Constant Depth Reducibility

We compare two types of automata for accepting picture languages to each other: the two-dimensional one-marker automaton and the sgraffito automaton. On the one hand, it is shown that deterministic sgraffito automata are strictly more powerful than deterministic two-dimensional one-marker automata. On the other hand, nondeterministic two-dimensional one-marker automata accept some picture languages that cannot be accepted by sgraffito automata. However, if nondeterministic two-dimensional one-marker automata were to accept all picture languages that are accepted by (deterministic) sgraffito automata, then the complexity classes NL (nondeterministic logarithmic space) and P (deterministic polynomial time) would coincide. Accordingly, it is likely that the classes of picture languages accepted by these two types of nondeterministic automata are incomparable under inclusion.

This paper presents a new probabilistie primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.

Groupoids are used instead of monoids to extend D.A. Barrington's (1988) successful polynomial length program over a monoid computation model to characterize complexity classes TC and LOGCFL. Further allowing groupoid families instead of fixed groupoids, deterministic and nondeterministic logarithmic space are also characterized. Several language classes arising from extended programs over Abelian monoid families are investigated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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In this paper, we study the set of thresholds that the protagonist can force in a zero-sum two-player multidimensional mean-payoff game. The set of maximal elements of such a set is called the Pareto curve, a classical tool to analyze trade-offs. As thresholds are vectors of real numbers in multiple dimensions, there exist usually an infinite number of such maximal elements. Our main results are as follow. First, we study the geometry of this set and show that it is definable as a finite union of convex sets given by linear inequations. Second, we provide a \(\varSigma _2\) P algorithm to decide if this set intersects a convex set defined by linear inequations, and we prove the optimality of our algorithm by providing a matching complexity lower bound for the problem. Furthermore, we show that, under natural assumptions, i.e. fixed number of dimensions and polynomially bounded weights in the game, the problem can be solved in deterministic polynomial time. Finally, we show that the Pareto curve can be effectively constructed, and under the former natural assumptions, this construction can be done in deterministic polynomial time.

Synthesis from examples enables non-expert users to generate programs by specifying examples of their behavior. A domain-specific form of such synthesis has been recently deployed in a widely used spreadsheet software product. In this paper we contribute to foundations of such techniques and present a complete algorithm for synthesis of a class of recursive functions defined by structural recursion over a given algebraic data type definition. The functions we consider map an algebraic data type to a string; they are useful for, e.g., pretty printing and serialization of programs and data. We formalize our problem as learning deterministic sequential top-down tree-to-string transducers with a single state (1STS). The first problem we consider is learning a tree-to-string transducer from any set of input/output examples provided by the user. We show that, given a set of input/output examples, checking whether there exists a 1STS consistent with these examples is NP-complete in general. In contrast, the problem can be solved in polynomial time under a (practically useful) closure condition that each subtree of a tree in the input/output example set is also part of the input/output examples. Because coming up with relevant input/output examples may be difficult for the user while creating hard constraint problems for the synthesizer, we also study a more automated active learning scenario in which the algorithm chooses the inputs for which the user provides the outputs. Our algorithm asks a worst-case linear number of queries as a function of the size of the algebraic data type definition to determine a unique transducer. To construct our algorithms we present two new results on formal languages. First, we define a class of word equations, called sequential word equations, for which we prove that satisfiability can be solved in deterministic polynomial time. This is in contrast to the general word equations for which the best known complexity upper bound is in linear space. Second, we close a long-standing open problem about the asymptotic size of test sets for context-free languages. A test set of a language of words L is a subset T of L such that any two word homomorphisms equivalent on T are also equivalent on L. We prove that it is possible to build test sets of cubic size for context-free languages, matching for the first time the lower bound found 20 years ago.

It is well known that among the three classes of the PRAM models, namely, CRCW, CREW, and EREW, the CRCW models are the weakest, in the sense that, they permit concurrent read/write by processors. Accordingly, algorithms on the CRCW model mainly concentrate on the core computations without much ado about data access. Consequently, this model, at least in principle, allows for the design of the fastest algorithm for a problem. It is intriguing to ask how fast prefixes can be computed on the CRCW models. Since CRCW models are equivalent to the unbounded fan-in circuits (refer to Chapter 2), the task of developing the fastest algorithms for the prefix problems is pursued in the context of the unbounded fan-in circuits. Recall from Chapter 2, that while the standard measures, such as, size and depth are still used to quantify the goodness of unbounded fanin circuits, the size of the circuit is measured by the total number of edges incident on all of its operation nodes, instead of by the number of operations nodes. It turns out that the size and depth of unbounded fanin circuits for computing prefixes, depends critically on the structure of the underlying semigroup from which the input elements are drawn. The principal result of this concluding Chapter may be stated as follows: There exists unbounded fan-in parallel prefix circuits of constant depth and polynomial size if, and only if, the underlying semigroup is group free. The proof of this result involves a very clever synthesis of a number of ideas drawn from different directions — structure of group free semigroups, their relations to a special class of regular sets, called non-counting regular sets, the relation of this latter class of regular sets to yet another class of regular sets defined by star-free regular expressions, and the design of a special class of finite state deterministic automata called RS machines that accept star-free regular expressions. In this context, it is convenient to define the notion of small circuits as the class of circuits with constant depth and polynomial size.

Finite-valued distance automata

Given an n-dimensional lattice L and some target vector, this paper studies the algorithms for approximate closest vector problem (CVPγ) by using an approximate shortest independent vectors problem oracle (SIVPγ). More precisely, if the distance between the target vector and the lattice is no larger than c \( {\scriptscriptstyle \frac{c}{\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \) for arbitrary large but finite constant c > 0, we give randomized and deterministic polynomial time algorithms to find a closest vector, while previous reductions were only known for \( {\scriptscriptstyle \frac{c}{2\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \). Moreover, if the distance between the target vector and the lattice is larger than some quantity with respect to λn(L), using SIVPγ oracle and Babai’s nearest plane algorithm, we can solve \( \mathrm{CVP}\upgamma \sqrt{n} \) in deterministic polynomial time. Specially, if the approximate factor γ ϵ (1, 2) in the SIVPγ oracle, we obtain a better reduction factor for CVP.

Optimal Depth, Very Small Size Circuits for Symmetrical Functions in AC0

We investigate the problem to decide for DFAs, NFAs and regular expressions whether they describe deterministic regular languages. For DFAs with limited alphabet size we show the NL-completeness (nondeterministic logarithmic space-completeness) of the problem. Bruggemann-Klein and Wood (Inf. Comput. 142(2), 182---206, (1998)) gave an algorithm deciding for a minimal DFA whether it recognizes a deterministic regular language. Their algorithm requires polynomial time. Based on this algorithm we construct an algorithm for those DFAs which are not necessarily minimal, but limited in the size of the alphabet. We first focus on the case that the DFAs are minimal and extend to general DFAs later on. The new algorithm is substantially modified in contrast to the original one and uses structural properties of minimal DFAs and of special subautomata of the DFAs, called orbit automata. This algorithm runs in nondeterministic logarithmic space. After that we show the NL-hardness of the problem. For NFAs and regular expressions with arbitrary alphabets the problem is shown to be in PSPACE by Czerwinski et al. (2013) (and, in fact, to be PSPACE-complete). Their approach resulted in an algorithm working with space ??(x4)$\mathcal {O}(x^{4})$. However, our approach for the NL-algorithm can be modified to get an algorithm for NFAs and regular expressions using just quadratic space.

In this paper, a global function is a function that computes a (local) function in each ordered structure of a specified vocabulary. We design algebras of global functions for a number of complexity classes for which such algebras have not been known, e.g. for the functions computable in nondeterministic logarithmic space, or in nondeterministic polynomial time. In addition, we present a functional analogue of first-order logic and give a new functional characterization of polynomial time.

We study deterministic one-way communication complexity of functions with Hankel communication matrices. In this paper some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient to consider only the communication direction from the party with the shorter input to the other party. These facts do not hold for arbitrary Boolean functions in general. Next, gaps between one-way and two-way communication complexity for symmetric Boolean functions are discussed. Finally, we give some generalizations to the case of multiple parties.

On the computational complexity of the languages of general symbolic dynamical systems and beta-shifts

The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n−Hℕ(n), where Hℕ(n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexity of symmetric functions and fractals derived from the parity of binomial coefficients.