9,941 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
Edsger Wybe Dijkstra (1930 -- 2002): A Portrait of a Genius
We discuss the scientific contributions of Edsger Wybe Dijkstra, his opinions
and his legacy.Comment: 10 pages. To appear in Formal Aspects of Computin
Narrow Proofs May Be Maximally Long
We prove that there are 3-CNF formulas over n variables that can be refuted
in resolution in width w but require resolution proofs of size n^Omega(w). This
shows that the simple counting argument that any formula refutable in width w
must have a proof in size n^O(w) is essentially tight. Moreover, our lower
bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams,
implying that the corresponding size upper bounds in terms of degree and rank
are tight as well. Our results do not extend all the way to Lasserre, however,
where the formulas we study have proofs of constant rank and size polynomial in
both n and w
Elementary Proof of Strong Normalization for Atomic F
We give an elementary proof (in the sense that it is formalizable in Peano arithmetic) of the strong normalization of the atomic polymorphic calculus Fₐₜ (a predicative restriction of Girard’s system F)
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
Narrow proofs may be maximally long
We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n(Omega(w)). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n(O(w)) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.Peer ReviewedPostprint (author's final draft
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