267 research outputs found
Lower Bound Techniques for QBF Proof Systems
How do we prove that a false QBF is inded false? How big a proof is needed? The special case when all quantifiers are existential is the well-studied setting of propositional proof complexity. Expectedly, universal quantifiers change the game significantly. Several proof systems have been designed in the last couple of decades to handle QBFs. Lower bound paradigms from propositional proof complexity cannot always be extended - in most cases feasible interpolation and consequent transfer of circuit lower bounds works, but obtaining lower bounds on size by providing lower bounds on width fails dramatically. A new paradigm with no analogue in the propositional world has emerged in the form of strategy extraction, allowing for transfer of circuit lower bounds, as well as obtaining independent
genuine QBF lower bounds based on a semantic cost measure.
This talk will provide a broad overview of some of these developments
MaxSAT Resolution and Subcube Sums
We study the MaxRes rule in the context of certifying unsatisfiability. We
show that it can be exponentially more powerful than tree-like resolution, and
when augmented with weakening (the system MaxResW), p-simulates tree-like
resolution. In devising a lower bound technique specific to MaxRes (and not
merely inheriting lower bounds from Res), we define a new proof system called
the SubCubeSums proof system. This system, which p-simulates MaxResW, can be
viewed as a special case of the semialgebraic Sherali-Adams proof system. In
expressivity, it is the integral restriction of conical juntas studied in the
contexts of communication complexity and extension complexity. We show that it
is not simulated by Res. Using a proof technique qualitatively different from
the lower bounds that MaxResW inherits from Res, we show that Tseitin
contradictions on expander graphs are hard to refute in SubCubeSums. We also
establish a lower bound technique via lifting: for formulas requiring large
degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums
Small space analogues of Valiant\u27s classes and the limitations of skew formula
In the uniform circuit model of computation, the width of a boolean
circuit exactly characterises the ``space\u27\u27 complexity of the
computed function. Looking for a similar relationship in Valiant\u27s
algebraic model of computation, we propose width of an arithmetic
circuit as a possible measure of space. We introduce the class
VL as an algebraic variant of deterministic log-space L. In
the uniform setting, we show that our definition coincides with that
of VPSPACE at polynomial width.
Further, to define algebraic variants of non-deterministic
space-bounded classes, we introduce the notion of ``read-once\u27\u27
certificates for arithmetic circuits. We show that polynomial-size
algebraic branching programs can be expressed as a read-once
exponential sum over polynomials in VL, ie
.
We also show that , ie
VBPs are stable under read-once exponential sums. Further, we
show that read-once exponential sums over a restricted class of
constant-width arithmetic circuits are within VQP, and this is the
largest known such subclass of poly-log-width circuits with this
property.
We also study the power of skew formulas and show that exponential
sums of a skew formula cannot represent the determinant polynomial
On sorting by 3-bounded transpositions
AbstractHeath and Vergara [Sorting by short block moves, Algorithmica 28 (2000) 323–352] proved the equivalence between sorting by 3-bounded transpositions and sorting by correcting skips and correcting hops. This paper explores various algorithmic as well as combinatorial aspects of correcting skips/hops, with the aim of understanding 3-bounded transpositions better.We show that to sort any permutation via correcting hops and skips, ⌊n/2⌋ correcting skips suffice. We also present a tighter analysis of the 43 approximation algorithm of Heath and Vergara, and a possible simplification. Along the way, we study the class Hn of those permutations of Sn which can be sorted using correcting hops alone, and characterize large subsets of this class. We obtain a combinatorial characterization of the set Gn⊆Sn of all correcting-hop-free permutations, and describe a linear-time algorithm to optimally sort such permutations. We also show how to efficiently sort a permutation with a minimum number of correcting moves
Longest paths in Planar DAGs in Unambiguous Logspace
We show via two different algorithms that finding the length of the longest
path in planar directed acyclic graph (DAG) is in unambiguous logspace UL, and
also in the complement class co-UL. The result extends to toroidal DAGs as
well
Computing the Maximum using (min, +) Formulas
We study computation by formulas over (min,+). We consider the
computation of max{x_1,...,x_n} over N as a difference of
(min,+) formulas, and show that size n + n log n is sufficient
and necessary. Our proof also shows that any (min,+) formula
computing the minimum of all sums of n-1 out of n variables must
have n log n leaves; this too is tight. Our proofs use a
complexity measure for (min,+) functions based on minterm-like
behaviour and on the entropy of an associated graph
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