈ {0,1}, as n approaches infinity, the probability that the random graph G(2n+i, p) satisfies φ approaches ai. Our results also extend appropriately to FO equipped with Modq quantifiers for prime q. In the process of deriving the above theorem, we explore a new question that may be of interest in its own right. Specifically, we study the joint distribution of the subgraph statistics modulo 2 of G(n,p): namely, the number of copies, mod 2, of a fixed number of graphs F1, ..., Fl of bounded size in G(n,p). We first show that every FOP property φ is almost surely determined by subgraph statistics modulo 2 of the above type. Next, we show that the limiting joint distribution of the subgraph statistics modulo 2 depends only on n Mod 2, and we determine this limiting distribution completely. Interestingly, both these steps are based on a common technique using multivariate polynomials over finite fields and, in particular, on a new generalization of the Gowers norm that we introduce. The first step above is analogous to the Razborov-Smolensky method for lower bounds for AC0 with parity gates, yet stronger in certain ways. For instance, it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every FOP sentence, which is something that is not known for AC.
- Babai, Nisan, and Szegedy. Multiparty protocols and logspace-hard pseudorandom sequences. In STOC: ACM Symposium on Theory of Computing (STOC), 1989. Google ScholarDigital Library
- A. Blass, Y. Gurevich, and D. Kozen. A zero-one law for logic with a fixed point operator. Information and Control, 67:70--90, 1985. Google ScholarDigital Library
- A. Bogdanov and E. Viola. Pseudorandom bits for polynomials. In FOCS, pages 41--51, 2007. Google ScholarDigital Library
- R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In R. M. Karp, editor, Complexity of Computation, SIAM--AMS Proceedings, Vol. 7, pages 43--73, 1974.Google Scholar
- R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41:50--58, 1976.Google ScholarCross Ref
- Y. V. Glebskii, D. I. Kogan, M. I. Liogonki, and V. A. Talanov. Range and degree of realizability of formulas in the restricted predicate calculus. Cybernetics, 5:142--154, 1969.Google ScholarCross Ref
- W. T. Gowers. A new proof of Szemerédi's theorem. Geom. Funct. Anal., 11(3):465--588, 2001.Google ScholarCross Ref
- B. Green and T. Tao. The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2), 167(2):481--547, 2008.Google ScholarCross Ref
- L. Hella, P. Kolaitis, and K. Luosto. Almost everywhere equivalence of logics in finite model theory. Bulletin of Symbolic Logic, 2(4):422--443, 1996.Google ScholarCross Ref
- P. G. Kolaitis and M. Y. Vardi. The decision problem for the probabilities of higher-order properties. In Proc. 19th ACM Symp. on Theory of Computing, pages 425--435, 1987. Google ScholarDigital Library
- P. G. Kolaitis and M. Y. Vardi. 0-1 laws and decision problems for fragments of second-order logic. Information and Computation, 87:302--338, 1990. Google ScholarDigital Library
- S. Lovett. Unconditional pseudorandom generators for low degree polynomials. In STOC, pages 557--562, 2008. Google ScholarDigital Library
- L. Pacholski and W. Szwast. The 0-1 law fails for the class of existential second-order Gödel sentences with equality. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 280--285, 1989. Google ScholarDigital Library
- Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. MATHNASUSSR: Mathematical Notes of the Academy of Sciences of the USSR, 41, 1987.Google Scholar
- S. Shelah and J. Spencer. Zero-one laws for sparse random graphs. J. Amer. Math. Soc., 1:97--115, 1988.Google ScholarCross Ref
- R. Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In STOC, pages 77--82, 1987. Google ScholarDigital Library
- J. Spencer and S. Shelah. Threshold spectra for random graphs. In Proc. 19th ACM Symp. on Theory of Computing, pages 421--424, 1987. Google ScholarDigital Library
- E. Viola. The sum of d small-bias generators fools polynomials of degree d. In IEEE Conference on Computational Complexity, pages 124--127, 2008. Google ScholarDigital Library
- E. Viola and A. Wigderson. Norms, xor lemmas, and lower bounds for gf(2) polynomials and multiparty protocols. In 22th IEEE Conference on Computational Complexity (CCC), 2007. Google ScholarDigital Library
Index Terms
- Random graphs and the parity quantifier
Recommendations
Random graphs and the parity quantifier
The classical zero-one law for first-order logic on random graphs says that for every first-order property φ in the theory of graphs and every p ∈ (0,1), the probability that the random graph G(n, p) satisfies φ approaches either 0 or 1 as n approaches ...
On derandomizing algorithms that err extremely rarely
STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computingDoes derandomization of probabilistic algorithms become easier when the number of "bad" random inputs is extremely small?
In relation to the above question, we put forward the following quantified derandomization challenge: For a class of circuits C (...
On the constant-depth complexity of k-clique
STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computingWe prove a lower bound of ω(nk/4) on the size of constant-depth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound of ω(nk/89d2) due to Beame where d is the circuit depth. Our lower bound has the ...
Comments