Circuit Complexity Classes Research Articles

Knapsack and the power word problem in solvable Baumslag–Solitar groups

We prove that the power word problem for certain metabelian subgroups of [Formula: see text] (including the solvable Baumslag–Solitar groups [Formula: see text]) belongs to the circuit complexity class [Formula: see text]. In the power word problem, the input consists of group elements [Formula: see text] and binary encoded integers [Formula: see text] and it is asked whether [Formula: see text] holds. Moreover, we prove that the knapsack problem for [Formula: see text] is [Formula: see text]-complete. In the knapsack problem, the input consists of group elements [Formula: see text] and it is asked whether the equation [Formula: see text] has a solution in [Formula: see text]. For the more general case of a system of so-called exponent equations, where the exponent variables [Formula: see text] can occur multiple times, we show that solvability is undecidable for [Formula: see text].

Low complexity algorithms in knot theory

Agol, Haas and Thurston showed that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. This shows that (unless P[Formula: see text]NP) the genus problem has high computational complexity even for knots in a 3-manifold. We initiate the study of classes of knots where the genus problem and even the equivalence problem have very low computational complexity. We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace[Formula: see text]. Alternating knots with some additional combinatorial structure will be referred to as standard. As expected, almost all alternating knots of a given genus are standard. We show that the genus problem for these knots belongs to [Formula: see text] circuit complexity class. We also show, that the equivalence problem for such knots with [Formula: see text] crossings has time complexity [Formula: see text] and is in Logspace[Formula: see text] and [Formula: see text] complexity classes.

OUP accepted manuscript

Separations among the first order logic Ring(0,+, ∗) of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures the logic of finite residue class rings and extensions are known to capture DLOGTIMEuniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. We further give general conditions under which a logic over the finite residue class rings has a sentence whose spectrum has no h-density. One application of this result is that Supported by MICINN project TIN2011-27479-C04-03 (BASMATI), MINECO project TIN201457226-P (APCOM) and Gen. Cat. project SGR2014-890 (MACDA).

On some derivation mechanisms and the complexity of their Szilard languages

A Szilard language is a language theoretical tool used to describe the derivation process in a formal grammar or grammar system. We investigate computational resources used by (alternating) Turing machines to accept Szilard languages of Chomsky grammars, regulated grammars and grammar systems. The results are related to the circuit complexity classes NC1 and NC2. The paper is a survey of the most important complexity results in the field, but it also brings several new insights into the parallel complexity of Szilard languages of grammar systems. We focus on parallel communicating grammar systems (PCGSs) with context-free components, and we prove that the class of Szilard languages of centralized (returning or non-returning) PCGSs is included in NC1.

Extensional Uniformity for Boolean Circuits

Imposing an extensional uniformity condition on a nonuniform circuit complexity class $\mathcal{C}$ means simply intersecting $\mathcal{C}$ with a uniform class $\mathcal{L}$. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining $\mathcal{C}$. We say that $(\mathcal{C},\mathcal{L})$ has the uniformity duality property if the extensionally uniform class $\mathcal{C}\cap\mathcal{L}$ can be captured intensionally by means of adding so-called $\mathcal{L}$-numerical predicates to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining $\mathcal{C}$. This paper exhibits positive instances and negative instances of the uniformity duality property.

Reversal complexity revisited

We study a generalized version of reversal bounded Turing machines where, apart from several tapes on which the number of head reversals is bounded by r ( n ) , there are several further tapes on which head reversals remain unrestricted, but size is bounded by s ( n ) (where n denotes the input length). Recently [M. Grohe, C. Koch, N. Schweikardt, Tight lower bounds for query processing on streaming and external memory data, Theoretical Computer Science 380 (1–2) (2007) 199–217; M. Grohe, N. Schweikardt, Lower bounds for sorting with few random accesses to external memory, in: Proc. PODS’05, ACM Press, 2005, pp. 238–249], such machines were introduced as a formalization of a computation model that restricts random access to external memory and internal memory space. Here, each of the tapes with a restriction on the head reversals corresponds to an external memory device, and the tapes of restricted size model internal memory. We use ST ( r ( n ) , s ( n ) , O ( 1 ) ) to denote the class of all problems that can be solved by deterministic Turing machines that comply to the above resource bounds. Similarly, NST ( ⋯ ) and RST ( ⋯ ) , respectively, are used for the corresponding nondeterministic and randomized classes. While previous papers focused on lower bounds for particular problems, including sorting, the set equality problem, and several query evaluation problems, the present paper addresses the relations between the (R,N) ST ( ⋯ ) -classes and classical complexity classes and investigates the structural complexity of the (R,N) ST ( ⋯ ) -classes. Our main results are (1) a trade-off between internal memory space and external memory head reversals, (2) correspondences between the (R,N) ST ( ⋯ ) classes and “classical” time-bounded, space-bounded, reversal-bounded, and circuit complexity classes, and (3) hierarchies of (R) ST ( ⋯ ) -classes in terms of increasing numbers of head reversals on external memory tapes.

Definability of Languages by Generalized First-Order Formulas over $(\mathbb{N},+)$

We consider an extension of first-order logic by modular quantifiers of a fixed modulus $q$. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1's is divisible by a prime $p$ that does not divide $q$. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class $ACC$ from $NC^1$. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.

Word Problems and Membership Problems on Compressed Words

We consider a compressed form of the word problem for finitely presented monoids, where the input consists of two compressed representations of words over the generators of a monoid M, and we ask whether these two words represent the same monoid element of M. Words are compressed using straight-line programs, i.e., context-free grammars that generate exactly one word. For several classes of finitely presented monoids we obtain completeness results for complexity classes in the range from P to EXPSPACE. As a by-product of our results on compressed word problems we obtain a fixed deterministic context-free language with a PSPACE-complete compressed membership problem. The existence of such a language was open so far. Finally, we will investigate the complexity of the compressed membership problem for various circuit complexity classes.

Dimension, entropy rates, and compression

This paper develops new relationships between resource-bounded dimension, entropy rates, and compression. New tools for calculating dimensions are given and used to improve previous results about circuit-size complexity classes. Approximate counting of SpanP functions is used to prove that the NP-entropy rate is an upper bound for dimension in Δ 3 E , the third level of the exponential-time hierarchy. This general result is applied to simultaneously improve the results of Mayordomo [E. Mayordomo, Contributions to the study of resource-bounded measure, PhD thesis, Universitat Politècnica de Catalunya, 1994] on the measure on P / poly in Δ 3 E and of Lutz [J.H. Lutz, Dimension in complexity classes, SIAM J. Comput. 32 (5) (2003) 1236–1259] on the dimension of exponential-size circuit complexity classes in ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied in conjunction with time-bounded dimension. It is shown that rankable entropy rates give upper bounds for time-bounded dimensions. We use this to improve results of Lutz [J.H. Lutz, Almost everywhere high nonuniform complexity, J. Comput. System Sci. 44 (2) (1992) 220–258] about polynomial-size circuit complexity classes from resource-bounded measure to dimension. Exact characterizations of the effective dimensions in terms of Kolmogorov complexity rates at the polynomial-space and higher levels have been established, but in the time-bounded setting no such equivalence is known. We introduce the concept of polynomial-time superranking as an extension of ranking. We show that superranking provides an equivalent definition of polynomial-time dimension. From this superranking characterization we show that polynomial-time Kolmogorov complexity rates give a lower bound on polynomial-time dimension.

Uniform constant-depth threshold circuits for division and iterated multiplication

It has been known since the mid-1980s (SIAM J. Comput. 15 (1986) 994; SIAM J. Comput. 21 (1992) 896) that integer division can be performed by poly-time uniform constant-depth circuits of Majority gates; equivalently, the division problem lies in P-uniform TC 0. Recently, this was improved to L-uniform TC 0 (RAIRO Theoret. Inform. Appl. 35 (2001) 259), but it remained unknown whether division can be performed by DLOGTIME-uniform TC 0 circuits. The DLOGTIME uniformity condition is regarded by many as being the most natural notion of uniformity to apply to small circuit complexity classes such as TC 0; DLOGTIME-uniform TC 0 is also known as FOM, because it corresponds to first-order logic with Majority quantifiers, in the setting of finite model theory. Integer division has been the outstanding example of a natural problem known to be in a P-uniform circuit complexity class, but not known to be in its DLOGTIME-uniform version. We show that indeed division is in DLOGTIME-uniform TC 0. First we show that division lies in the complexity class FOM+POW obtained by augmenting FOM with a predicate for powering modulo small primes. Then we show that the predicate POW itself lies in FOM. (In fact, it lies in FO, or DLOGTIME-uniform AC 0.) The essential idea in the fast parallel computation of division and related problems is that of Chinese remainder representation (CRR)—storing a number in the form of its residues modulo many small primes. The fact that CRR operations can be carried out in log space has interesting implications for small space classes. We define two versions of s( n) space for s(n)=o( log n) : dspace( s( n)) as the traditional version where the worktape begins blank, and DSPACE( s( n)) where the space bound is established by endmarkers before the computation starts. We present a new translational lemma characterizing the unary languages in the DSPACE classes. It is known (Theoret. Comput. Sci. 3 (1976) 213) that {0 n : n is prime} ∉ dspace( log log n) . We show that if this can be improved to {0 n : n is prime} ∉ DSPACE( log log n) , it follows that L≠NP.

Languages Defined with Modular Counting Quantifiers

We prove that a regular language defined by a boolean combination of generalized Σ 1 -sentences built using modular counting quantifiers can be defined by a boolean combination of Σ 1 -sentences in which only regular numerical predicates appear. The same statement, with “Σ 1 ” replaced by “first-order,” is equivalent to the conjecture that the nonuniform circuit complexity class ACC is strictly contained in NC 1 . The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case.

Finite semigroup varieties defined by programs

We study the regular languages recognized by polynomial-length programs over finite semigroups belonging to product varieties V ∗ LI, where V is a variety of finite monoids, and LI is the variety of finite locally trivial semigroups. In the case where the semigroup variety has a particular closure property with respect to programs, we are able to give precise characterizations of these regular languages. As a corollary we obtain new proofs of the results of Barrington, Compton, Straubing and Therien characterizing the regular languages in certain circuit complexity classes.

On the Circuit Complexity of Sigmoid Feedforward Neural Networks

This paper aims to examine the circuit complexity of sigmoid activation feedforward artificial neural networks by placing them amongst several classic Boolean and threshold gate circuit complexity classes. The starting point is the class NNk defined by Shawe-Taylor et al. (1992) Classes of feedforward neural nets and their circuit complexity. Neural Networks 5(6), 971–977. For a better characterisation, we introduce two additional classes NNkΔ and NNkΔ,ϵ having less restrictive conditions than NNk concerning fan-in and accuracy, and proceed to prove relations amongst these three classes and well established circuit complexity classes. For doing that, a particular class of Boolean functions FΔ is first introduced and we show how a threshold gate circuit can be recursively built for any fΔ belonging to FΔ. As the G-functions (computing the carries) are fΔ functions, a class of solutions is obtained for threshold gate adders. We then constructively prove the inclusions amongst circuit complexity classes. This is done by converting the sigmoid feedforward artificial neural network into an equivalent threshold gate circuit Shawe-Taylor et al. (1992). Each threshold gate is then replaced by a multiple input adder having a binary tree structure, relaxing the logarithmic fan-in condition from (Shawe-Taylor et al. 1992) to (almost) polynomial. This means that larger classes of sigmoid activation feedforward neural networks can be implemented in polynomial size Boolean circuits with a small constant fan-in at the expense of a logarithmic factor increase in the number of layers. Similar results are obtained for threshold circuits, and are liked with the previous ones. The main conclusion is that there are interesting fan-in dependent depth-size tradeoffs when trying to digitally implement sigmoid activation feedforward neural networks. Copyright © 1996 Elsevier Science Ltd

Superlinear Lower Bounds for Bounded-Width Branching Programs

We use algebraic techniques to obtain superlinear lower bounds on the size of bounded-width branching programs to solve a number of problems. In particular, we show that any bounded-width branching program computing a nonconstant threshold function has length Ω(n log log n), improving on the previous lower bounds known to apply to all such threshold functions. We also show that any program over a finite solvable monoid computing a product in a nonsolvable group has length Ω(n log log n). This result is a step toward proving the conjecture that the circuit complexity class ACC0 is properly contained in NC1.

Regular languages in NC1

We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC 0, thus answering a question of Chandra, Fortune, and Lipton. As a by-product, we are able to determine effectively whether a given regular language is in AC 0 and to solve in part an open problem originally posed by McNaughton. Using recent lower-bound results of Razborov and Smolensky, we obtain similar characterizations of the family of regular languages recognized by constant-depth circuit families that include unbounded fan-in mod p addition gates for a fixed prime p along with unbounded fan-in boolean gates. We also obtain logical characterizations for the class of all languages recognized by nonuniform circuit families in which mod m gates (where m is not necessarily prime) are permitted. Comparison of this characterization with our previous results provides evidence for a conjecture concerning the regular languages in this class. A proof of this conjecture would show that computing the bit sum modulo p, where p is a prime not dividing m, is not AC 0-reducible to addition mod m, and thus that MAJORITY is not AC 0-reducible to addition mod m.

On uniformity within NC1

In order to study circuit complexity classes within NC 1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC 1 uniformity, have appeared in recent research: Immerman's families of circuits defined by first-order formulas and a uniformity corresponding to Buss' deterministic log-time reductions. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC 1 still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Thérien, and Thomas. A preliminary version of this work appeared in “Structure of Complexity Theory: Third Annual Conference” pp. 47–59, IEEE Comput. Soc., Washington, DC, 1988.