A zero-one law for logic with a fixed-point operator
Information and Control
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Threshold spectra for random graphs
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
The decision problem for the probabilities of higher-order properties
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Multiparty protocols and logspace-hard pseudorandom sequences
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
0-1 laws and decision problems for fragments of second-order logic
Information and Computation - Selections from 1988 IEEE symposium on logic in computer science
Norms, XOR Lemmas, and Lower Bounds for GF(2) Polynomials and Multiparty Protocols
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Pseudorandom Bits for Polynomials
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Unconditional pseudorandom generators for low degree polynomials
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The Sum of d Small-Bias Generators Fools Polynomials of Degree d
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
The 0-1 law fails for the class of existential second order Godel sentences with equality
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Degree lower bounds of tower-type for approximating formulas with parity quantifiers
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
Degree lower bounds of tower-type for approximating formulas with parity quantifiers
ACM Transactions on Computational Logic (TOCL)
Hi-index | 0.00 |
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.