The Unit Gap: How Sharing Works in Boolean Circuits
Kirill Krinkin
备注：
Withdrawn: Theorem 2 (the Unit Gap Theorem) is false. For 3-variable parity Khrapchenko forces >= 9 formula leaves (>= 8 gates), yet a 6-gate circuit computes it, so gap >= 2; the true value is 3. The Section 2 recursion uses the DAG measure opt in the children where formula size needs tree. Counterexample and certificates: L. A. Busnello, doi:10.5281/zenodo.21630762. Full notice in v3
We study the gap between the minimum size of a Boolean circuit (DAG) and the minimum size of a formula (tree circuit) over the And-Inverter Graph (AIG) basis {AND, NOT} with free inversions. We prove that this gap is always 0 or 1 (Unit Gap Theorem), that sharing requires opt(f) >= n essential variables (Threshold Theorem), and that no sharing is needed when opt(f) <= 3 (Tree Theorem). Gate counts in optimal circuits satisfy an exact decomposition formula with a binary sharing term. When the gap equals 1, it arises from exactly one gate with fan-out 2, employing either dual-polarity or same-polarity reuse; we prove that no other sharing structure can produce a unit gap.
In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is called the inversion complexity of $f$. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that $\lceil \log_2(n+1) \rceil$ NOT gates are sufficient to compute any Boolean function on $n$ variables. As far as we know, no result is known for inversion complexity in Boolean formulas, i.e., the minimum number of NOT…
Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class 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 C. We say that (C,L) has the "Uniformity Duality Property" if the extensionally uniform class C \cap L can be captured intensionally by means of adding so-called "L-numerical predicates" to the first-order descriptive complexity apparatus describing the connection language of the circuit family def…
We study Boolean circuits as a representation of Boolean functions and consider different equivalence, audit, and enumeration problems. For a number of restricted sets of gate types (bases) we obtain efficient algorithms, while for all other gate types we show these problems are at least NP-hard.
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connecti…
The problem of constructing hazard-free Boolean circuits (those avoiding electronic glitches) dates back to the 1940s and is an important problem in circuit design and even in cybersecurity. We show that a DeMorgan circuit is hazard-free if and only if the circuit produces (purely syntactically) all prime implicants as well as all prime implicates of the Boolean function it computes. This extends to arbitrary DeMorgan circuits a classical result of Eichelberger [IBM J. Res. Develop., 9 (1965)] showing this property for special depth-two circuits. Via an amazingly simple proof, we also strength…
In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and $\textrm{AC}^1$ as well as their arithmetic counterparts $\#\textrm{NC}^1$, $\#\textrm{SAC}^1$ and $\#\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\textrm{AC}^0 = \textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering sli…
Using a proofs-as-programs correspondence, Terui was able to compare two models of parallel computation: Boolean circuits and proof nets for multiplicative linear logic. Mogbil et. al. gave a logspace translation allowing us to compare their computational power as uniform complexity classes. This paper presents a novel translation in AC0 and focuses on a simpler restricted notion of uniform Boolean proof nets. We can then encode constant-depth circuits and compare complexity classes below logspace, which were out of reach with the previous translations.
Computer Science Department, University of California, Los Angeles（加州大学洛杉矶分校计算机科学系）
Tractable Boolean and arithmetic circuits have been studied extensively in AI for over two decades now. These circuits were initially proposed as "compiled objects," meant to facilitate logical and probabilistic reasoning, as they permit various types of inference to be performed in linear-time and a feed-forward fashion like neural networks. In more recent years, the role of tractable circuits has significantly expanded as they became a computational and semantical backbone for some approaches that aim to integrate knowledge, reasoning and learning. In this article, we review the foundations …
Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate the analysis of optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS based certificates remain valid: First, for polynomials, which are nonnegative over the $n$…
We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of $\mathsf{AND}$, $\mathsf{OR}$, and $\mathsf{NOT}$ gates. Our hierarchy theorem says that for every $d \geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ circuit that agrees with $f$ on $(1/2 + o_n(1))$ fraction of all inputs must have size $\exp({n^{Ω(1/d)}}).$ This answers an open question posed by Håstad in his Ph.D. thesis. Our average-case depth hierarchy theorem implies that the polynomial hierarchy…
Fungal automata are a variation of the two-dimensional sandpile automaton of Bak, Tang, and Wiesenfeld (Phys. Rev. Lett. 1987). In each step toppling cells emit grains only to some of their neighbors chosen according to a specific update sequence. We show how to embed any Boolean circuit into the initial configuration of a fungal automaton with update sequence $HV$. In particular we give a constructor that, given the description $B$ of a circuit, computes the states of all cells in the finite support of the embedding configuration in $O(\log |B|)$ space. As a consequence the prediction problem…
We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits …
We study the circuit complexity of boolean functions in a certain infinite basis. The basis consists of all functions that take value $1$ on antichains over the boolean cube. We prove that the circuit complexity of the parity function and the majority function of $n$ variables in this basis is $\lfloor \frac{n+1}{2} \rfloor$ and $\left\lfloor \frac{n}{2} \right \rfloor +1$ respectively. We show that the asymptotic of the maximum complexity of $n$-variable boolean functions in this basis equals $n.$
IMFD Chile（智利基础数据研究所）；Inria Lille – Nord Europe（法国国家信息与自动化研究所里尔-北欧分部）；UMR 9189 CRIStAL（CRIStAL联合研究单元）
Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAP-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean mo…
Random instances of feedforward Boolean circuits are studied both analytically and numerically. Evaluating these circuits is known to be a P-complete problem and thus, in the worst case, believed to be impossible to perform, even given a massively parallel computer, in time much less than the depth of the circuit. Nonetheless, it is found that for some ensembles of random circuits, saturation to a fixed truth value occurs rapidly so that evaluation of the circuit can be accomplished in much less parallel time than the depth of the circuit. For other ensembles saturation does not occur and circ…
University of Waterloo（滑铁卢大学）；Massachusetts Institute of Technology（麻省理工学院）；Columbia University（哥伦比亚大学）
Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. O…
We continue to study the notion of cancellation-free linear circuits. We show that every matrix can be computed by a cancellation- free circuit, and almost all of these are at most a constant factor larger than the optimum linear circuit that computes the matrix. It appears to be easier to prove statements about the structure of cancellation-free linear circuits than for linear circuits in general. We prove two nontrivial superlinear lower bounds. We show that a cancellation-free linear circuit computing the $n\times n$ Sierpinski gasket matrix must use at least 1/2 n logn gates, and that this…
Generalized circuits are an important tool in the study of the computational complexity of equilibrium approximation problems. However, in this paper, we reveal that they have a conceptual flaw, namely that the solution concept is not monotonic. By this we mean that if $\varepsilon < \varepsilon'$, then an $\varepsilon$-approximate solution for a certain generalized circuit is not necessarily also an $\varepsilon'$-approximate solution. The reason for this non-monotonicity is the way Boolean operations are modeled. We illustrate that non-monotonicity creates subtle technical issues in prior wo…
We study properties of relational structures such as graphs that are decided by families of Boolean circuits. Circuits that decide such properties are necessarily invariant to permutations of the elements of the input structures. We focus on families of circuits that are symmetric, i.e., circuits whose invariance is witnessed by automorphisms of the circuit induced by the permutation of the input structure. We show that the expressive power of such families is closely tied to definability in logic. In particular, we show that the queries defined on structures by uniform families of symmetric B…
We consider the power of Boolean circuits with MOD$_{6}$ gates. First, we introduce a few basic notions of computational complexity, and describe the standard models with which we study the complexity of problems. We then define the model of Boolean circuits, equate a restricted class of circuits with an algebraic model, and present some results from working with this algebra.
Any monotone Boolean circuit computing the $n$-dimensional Boolean convolution requires at least $n^2$ and-gates. This precisely matches the obvious upper bound.
$\newcommand{\EC}{\mathsf{EC}}\newcommand{\KW}{\mathsf{KW}}\newcommand{\DT}{\mathsf{DT}}\newcommand{\psens}{\mathsf{psens}} \newcommand{\calB}{\cal B} $ For a Boolean function $f:\{0,1\}^n \to \{0,1\}$ computed by a circuit $C$ over a finite basis $\mathcal{B}$, the energy complexity of $C$ (denoted by $\EC_{\calB}(C)$) is the maximum over all inputs $\{0,1\}^n$ the numbers of gates of the circuit $C$ (excluding the inputs) that output a one. Energy Complexity of a Boolean function over a finite basis $\calB$ denoted by $\EC_\calB(f):= \min_C \EC_{\calB}(C)$ where $C$ is a circuit over $\calB$…
The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free …
We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising $p$-spin glass model, and (b) finding a large independent set in a sparse Erdős-Rényi graph. The following families of algorithms are considered: (a) low-degree polynomials of the input; (b) low-depth Boolean circuits; (c) the Langevin dynamics algorithm. We show that these families of algorithms fail to produce nearly optimal solutions with high probability. For the case of Boolean cir…
We consider boolean circuits computing n-operators f:{0,1}^n --> {0,1}^n. As gates we allow arbitrary boolean functions; neither fanin nor fanout of gates is restricted. An operator is linear if it computes n linear forms, that is, computes a matrix-vector product y=Ax over GF(2). We prove the existence of n-operators requiring about n^2 wires in any circuit, and linear n-operators requiring about n^2/\log n wires in depth-2 circuits, if either all output gates or all gates on the middle layer are linear.
Measures of circuit complexity are usually analyzed to ensure the computation of Boolean functions with economy and efficiency. One of these measures is energy complexity, which is related to the number of gates that output true in a circuit for an assignment. The idea behind energy complexity comes from the counting of `firing' neurons in a natural neural network. The initial model is based on threshold circuits, but recent works also have analyzed the energy complexity of traditional Boolean circuits. In this work, we discuss the time complexity needed to compute the best-case energy complex…
In a previous paper we have suggested a number of ideas to attack circuit size complexity with cohomology. As a simple example, we take circuits that can only compute the AND of two inputs, which essentially reduces to SET COVER. We show a very special case of the cohomological approach (one particular free category, using injective and superskyscraper sheaves) gives the linear programming bound coming from the relaxation of the standard integer programming reformulation of SET COVER.
We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called $ΣΠΣ$ circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Combining this with known coding theory results, we show that functions with $n$ inputs and $n$ outputs with the highest possible nonlinearity must have at least $2.32n$ AND gates. We further show that one cannot prove stronger lower boun…
We show that derandomizing polynomial identity testing over an arbitrary finite field implies that NEXP does not have polynomial size boolean circuits. In other words, for any finite field F(q) of size q, $PIT_q\in NSUBEXP\Rightarrow NEXP\not\subseteq P/poly$, where $PIT_q$ is the polynomial identity testing problem over F(q), and NSUBEXP is the nondeterministic subexpoential time class of languages. Our result is in contract to Kabanets and Impagliazzo's existing theorem that derandomizing the polynomial identity testing in the integer ring Z implies that NEXP does have polynomial size boolea…
It is already shown that a Boolean function for a NP-complete problem can be computed by a polynomial-sized circuit if its variables have enough number of automorphisms. Looking at this previous study from the different perspective gives us the idea that the small number of automorphisms might be a barrier for a polynomial time solution for NP-complete problems. Here I show that by interpreting a Boolean circuit as a graph, the small number of graph automorphisms and the large number of subgraph automorphisms in the circuit establishes the exponential circuit lower bound for NP-complete proble…
In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However the problem for nilpotent (which had not been shown to be NP-hard) but not supernilpotent algebras (which had been shown to be polynomial time) remained open. In this paper we provide a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circui…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebra…
We give a simple construction of $n\times n$ Boolean matrices with $Ω(n^{4/3})$ zero entries that are free of $2 \times 2$ all-zero submatrices and have covering number $O(\log^4(n))$. This construction provides an explicit counterexample to a conjecture of Pudlák, Rödl and Savický and Research Problems 1.33, 4.9, 11.17 of Jukna [Boolean function complexity]. These conjectures were previously refuted by Katz using a probabilistic construction.
We construct reversible Boolean circuits efficiently simulating reversible Turing machines. Both the circuits and the simulation proof are rather simple. Then we give a fairly straightforward generalization of the circuits and the simulation proof to the quantum case.
Efficient solutions to NP-complete problems would significantly benefit both science and industry. However, such problems are intractable on digital computers based on the von Neumann architecture, thus creating the need for alternative solutions to tackle such problems. Recently, a deterministic, continuous-time dynamical system (CTDS) was proposed (Nat.Phys. {\bf 7}(12), 966 (2011)) to solve a representative NP-complete problem, Boolean Satisfiability (SAT). This solver shows polynomial analog time-complexity on even the hardest benchmark $k$-SAT ($k \geq 3$) formulas, but at an energy cost …
In this work we investigate into energy complexity, a Boolean function measure related to circuit complexity. Given a circuit $\mathcal{C}$ over the standard basis $\{\vee_2,\wedge_2,\neg\}$, the energy complexity of $\mathcal{C}$, denoted by $\mathrm{EC}(\mathcal{C})$, is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function $f$, denoted by $\mathrm{EC}(f)$, is the minimum of $\mathrm{EC}(\mathcal{C})$ over all circuits $\mathcal{C}$ computing $f$. This concept has attracted lots of attention in literature. Recently, Dinesh, Otiv, and Sar…
Unrestricted Boolean Multiplicative Complexity of Four-Term Binary Polynomial Multiplication: Rational Places, Hasse Jets, and the Failure of Nonlinear Feedback
We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the decomposition enforces the same level of consistency as a specialized propagation algorithm. We prove that a constraint propagator has a a polynomial size decomposition if and only if it can be computed by a polynomial size monotone Boolean circuit. Lower bounds on the size of monotone Boolean circuits thus translate to lower bounds on the size of decompositi…
Embedding arbitrary Boolean circuits into fungal automata with arbitrary update sequences
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学工程与理学院）；Aalto University（阿尔托大学）；Facultad de Ingeniería y Ciencias, Universidad de Chile（智利大学工程与理学院）；Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）
Detecting and eliminating logic hazards in Boolean circuits is a fundamental problem in logic circuit design. We show that there is no $O(3^{(1-ε)n} \text{poly}(s))$ time algorithm, for any $ε> 0$, that detects logic hazards in Boolean circuits of size $s$ on $n$ variables under the assumption that the strong exponential time hypothesis is true. This lower bound holds even when the input circuits are restricted to be formulas of depth four. We also present a polynomial time algorithm for detecting $1$-hazards in DNF (or, $0$-hazards in CNF) formulas. Since $0$-hazards in DNF (or, $1$-hazards i…
Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. This framework paves the way for obtaining new results on error-rates, function-depth and sensitivity, and their dependence on the gate-type and noise model used.
The class P is in fact a proper sub-class of NP. We explore topological properties of the Hamming space 2^[n] where [n]={1, 2,..., n}. With the developed theory, we show: (i) a theorem that is closely related to Erdos and Rado's sunflower lemma, and claims a stronger statement in most cases, (ii) a new approach to prove the exponential monotone circuit complexity of the clique problem, (iii) NC \ne NP through the impossibility of a Boolean circuit with poly-log depth to compute cliques, based on the construction of (ii), and (iv) P \ne NP through the exponential circuit complexity of the cliqu…
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by Fürer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg n \cdot 4^{\lg^*n})$, where $\lg^*n$ is the very slowly growing iterated logarithm. In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size $Ω(n \lg n)$, thus almost completely settling the complexity of multiplication circuits. We additionally revisit …
We investigate computing models that are presented as families of finite computing devices with a uniformity condition on the entire family. Examples of such models include Boolean circuits, membrane systems, DNA computers, chemical reaction networks and tile assembly systems, and there are many others. However, in such models there are actually two distinct kinds of uniformity condition. The first is the most common and well-understood, where each input length is mapped to a single computing device (e.g. a Boolean circuit) that computes on the finite set of inputs of that length. The second, …
The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Sigma-2-complete under Turing reductions, but restricted versions are tractable. We study the complexity of minimization for formulas in two established frameworks for restricted propositional logic: The Post framework allowing arbitrarily nested formulas over a set of Boolean connectors, and the constraint setting, allowing generalizations of CNF formulas. In the Post case, we obtain a dichotomy result: Minimization is solvable in…
