Hilbert’s Thirteenth Problem and Circuit Complexity

Lecture Notes in Computer Science, 1999

Constant-depth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC 0 and GapAC 0. These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC 0 (where many lower bounds are known) and TC 0 (where essentially no lower bounds are known). In this paper, we resolve several questions regarding the closure properties of #AC 0 and GapAC 0. Counting classes are usually characterized in terms of problems of counting paths in a class of graphs (simple paths in directed or undirected graphs for #P, simple paths in directed acyclic graphs for #L, or paths in bounded-width graphs for GapNC 1). It was shown in [BLMS98] that complete problems for depth k Boolean AC 0 can be obtained by considering the reachability problem for width k grid graphs. It would be tempting to conjecture that #AC 0 could be characterized by counting paths in bounded-width grid graphs. We disprove this, but nonetheless succeed in characterizing #AC 0 by counting paths in another family of bounded-width graphs.

Journal of Computer and System Sciences, 1981

We argue that uniform circuit complexity introduced by Borodin is a reasonable model of parallel complexity. Three main results are presented. First, we show that alternating Turing machines are also a surprisingly good model of parallel complexity, by showing that simultaneous size/depth of uniform circuits is the same as space/time of alternating Turing machines, with depth and time within a constant factor and likewise log(size) and space. Second, we apply this to characterize NC, the class of polynomial size and polynomial-in-log depth circuits, in terms of tree-size bounded alternating TMs and other models. In particular, this enables us to show that context-free language recognition is in NC. Third, we investigate various definitions of uniform circuit complexity, showing that it is fairly insensitive to the choice of definition.

Journal of Computer and System Sciences, 1998

We show that all sets that are complete for NP under non-uniform AC 0 reductions are isomorphic under non-uniform AC 0 -computable isomorphisms. Furthermore, these sets remain NP-complete even under non-uniform NC 0 reductions.

1997

Journal of Computer and System Sciences, 2000

Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC 1 [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC 1 [CMTV96], we study the class of functions #AC 0. One way to define #AC 0 is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. In contrast to the preceding * Part of this research was done while visiting the University of Ulm under an Alexander von Humboldt Fellowship.

2021

In this thesis, we study the descriptive complexity of counting classes based on Boolean circuits. In descriptive complexity, the complexity of problems is studied in terms of logics required to describe them. The focus of research in this area is on identifying logics that can express exactly the problems in specific complexity classes. For example, problems are definable in ESO, existential second-order logic, if and only if they are in NP, the class of problems decidable in nondeterministic polynomial time. In the computation model of Boolean circuits, individual circuits have a fixed number of inputs. Circuit families are used to allow for an arbitrary number of input bits. A priori, the circuits in a family are not uniformly described, but one can impose this as an additional condition, e.g., requiring that there is an algorithm constructing them. For any circuit there is a function counting witnesses (or proofs) for the circuit producing the output 1. Consequently, any class o...

Theory of Computing Systems / Mathematical Systems Theory, 2007

Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, what Boolean functions can be defined depends on these gate functions: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes have been known since the 1920s, their computational complexity was never investigated. In this paper we will study how difficult it is to decide for a Boolean function f and a class B, whether f is in B. Moreover, we will provide such a decision algorithm with additional information: How difficult is it to decide whether or not f is in B, provided we already know a circuit for f, but with gates from another class A? Given such a circuit, we know that f is in A. Is the problem harder if we do not have a concrete representation for f, but still know that it is from A? For nearly all possible combinations, we show that this is not the case, and that the problem is either in P or coNP-complete.

Information Processing Letters, 2005

In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boolean function on n input bits. The best known bounds appear to be However, the bounds do not seem to be explicitly stated in the literature. We give a simple direct elementary proof of the lower bound valid for the full binary basis, and we give an explicit proof of the upper bound valid for the basis {¬, ∧, ∨}.

Mathematical Systems Theory, 1994

It is well-known that the class P=poly can be characterized in terms of polynomial size circuits. We obtain a characterization of the class P= log using polynomial size circuits with low resource-bounded Kolmogorov Complexity. The concept of \small circuits with easy descriptions" has been introduced in the literature as a candidate to characterizing P= log. We prove that this concept corresponds exactly to the class P=O(log n log(log n)), and that this is di erent from P= log. Generalizations of this result are also obtained.

Proceedings of the twenty-sixth annual ACM …, 1994

We investigate the computational power of depth-2 circuits consisting of MOD' gates at the bottom and a threshold gate with arbitrary weights at the top (for short, threshold-MOD' circuits) and circuits with two levels of MOD gates (MODp-MOD4 circuits). In particular, we will show the following results: (i) For all prime numbers p and integers q,r, it holds that if p divides r but not q then all threshold-MOD4 circuits for MOD' have exponentially many nodes. (ii) For all integers r, all problems computable by depth-2 {AND,OR,NOT} circuits of polynomial size have threshold-MOD' circuits with polynomially many edges. (iii) There is a problem computable by depth 3 {AND, OR, NOT} circuits of linear size and constant bottom fan-in which for all r needs threshold-MOD' circuits with exponentially many nodes.

2009

We study the problem of polynomial identity testing (PIT) in arithmetic circuits. is is a fundamental problem in computational algebra and has been very well studied in the past few decades. Despite many ešorts, a deterministic polynomial time algorithm is known only for restricted circuits of depth 3. A recent result of Agrawal and Vinay show that PIT for depth 4 circuit is almost as hard as the general case, and hence explains why there is no progress beyond depth 3. e main contribution of this thesis is a new approach to designing a polynomial time algorithm for depth 3 circuits. We rst provide the background and related results to motivate the problem. We discuss the connections of PIT with arithmetic circuit lower bounds, and also briežy the results in depth reduction. We then look at the deterministic algorithms for PIT on restricted circuits. We then proceed to the main contribution of the thesis which studies the power of arithmetic circuits over higher dimensional algebras...

2007

The classes of the W-hierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. The analysis of the parameterized majority vertex cover problem and other parameterized problems led us to study circuits that contain connectives such as majority, not-all-equal, and unique, instead of (or in addition to) the Boolean connectives. For example, a gate labelled by the majority connective outputs TRUE if more than half of the inputs are TRUE. For any finite set C of connectives we construct the corresponding W(C)-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W(C)-hierarchy coincide levelwise. Surprisingly, if C contains only the majority connective (i.e., no boolean connectives), then the first levels coincide. We use this to show that the majority vertex cover problem is W[1]-complete.

Lecture Notes in Computer Science, 2012

Spira [36] showed that any Boolean formula of size s can be simulated in depth O(log s). We generalize Spira's theorem and show that any Boolean circuit of size s with segregators of size f (s) can be simulated in depth O(f (s) log s). If the segregator size is at least s ε for some constant ε > 0, then we can obtain a simulation of depth O(f (s)). This improves and generalizes a simulation of polynomial-size Boolean circuits of constant treewidth k in depth O(k 2 log n) by Jansen and Sarma [21]. Since the existence of small balanced separators in a directed acyclic graph implies that the graph also has small segregators, our results also apply to circuits with small separators. Our results imply that the class of languages computed by non-uniform families of polynomial-size circuits that have constant size segregators equals non-uniform N C 1. Considering space bounded Turing machines to generate the circuits, for f (s) log 2 sspace uniform families of Boolean circuits our small-depth simulations are also f (s) log 2 sspace uniform. As a corollary, we show that the Boolean Circuit Value problem for circuits with constant size segregators (or separators) is in deterministic SP ACE(log 2 n). Our results also imply that the Planar Circuit Value problem, which is known to be P-Complete [19], is in SP ACE(√ n log n). We also show that the Layered Circuit Value and Synchronous Circuit Value problems, which are both P-complete [20], are in SP ACE(√ n).

2008

Algebraic circuits combine operations drawn from an algebraic system. In this chapter we develop algebraic and combinatorial circuits for a variety of generally non-Boolean problems, including multiplication and inversion of matrices, convolution, the discrete Fourier transform, and sorting networks. These problems are used primarily to illustrate concepts developed in later chapters, so that this chapter may be used for reference when studying those chapters. For each of the problems examined here the natural algorithms are straight-line and the graphs are directed and acyclic; that is, they are circuits. Not only are straight-line algorithms the ones typically used for these problems, but in some cases they are the best possible. The quality of the circuits developed here is measured by circuit size, the number of circuit operations, and circuit depth, the length of the longest path between input and output vertices. Circuit size is a measure of the work necessary to execute the c...

Information and Computation, 1987

A finite function f is a mapping of {1 , 2 ,. .. , m } into {1 , 2 ,. .. , m } ∪ { # } where # is a symbol to be thought of as ''undefined.'' This paper defines a measure M(f) of the difficulty of inverting a finite function f, which is given by M(f) = MIN    log 2 C(f) log 2 C(g) _ _________ : g an inverse of f    where C(f) is a circuit complexity measure of the difficulty of computing f. We say that one-way functions exist (in a circuit complexity sense) if and only if M(f) is unbounded. We prove that one-way functions exist if and only if the satisfiability problem SAT has polynomial sized circuits. This paper also defines an analogous measure M d (f) in which only circuits of depth ≤ d are allowed. We show that one-way functions exist in this bounded-depth circuit complexity model, by showing for the permutations σ n on {1 , 2 ,. .. , 2 n } defined by σ n (k) ≡ 3k (mod 2 n) that for d ≥ 4 there is a positive constant c d such that M d (σ n) > c d log n as n → ∞.