10 research outputs found
On the existence of complete disjoint NP-pairs
Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity. The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field. In this paper we prove that there exists a many-one hard disjoint NP-pair which is computed with access to a very weak oracle (a tally NP-oracle). In addition, we exhibit candidates for complete NP-pairs and apply our results to a recent line of research on the construction of hard tautologies from pseudorandom generators
Tri-State Boolean Satisfiability with Commit: An Efficient Partial Solution Using Hyperlogic
We present two implementation enhancements for the Boolean satisfiability problem and one visualization technique. The first is an expansion to a tri-nary logic system with a commit phase. The three states are (1) true, (2) false, and (3) don\u27t care. We abstracted the operations of AND and OR to this hyperlogic system in a novel way. The commit phase works on one variable at a time and transitions values from temporary to permanent whenever possible. We viewed tri-state logic as a hyperspace above the binary (Boolean) logic. The second improvement is algorithmic. We modified the semantics of the classic 3 Conjunctive Normal Form Problem in order to develop a polynomial time algorithm for a simplified normal form - avoiding the need to examine all combinatoric limitations. In particular, we abandoned 3 CNF and used an unstructured left to right associativity. We do not claim that this new semantic is comprehensive. We do claim that it is simpler. Lastly, we introduced a node analogy to help us understand the algorithm itself
Complexity of Propositional Proofs under a Promise
We study -- within the framework of propositional proof complexity -- the
problem of certifying unsatisfiability of CNF formulas under the promise that
any satisfiable formula has many satisfying assignments, where ``many'' stands
for an explicitly specified function \Lam in the number of variables . To
this end, we develop propositional proof systems under different measures of
promises (that is, different \Lam) as extensions of resolution. This is done
by augmenting resolution with axioms that, roughly, can eliminate sets of truth
assignments defined by Boolean circuits. We then investigate the complexity of
such systems, obtaining an exponential separation in the average-case between
resolution under different size promises:
1. Resolution has polynomial-size refutations for all unsatisfiable 3CNF
formulas when the promise is \eps\cd2^n, for any constant 0<\eps<1.
2. There are no sub-exponential size resolution refutations for random 3CNF
formulas, when the promise is (and the number of clauses is
), for any constant .Comment: 32 pages; a preliminary version appeared in the Proceedings of
ICALP'0
Sum-of-Squares Lower Bounds for the Minimum Circuit Size Problem
We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the
Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean
function , SoS requires degree
to prove that does not have circuits of size
(for any ). As a corollary we obtain that there are no
low degree SoS proofs of the statement NP P/poly.
We also show that for any there are Boolean functions with
circuit complexity larger than but SoS requires size
to prove this. In addition we prove analogous
results on the minimum \emph{monotone} circuit size for monotone Boolean slice
functions.
Our approach is quite general. Namely, we show that if a proof system has
strong enough constraint satisfaction problem lower bounds that only depend on
good expansion of the constraint-variable incidence graph and, furthermore,
is expressive enough that variables can be substituted by local Boolean
functions, then the MCSP problem is hard for .Comment: A conference version appeared previously in CCC'2
Time-Space Tradeoffs for Distinguishing Distributions and Applications to Security of Goldreich's PRG
In this work, we establish lower-bounds against memory bounded algorithms for
distinguishing between natural pairs of related distributions from samples that
arrive in a streaming setting.
In our first result, we show that any algorithm that distinguishes between
uniform distribution on and uniform distribution on an
-dimensional linear subspace of with non-negligible advantage
needs samples or memory.
Our second result applies to distinguishing outputs of Goldreich's local
pseudorandom generator from the uniform distribution on the output domain.
Specifically, Goldreich's pseudorandom generator fixes a predicate
and a collection of subsets of size . For any seed , it
outputs where is the
projection of to the coordinates in . We prove that whenever is
-resilient (all non-zero Fourier coefficients of are of degree
or higher), then no algorithm, with memory, can distinguish the
output of from the uniform distribution on with a large inverse
polynomial advantage, for stretch (barring some
restrictions on ). The lower bound holds in the streaming model where at
each time step , is a randomly chosen (ordered) subset of
size and the distinguisher sees either or a uniformly random
bit along with .
Our proof builds on the recently developed machinery for proving time-space
trade-offs (Raz 2016 and follow-ups) for search/learning problems.Comment: 35 page
Feasibly constructive proofs of succinct weak circuit lower bounds
We ask for feasibly constructive proofs of known circuit lower bounds for explicit functions on bit strings of length n. In 1995 Razborov showed that many can be proved in PV1, a bounded arithmetic formalizing polynomial time reasoning. He formalized circuit lower bound statements for small n of doubly logarithmic order.
It is open whether PV1 proves known lower bounds in succinct formalizations for n of logarithmic order. We give such proofs in APC1, an extension of PV1 formalizing probabilistic polynomial time reasoning: for parity and AC0, for mod q and AC0[p] (only for n slightly smaller than logarithmic), and for k-clique and monotone circuits.
We also formalize Razborov and Rudich’s natural proof barrier.
We ask for short propositional proofs of circuit lower bounds expressed succinctly by propositional formulas of size nO(1) or at least much smaller than the 2O(n) size of the common “truth table” formula. We discuss two such expressions: one via feasible functions witnessing errors of circuits, and one via the anticheckers of Lipton and Young 1994. Our APC1 formalizations yield conditional upper bounds for the succinct formulas obtained by witnessing: we get short Extended Frege proofs from general circuit lower bounds expressed by the common “truth-table” formulas. We also show how to construct in quasipolynomial time propositional proofs of quasipolynomial size tautologies expressing AC0[p] quasipolynomial size
lower bounds; these proofs are in Jerábek’s system WF.Peer ReviewedPostprint (author's final draft
Pseudorandom Generators in Propositional Proof Complexity
We call a pseudorandom generator G n : f0; 1g hard for a propositional proof system P if P can not eciently prove the (properly encoded) statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 f0; 1g