56,822 research outputs found
Algorithms and Lower Bounds in Circuit Complexity
Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits
Symmetric Formulas for Products of Permutations
We study the formula complexity of the word problem : given -by- permutation matrices
, compute the -entry of the matrix product . An important feature of this function is that it is invariant under
action of given by This symmetry is also exhibited in the smallest known unbounded fan-in
-formulas for
, which have size .
In this paper we prove a matching lower bound for
-invariant formulas computing . This result
is motivated by the fact that a similar lower bound for unrestricted
(non-invariant) formulas would separate complexity classes and
.
Our more general main theorem gives a nearly tight lower
bound on the -invariant depth-
-formula size of
for any finite simple group whose minimum permutation
representation has degree~. We also give nearly tight lower bounds on the
-invariant depth-
-formula size in the case where
is an abelian group.Comment: ITCS 202
Permutation Games for the Weakly Aconjunctive -Calculus
We introduce a natural notion of limit-deterministic parity automata and
present a method that uses such automata to construct satisfiability games for
the weakly aconjunctive fragment of the -calculus. To this end we devise a
method that determinizes limit-deterministic parity automata of size with
priorities through limit-deterministic B\"uchi automata to deterministic
parity automata of size and with
priorities. The construction relies on limit-determinism to avoid the full
complexity of the Safra/Piterman-construction by using partial permutations of
states in place of Safra-Trees. By showing that limit-deterministic parity
automata can be used to recognize unsuccessful branches in pre-tableaux for the
weakly aconjunctive -calculus, we obtain satisfiability games of size
with priorities for weakly aconjunctive
input formulas of size and alternation-depth . A prototypical
implementation that employs a tableau-based global caching algorithm to solve
these games on-the-fly shows promising initial results
Succinct Graph Representations of ?-Calculus Formulas
Many algorithmic results on the modal mu-calculus use representations of formulas such as alternating tree automata or hierarchical equation systems. At closer inspection, these results are not always optimal, since the exact relation between the formula and its representation is not clearly understood. In particular, there has been confusion about the definition of the fundamental notion of the size of a mu-calculus formula.
We propose the notion of a parity formula as a natural way of representing a mu-calculus formula, and as a yardstick for measuring its complexity. We discuss the close connection of this concept with alternating tree automata, hierarchical equation systems and parity games. We show that well-known size measures for mu-calculus formulas correspond to a parity formula representation of the formula using its syntax tree, subformula graph or closure graph, respectively. Building on work by Bruse, Friedmann & Lange we argue that for optimal complexity results one needs to work with the closure graph, and thus define the size of a formula in terms of its Fischer-Ladner closure. As a new observation, we show that the common assumption of a formula being clean, that is, with every variable bound in at most one subformula, incurs an exponential blow-up of the size of the closure.
To realise the optimal upper complexity bound of model checking for all formulas, our main result is to provide a construction of a parity formula that (a) is based on the closure graph of a given formula, (b) preserves the alternation-depth but (c) does not assume the input formula to be clean
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