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A theory of resolution
We review the fundamental resolution-based methods for first-order theorem proving and present them in a uniform framework. We show that these calculi can be viewed as specializations of non-clausal resolution with simplification. Simplification techniques are justified with the help of a rather general notion of redundancy for inferences. As simplification and other techniques for the elimination of redundancy are indispensable for an acceptable behaviour of any practical theorem prover this work is the first uniform treatment of resolution-like techniques in which the avoidance of redundant computations attains the attention it deserves. In many cases our presentation of a resolution method will indicate new ways of how to improve the method over what was known previously. We also give answers to several open problems in the area
Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds
For a toric Calabi-Yau (CY) orbifold whose underlying toric
variety is semi-projective, we construct and study a non-toric Lagrangian torus
fibration on , which we call the Gross fibration. We apply the
Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of to
construct its mirror with the instanton corrections coming from genus 0 open
orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic
orbi-disks in bounded by fibers of the Gross fibration.
We explicitly evaluate all these invariants by first proving an open/closed
equality and then employing the toric mirror theorem for suitable toric
(partial) compactifications of . Our calculations are then applied
to
(1) prove a conjecture of Gross-Siebert on a relation between genus 0 open
orbifold GW invariants and mirror maps of -- this is called the
open mirror theorem, which leads to an enumerative meaning of mirror maps, and
(2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds
change under toric crepant resolutions -- an open analogue of Ruan's crepant
resolution conjecture.Comment: v4: 65 pages, significantly shortened to avoid too much overlap with
arXiv:1006.3830 and arXiv:1206.3994, to appear in JD
Deduction modulo theory
This paper is a survey on Deduction modulo theor
A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds
We study the problem of obtaining lower bounds for polynomial calculus (PC)
and polynomial calculus resolution (PCR) on proof degree, and hence by
[Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03]
established that if the clause-variable incidence graph of a CNF formula F is a
good enough expander, then proving that F is unsatisfiable requires high PC/PCR
degree. We further develop the techniques in [AR03] to show that if one can
"cluster" clauses and variables in a way that "respects the structure" of the
formula in a certain sense, then it is sufficient that the incidence graph of
this clustered version is an expander. As a corollary of this, we prove that
the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree
when restricted to constant-degree expander graphs. This answers an open
question in [Razborov '02], and also implies that the standard CNF encoding of
the FPHP formulas require exponential proof size in polynomial calculus
resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as
shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when
restricted to bounded-degree expanders.Comment: Full-length version of paper to appear in Proceedings of the 30th
Annual Computational Complexity Conference (CCC '15), June 201
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