Abstract: In this article we introduce a method of constructing binary linear codes and computing their weights by means of Boolean functions arising from mathematical objects called simplicial complexes. Inspired by Adamaszek (Am Math Mon 122:367–370, 2015) we introduce n-variable generating functions associated with simplicial complexes and derive explicit formulae. Applying the construction (Carlet in Finite Field Appl 13:121–135, 2007; Wadayama in Des Codes Cryptogr 23:23–33, 2001) of binary linear codes to Boolean functions arising from simplicial complexes, we obtain a class of optimal linear codes and a class of minimal linear codes.

Abstract: We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time decision procedure for checking the equivalence of Clifford group circuits. We evaluate our methods by performing automated verification of optimized Clifford+T circuits with up to 100 qubits and thousands of T gates, as well as the functional verification of quantum algorithms using hundreds of qubits. Our experiments culminate in the automated verification of the Hidden Shift algorithm for a class of Boolean functions in a fraction of the time it has taken recent algorithms to simulate.

Abstract: Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli $Z$ operators (Ising spin operators) with the terms of the sum corresponding to the function's Fourier expansion. For many classes of functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses. We give composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks. We apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results. A primary goal of this paper is to provide a $\textit{design toolkit for quantum optimization}$ which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to demystify the various constructions appearing in the literature.

Abstract: In addition to their applications in data communication and storage, linear codes also have nice applications in combinatorics and cryptography. Minimal linear codes, a special type of linear codes, are preferred in secret sharing. In this paper, a necessary and sufficient condition for a binary linear code to be minimal is derived. This condition enables us to obtain three infinite families of minimal binary linear codes with $w_{\min }/w_{\max } \leq 1/2$ from a generic construction, where $w_{\min }$ and $w_{\max }$ , respectively, denote the minimum and maximum nonzero weights in a code. The weight distributions of all these minimal binary linear codes are also determined.

Abstract: We give a simple, fast algorithm for hyperparameter optimization inspired by techniques from the analysis of Boolean functions. We focus on the high-dimensional regime where the canonical example is training a neural network with a large number of hyperparameters. The algorithm --- an iterative application of compressed sensing techniques for orthogonal polynomials --- requires only uniform sampling of the hyperparameters and is thus easily parallelizable. Experiments for training deep neural networks on Cifar-10 show that compared to state-of-the-art tools (e.g., Hyperband and Spearmint), our algorithm finds significantly improved solutions, in some cases better than what is attainable by hand-tuning. In terms of overall running time (i.e., time required to sample various settings of hyperparameters plus additional computation time), we are at least an order of magnitude faster than Hyperband and Bayesian Optimization. We also outperform Random Search 8x. Additionally, our method comes with provable guarantees and yields the first improvements on the sample complexity of learning decision trees in over two decades. In particular, we obtain the first quasi-polynomial time algorithm for learning noisy decision trees with polynomial sample complexity.

Abstract: An explicit quantum design of AES-128 is presented in this paper. The design is structured to utilize the lowest number of qubits. First, the main components of AES-128 are designed as quantum circuits and then combined to construct the quantum version of AES-128. Some of the most efficient approaches in classical hardware implementations are adopted to construct the circuits of the multiplier and multiplicative inverse in $${\mathbb {F}}_{2}[x]/(x^8+x^4+x^3+x+1)$$F2[x]/(x8+x4+x3+x+1). The results show that 928 qubits are sufficient to implement AES-128 as a quantum circuit. Moreover, to maintain the key uniqueness when the quantum AES-128 is employed as a Boolean function within a Black-box in other key searching quantum algorithms, a method with a cost of 930 qubits is also proposed.

Abstract: A Boolean network is a finite-state discrete-time dynamical system. At each step, each variable takes a value from a binary set. The value update rule for each variable is a local function which depends only on a selected subset of variables. Boolean networks have been used in modeling gene regulatory networks. In this paper, we focus on a special class of Boolean networks, namely, the conjunctive Boolean networks (CBNs), whose value update rule is comprised of only logic AND operations. It is known that any trajectory of a Boolean network will enter a periodic orbit. Periodic orbits of a CBN have been completely understood. In this paper, we investigate the orbit-controllability and state-controllability of a CBN: We ask the question of how one can steer a CBN to enter any periodic orbit or to reach any final state, from any initial state. We establish necessary and sufficient conditions for a CBN to be orbit-controllable and state-controllable. Furthermore, explicit control laws are presented along the analysis.

Abstract: Binary decision diagrams provide a data structure for representing and manipulating Boolean functions in symbolic form. They have been especially effective as the algorithmic basis for symbolic model checkers. A binary decision diagram represents a Boolean function as a directed acyclic graph, corresponding to a compressed form of decision tree. Most commonly, an ordering constraint is imposed among the occurrences of decision variables in the graph, yielding ordered binary decision diagrams (OBDD). Representing all functions as OBDDs with a common variable ordering has the advantages that (1) there is a unique, reduced representation of any function, (2) there is a simple algorithm to reduce any OBDD to the unique form for that function, and (3) there is an associated set of algorithms to implement a wide variety of operations on Boolean functions represented as OBDDs. Recent work in this area has focused on generalizations to represent larger classes of functions, as well as on scaling implementations to handle larger and more complex problems.

Abstract: We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [-1, 1]n. Next, we use a fractional pseudorandom generator as steps of a random walk in [-1, 1]n that converges to {-1, 1}n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.

Abstract: Golay complementary sets (GCSs) have been proposed to reduce the peak-to-average power ratios (PAPRs) in orthogonal frequency-division multiplexing (OFDM). They have upper bounds depending on the set sizes. The constructions of GCSs based on generalized Boolean functions have been proposed in the literature. However, most of these constructed GCSs have limited lengths, and hence they are not feasible for practical OFDM communication systems. This letter proposes a new construction of GCSs with flexible lengths. The proposed construction is a direct construction based on generalized Boolean functions. In addition, the constructed GCSs have various constellation sizes, set sizes, and upper bounds on PAPR.

Abstract: Memristor is considered as a promising circuit element which can be used in many applications. Various synthesis methods for Boolean functions have been explored in the literature using memristor-based design styles. Memristor crossbar is considered as one of the most preferred structures for implementing logic functions as well as memory. In this paper, a general synthesis flow has been proposed using the MAGIC logic design style to map multioutput Boolean functions to memristor crossbars. The functions are realized as a netlist of NOR and NOT gates. Two alternate methods of evaluating the gates are used, serial and parallel, which give a tradeoff between the number of cycles and the size of the crossbar. A strategy for scheduling the gates to time steps has also been proposed to reduce the hardware overhead. The switching delays and energy requirements are estimated using SPICE simulation. Synthesis results are reported for ISCAS’85 benchmark functions that show an average reduction of 68.8% in the number of cycles, 52.8% in energy consumption, and 96.4% in the number of memristors required as compared to a very recently published work.

Abstract: The goal of this article is to identify fundamental limitations on how efficiently algorithms implemented on platforms such as MapReduce and Hadoop can compute the central problems in motivating application domains, such as graph connectivity problems. We introduce an abstract model of massively parallel computation, where essentially the only restrictions are that the “fan-in” of each machine is limited to s bits, where s is smaller than the input size n, and that computation proceeds in synchronized rounds, with no communication between different machines within a round. Lower bounds on the round complexity of a problem in this model apply to every computing platform that shares the most basic design principles of MapReduce-type systems. We prove that computations in our model that use few rounds can be represented as low-degree polynomials over the reals. This connection allows us to translate a lower bound on the (approximate) polynomial degree of a Boolean function to a lower bound on the round complexity of every (randomized) massively parallel computation of that function. These lower bounds apply even in the “unbounded width” version of our model, where the number of machines can be arbitrarily large. As one example of our general results, computing any nontrivial monotone graph property—such as connectivity—requires a super-constant number of rounds when every machine receives only a subpolynomial (in n) number of input bits s. Finally, we prove that, in two senses, our lower bounds are the best one could hope for. For the unbounded-width model, we prove a matching upper bound. Restricting to a polynomial number of machines, we show that asymptotically better lower bounds would separate P from NC1.

Abstract: Controllability is one of the most important properties of a Boolean control network (BCN). Essentially there are two kinds of controls: 1) networked inputs and 2) free logical inputs. This letter considers controllability of BCNs under mixed controls. The technique proposed is to convert a BCN with two kinds of controls into a set controllability problem. Using results obtained for set controllability, a necessary and sufficient condition for controllability of BCNs with mixed inputs is obtained. Some examples are presented to depict the theoretical results.

Abstract: A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH. As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.

Abstract: A Boolean generator for a large number of standard complementary QAM sequences of length $2^{K}$ is proposed. This Boolean generator is derived from the authors’ earlier paraunitary generator, which is based on matrix multiplications. Both generators are based on unitary matrices. In contrast to previous Boolean QAM algorithms which represent complementary sequences as a weighted sum, our algorithm has a multiplicative form. Any element of a sequence can be generated efficiently by indexing the entries of unitary matrices with the binary representation of the discrete time index (which is easily implemented as a binary counter). Our 1Qum (based on one QAM unitary matrix) and 2Qum (based on two QAM unitary matrices) algorithms generate generalized Case I – III sequences and generalized Case IV and V sequences, respectively, as specified by Liu et al. in 2013, in addition to many new 2Qum sequences. The ratio of the numbers of sequences that are generated by our new construction and the previous construction increases with the constellation size. For example, for a 1024-QAM sequence of length 1024, this ratio is 4.4. However, if we compare only 2Qum sequences to Case IV and V sequences, this ratio is 267.

Abstract: We give an example of a boolean function whose information complexity is exponentially smaller than its communication complexity. Our result simplifies recent work of Ganor, Kol and Raz [GKR14a, GKR14b].

Abstract: Tools to systematically reprogram cellular behavior are crucial to address pressing challenges in manufacturing, environment, or healthcare. Recombinases can very efficiently encode Boolean and history-dependent logic in many species, yet current designs are performed on a case-by-case basis, limiting their scalability and requiring time-consuming optimization. Here we present an automated workflow for designing recombinase logic devices executing Boolean functions. Our theoretical framework uses a reduced library of computational devices distributed into different cellular subpopulations, which are then composed in various manners to implement all desired logic functions at the multicellular level. Our design platform called CALIN (Composable Asynchronous Logic using Integrase Networks) is broadly accessible via a web server, taking truth tables as inputs and providing corresponding DNA designs and sequences as outputs (available at http://synbio.cbs.cnrs.fr/calin). We anticipate that this automated desi...

Abstract: A Boolean model is a simple, discrete and dynamic model without the need to consider the effects at the intermediate levels. However, little effort has been made into constructing activation, inhibition, and protein decay networks, which could indicate the direct roles of a gene (or its synthesized protein) as an activator or inhibitor of a target gene. Therefore, we propose to focus on the general Boolean functions at the subfunction level taking into account the effectiveness of protein decay, and further split the subfunctions into the activation and inhibition domains. As a consequence, we developed a novel data-driven Boolean model; namely, the Fundamental Boolean Model (FBM), to draw insights into gene activation, inhibition, and protein decay. This novel Boolean model provides an intuitive definition of activation and inhibition pathways and includes mechanisms to handle protein decay issues. To prove the concept of the novel model, we implemented a platform using R language, called FBNNet. Our experimental results show that the proposed FBM could explicitly display the internal connections of the mammalian cell cycle between genes separated into the connection types of activation, inhibition and protein decay. Moreover, the method we proposed to infer the gene regulatory networks for the novel Boolean model can be run in parallel and; hence, the computation cost is affordable. Finally, the novel Boolean model and related Fundamental Boolean Networks (FBNs) could show significant trajectories in genes to reveal how genes regulated each other over a given period. This new feature could facilitate further research on drug interventions to detect the side effects of a newly-proposed drug.

Abstract: This paper studies the problem of exactly identifying the structure of a probabilistic Boolean network (PBN) from a given set of samples, where PBNs are probabilistic extensions of Boolean networks. Cheng et al. studied the problem while focusing on PBNs consisting of pairs of AND/OR functions. This paper considers PBNs consisting of Boolean threshold functions while focusing on those threshold functions that have unit coefficients. The treatment of Boolean threshold functions, and triplets and ${n}$ -tuplets of such functions, necessitates a deepening of the theoretical analyses. It is shown that wide classes of PBNs with such threshold functions can be exactly identified from samples under reasonable constraints, which include: 1) PBNs in which any number of threshold functions can be assigned provided that all have the same number of input variables and 2) PBNs consisting of pairs of threshold functions with different numbers of input variables. It is also shown that the problem of deciding the equivalence of two Boolean threshold functions is solvable in pseudopolynomial time but remains co-NP complete.

Abstract: We study the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over l0,1rn. Our first main result is that distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that uses O(k2)/e queries (independent of n). Complementing this, our second main result is a lower bound showing that any non-adaptive distribution-free k-junta testing algorithm must make Ω(2k/3) queries even to test to accuracy e = 1/3. These bounds establish that while the optimal query complexity of non-adaptive k-junta testing is 2Θ(k), for adaptive testing it is poly(k), and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.

Abstract: We prove that any non-adaptive algorithm that tests whether an unknown Boolean function f:{0,1}n→ {0,1} is a k-junta or e-far from every k-junta must make Ω˜(k3/2) / e) many queries for a wide range of parameters k and e. Our result dramatically improves previous lower bounds and is essentially optimal since there is a known non-adaptive junta tester which makes Ω˜(k3/2) / e queries. Combined with the known existence of an adaptive tester which makes O(klog k + k /e) queries, our result shows that adaptivity enables polynomial savings in query complexity for junta testing.

Abstract: We introduce a new computer-aided design approach based on free binary decision diagrams (FBDDs) for implementing Boolean functions on crossbars using flow-based computing. Our crossbar synthesis procedure uses generalized FBDDs to design crossbars for a Boolean formula such that there is a flow of current from an input nanowire to an output nanowire through the sneak paths in the crossbar if and only if the Boolean formula evaluates to true. Generalized FBDDs are more succinct representations of Boolean formulas than traditional reduced ordered binary decision diagrams (ROBDDs) because they do not require the same variable ordering along all paths of the decision diagram. Our experimental results with the middle bit of a multiplier show that our designs are 69.9% more succinct than flow-based crossbar computing approaches designed using ROBDDs.

Abstract: Majority-inverter graphs (MIGs) are a logic representation with remarkable algebraic and Boolean properties that enable efficient logic optimizations beyond the capabilities of traditional logic representations. Further, since many nano-emerging technologies, such as quantum-dot cellular automata (QCA) or spin torque majority gates (STMG), are inherently majority-based, MIGs serve as a natural logic representation to map into these technologies. So far, MIG optimization methods predominantly target to reduce the depth of the logic networks, corresponding to low delay implementations in the respective technologies. In this paper, we introduce several methods to optimize the size of MIGs. They can be applied such that the depth of the logic network is preserved; therefore our methods have a direct effect on the physical area, without worsening the delay. Some methods are inspired by existing size optimization algorithms for non-majority-based logic networks, others make explicit use of the majority function and its properties. All methods are Boolean---in contrast to algebraic optimization methods---which has a positive effect on the quality but challenges their implementation. Our experiments show that using our methods the size of MIGs in the EPFL combinational benchmark suite can be reduced by up to 7.12%. When mapped to QCA and STMG technologies we reduce the average area-delay-energy product by 2.31% and 2.07%, respectively.

Abstract: In this paper, we discuss recent advances in exact synthesis, considering both their efficient implementation and various applications in which they can be employed. We emphasize on solving exact synthesis through Boolean satisfiability (SAT) encodings. Different SAT encodings for exact synthesis are compared, and examined the applications to multi-level logic synthesis, in both area and depth optimization. Another application of SAT based exact synthesis is optimization under many constraints. These constraints can, e.g., be a fixed fanout or delay constraints. Finally, we end our discussion by proposing directions for future research in exact synthesis.

Abstract: Boolean functional synthesis is the process of constructing a Boolean function from a Boolean specification that relates input and output variables. Despite significant recent developments in synthesis algorithms, Boolean functional synthesis remains a challenging problem even when state-of-the-art methods are used for decomposing the specification. In this work we bring a fresh decomposition approach, orthogonal to existing methods, that explores the decomposition of the specification into separate input and output components. We make use of an input-output decomposition of a given specification described as a CNF formula, by alternatingly analyzing the separate input and output components. We exploit well-defined properties of these components to ultimately synthesize a solution for the entire specification. We first provide a theoretical result that, for input components with specific structures, synthesis for CNF formulas via this framework can be performed more efficiently than in the general case. We then show by experimental evaluations that our algorithm performs well also in practice on instances which are challenging for existing state-of-the-art tools, serving as a good complement to modern synthesis techniques.

Abstract: A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH. As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.

Abstract: We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over R) of functions from some "simple" class C. In particular, given C we are interested in finding low-complexity functions lacking sparse representations. When C forms a basis for the space of Boolean functions (e.g., the set of PARITY functions or the set of conjunctions) this sort of problem has a well-understood answer; the problem becomes interesting when C is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where C is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts.We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. Let α(n) be an unbounded function such that nα(n) is time constructible (e.g. α(n) = log*(n)). We show:• Functions in NTIME[nα(n)] that require super-polynomially many linear threshold functions to represent (depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds).• Functions in NTIME[nα(n)] that require super-polynomially many ReLU gates to represent (depth-two neural networks with ReLU activation function).• Functions in NTIME[nα(n)] that require super-polynomially many O(1)-degree Fp-polynomials to represent exactly, for every prime p (related to problems regarding Higher-Order "Uncertainty Principles"). We also obtain a function in ENP requiring 2ω(n) linear combinations.• Functions in NTIME[npoly(log n)] that require super-polynomially many ACC c THR circuits to represent exactly (further generalizing the recent lower bounds of Murray and the author).We also obtain "fixed-polynomial" lower bounds for functions in NP, for the first three representation classes. All our lower bounds are obtained via algorithms for analyzing linear combinations of simple functions in the above scenarios, in ways which substantially beat exhaustive search.

Abstract: The correlation immunity of Boolean functions is a property related to cryptography, to error correcting codes, to orthogonal arrays (in combinatorics), and in a slightly looser way to sequences. Correlation-immune Boolean functions (in short, CI functions) have the property of keeping the same output distribution when some input variables are fixed. They have been widely used as combiners in stream ciphers to allow resistance to the Siegenthaler correlation attack. Very recently, a new use of CI functions has appeared in the framework of side channel attacks (SCA). To reduce the cost overhead of counter-measures to SCA, CI functions need to have low Hamming weights. This actually poses new challenges since the known constructions which are based on properties of the Walsh–Hadamard transform, do not allow to build unbalanced CI functions. In this paper, we propose constructions of low-weight $d$ th-order CI functions based on the Fourier–Hadamard transform, while the known constructions of resilient functions are based on the Walsh–Hadamard transform. These two transforms are closely related but the resulting constructions are very different. We first prove a simple but powerful result, which makes that one only need to consider the case where $d$ is odd in further research. Then, we investigate how constructing low Hamming weight CI functions through the Fourier–Hadamard transform (which behaves well with respect to the multiplication of Boolean functions). We use the characterization of CI functions by the Fourier–Hadamard transform and introduce a related general construction of CI functions by multiplication. By using the Kronecker product of vectors, we obtain more constructions of low-weight $d$ -CI Boolean functions. Furthermore, we present a method to construct low-weight d-CI Boolean functions by making additional restrictions on the supports built from the Kronecker product.

Abstract: We study the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}n. Our first main result is that distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that uses O(k2)/є queries (independent of n). Complementing this, our second main result is a lower bound showing that any non-adaptive distribution-free k-junta testing algorithm must make Ω(2k/3) queries even to test to accuracy є=1/3. These bounds establish that while the optimal query complexity of non-adaptive k-junta testing is 2Θ(k), for adaptive testing it is poly(k), and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.