1,931 research outputs found
Enumerating proofs of positive formulae
We provide a semi-grammatical description of the set of normal proofs of
positive formulae in minimal predicate logic, i.e. a grammar that generates a
set of schemes, from each of which we can produce a finite number of normal
proofs. This method is complete in the sense that each normal proof-term of the
formula is produced by some scheme generated by the grammar. As a corollary, we
get a similar description of the set of normal proofs of positive formulae for
a large class of theories including simple type theory and System F
On Tackling the Limits of Resolution in SAT Solving
The practical success of Boolean Satisfiability (SAT) solvers stems from the
CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a
propositional proof complexity perspective, CDCL is no more powerful than the
resolution proof system, for which many hard examples exist. This paper
proposes a new problem transformation, which enables reducing the decision
problem for formulas in conjunctive normal form (CNF) to the problem of solving
maximum satisfiability over Horn formulas. Given the new transformation, the
paper proves a polynomial bound on the number of MaxSAT resolution steps for
pigeonhole formulas. This result is in clear contrast with earlier results on
the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper
also establishes the same polynomial bound in the case of modern core-guided
MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard
for CDCL SAT solvers, show that these can be efficiently solved with modern
MaxSAT solvers
Is Complexity a Source of Incompleteness?
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems
of a finitely-specified theory cannot be significantly more complex than the
theory itself}, for an appropriate measure of complexity. We show that the
measure is invariant under the change of the G\"odel numbering. For this
measure, the theorems of a finitely-specified, sound, consistent theory strong
enough to formalize arithmetic which is arithmetically sound (like
Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded
complexity, hence every sentence of the theory which is significantly more
complex than the theory is unprovable. Previous results showing that
incompleteness is not accidental, but ubiquitous are here reinforced in
probabilistic terms: the probability that a true sentence of length is
provable in the theory tends to zero when tends to infinity, while the
probability that a sentence of length is true is strictly positive.Comment: 15 pages, improved versio
Remarks on the --permanent
We recall Vere-Jones's definition of the --permanent and describe the
connection between the (1/2)--permanent and the hafnian. We establish expansion
formulae for the --permanent in terms of partitions of the index set,
and we use these to prove Lieb-type inequalities for the --permanent
of a positive semi-definite Hermitian matrix and the
--permanent of a positive semi-definite real symmetric
matrix if is a nonnegative integer or . We are unable
to settle Shirai's nonnegativity conjecture for --permanents when
, but we verify it up to the case, in addition to
recovering and refining some of Shirai's partial results by purely
combinatorial proofs.Comment: 9 page
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