2,477 research outputs found
Lower Bounds for (Non-Monotone) Comparator Circuits
Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make "copies" of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and Boolean formulas. In this paper we prove the first superlinear lower bounds in the general (non-monotone) version of this model for an explicitly defined function. More precisely, we prove that the n-bit Element Distinctness function requires ?((n/ log n)^(3/2)) size comparator circuits
Proof complexity of positive branching programs
We investigate the proof complexity of systems based on positive branching
programs, i.e. non-deterministic branching programs (NBPs) where, for any
0-transition between two nodes, there is also a 1-transition. Positive NBPs
compute monotone Boolean functions, just like negation-free circuits or
formulas, but constitute a positive version of (non-uniform) NL, rather than P
or NC1, respectively.
The proof complexity of NBPs was investigated in previous work by Buss, Das
and Knop, using extension variables to represent the dag-structure, over a
language of (non-deterministic) decision trees, yielding the system eLNDT. Our
system eLNDT+ is obtained by restricting their systems to a positive syntax,
similarly to how the 'monotone sequent calculus' MLK is obtained from the usual
sequent calculus LK by restricting to negation-free formulas.
Our main result is that eLNDT+ polynomially simulates eLNDT over positive
sequents. Our proof method is inspired by a similar result for MLK by Atserias,
Galesi and Pudl\'ak, that was recently improved to a bona fide polynomial
simulation via works of Je\v{r}\'abek and Buss, Kabanets, Kolokolova and
Kouck\'y. Along the way we formalise several properties of counting functions
within eLNDT+ by polynomial-size proofs and, as a case study, give explicit
polynomial-size poofs of the propositional pigeonhole principle.Comment: 31 pages, 5 figure
Model Checking CTL is Almost Always Inherently Sequential
The model checking problem for CTL is known to be P-complete (Clarke,
Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of
CTL obtained by restricting the use of temporal modalities or the use of
negations---restrictions already studied for LTL by Sistla and Clarke (1985)
and Markey (2004). For all these fragments, except for the trivial case without
any temporal operator, we systematically prove model checking to be either
inherently sequential (P-complete) or very efficiently parallelizable
(LOGCFL-complete). For most fragments, however, model checking for CTL is
already P-complete. Hence our results indicate that, in cases where the
combined complexity is of relevance, approaching CTL model checking by
parallelism cannot be expected to result in any significant speedup. We also
completely determine the complexity of the model checking problem for all
fragments of the extensions ECTL, CTL+, and ECTL+
Arithmetic Circuit Lower Bounds via MaxRank
We introduce the polynomial coefficient matrix and identify maximum rank of
this matrix under variable substitution as a complexity measure for
multivariate polynomials. We use our techniques to prove super-polynomial lower
bounds against several classes of non-multilinear arithmetic circuits. In
particular, we obtain the following results :
As our main result, we prove that any homogeneous depth-3 circuit for
computing the product of matrices of dimension requires
size. This improves the lower bounds by Nisan and
Wigderson(1995) when .
There is an explicit polynomial on variables and degree at most
for which any depth-3 circuit of product dimension at most
(dimension of the space of affine forms feeding into each
product gate) requires size . This generalizes the lower bounds
against diagonal circuits proved by Saxena(2007). Diagonal circuits are of
product dimension 1.
We prove a lower bound on the size of product-sparse
formulas. By definition, any multilinear formula is a product-sparse formula.
Thus, our result extends the known super-polynomial lower bounds on the size of
multilinear formulas by Raz(2006).
We prove a lower bound on the size of partitioned arithmetic
branching programs. This result extends the known exponential lower bound on
the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
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