On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

Parameterized complexity of abduction in Schaefer’s framework

Abductive reasoning is a non-monotonic formalism stemming from the work of Peirce. It describes the process of deriving the most plausible explanations of known facts. Considering the positive version, asking for sets of variables as explanations, we study, besides the problem of wether there exists a set of explanations, two explanation size limited variants of this reasoning problem (less than or equal to, and equal to a given size bound). In this paper, we present a thorough two-dimensional classification of these problems: the first dimension is regarding the parameterized complexity under a wealth of different parameterizations, and the second dimension spans through all possible Boolean fragments of these problems in Schaefer’s constraint satisfaction framework with co-clones (T. J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1–3, 1978, San Diego, California, USA, R.J. Lipton, W.A. Burkhard, W.J. Savitch, E.P. Friedman, A.V. Aho eds, pp. 216–226. ACM, 1978). Thereby, we almost complete the parameterized complexity classification program initiated by Fellows et al. (The parameterized complexity of abduction. In Proceedings of the Twenty-Sixth AAAI Conference on Articial Intelligence, July 22–26, 2012, Toronto, Ontario, Canada, J. Homann, B. Selman eds. AAAI Press, 2012), partially building on the results by Nordh and Zanuttini (What makes propositional abduction tractable. Artificial Intelligence, 172, 1245–1284, 2008). In this process, we outline a fine-grained analysis of the inherent parameterized intractability of these problems and pinpoint their FPT parts. As the standard algebraic approach is not applicable to our problems, we develop an alternative method that makes the algebraic tools partially available again.

Non-Malleable Zero-Knowledge Arguments with Lower Round Complexity

Round complexity is one of the fundamental problems in zero-knowledge (ZK) proof systems. Non-malleable zero-knowledge (NMZK) protocols are ZK protocols that provide security even when man-in-the-middle adversaries interact with a prover and a verifier simultaneously. It is known that the first constant-round public-coin NMZK arguments for NP can be constructed by assuming the existence of collision-resistant hash functions (Pass, R. and Rosen, A. (2005) New and Improved Constructions of Non-Malleable Cryptographic Protocols. In Gabow, H.N. and Fagin, R. (eds) Proc. 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 2224, 2005, pp. 533542. ACM) and has relatively high round complexity; the first four-round private-coin NMZK arguments for NP can be constructed in the plain model by assuming the existence of one-way functions (Goyal, V., Richelson, S., Rosen, A. and Vald, M. (2014) An Algebraic Approach to Non-Malleability. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 1821, 2014, pp. 4150. IEEE Computer Society and Ciampi, M., Ostrovsky, R., Siniscalchi, L. and Visconti, I. (2017) Delayed-Input Non-Malleable Zero Knowledge and Multi-Party Coin Tossing in Four Rounds. In Kalai, Y. and Reyzin, L. (eds) Theory of Cryptography15th Int. Conf., TCC 2017. Lecture Notes in Computer Science, Baltimore, MD, USA, November 1215, 2017, Part I, Vol. 10677, pp. 711742. Springer). In this paper, we present a six-round public-coin NMZK argument of knowledge system assuming the existence of collision-resistant hash functions and a three-round private-coin NMZK argument system from multi-collision resistance of hash functions assumption in the keyless setting.

Prolegomena to an Operator Theory of Computation

Defining computation as information processing (information dynamics) with information as a relational property of data structures (the difference in one system that makes a difference in another system) makes it very suitable to use operator formulation, with similarities to category theory. The concept of the operator is exceedingly important in many knowledge areas as a tool of theoretical studies and practical applications. Here we introduce the operator theory of computing, opening new opportunities for the exploration of computing devices, processes, and their networks.

Modeling Computing Devices and Processes by Information Operators

The concept of operator is exceedingly important in many areas as a tool of theoretical studies and practical applications. Here, we introduce the operator theory of computing, opening new opportunities for the exploration of computing devices, networks, and processes. In particular, the operator approach allows for the solving of many computing problems in a more general context of operating spaces. In addition, operator representation of computing devices and their networks allows for the construction of a variety of operator compositions and the development of new schemas of computation as well as network and computer architectures using operations with operators. Besides, operator representation allows for the efficient application of the axiomatic technique for the investigation of computation.

Modeling Computing Devices and Processes by Information Operators

The concept of operator is exceedingly important in many areas as a tool of theoretical studies and practical applications. Here, we introduce the operator theory of computing, opening new opportunities for the exploration of computing devices, networks, and processes. In particular, the operator approach allows for the solving of many computing problems in a more general context of operating spaces. In addition, operator representation of computing devices and their networks allows for the construction of a variety of operator compositions and the development of new schemas of computation as well as network and computer architectures using operations with operators. Besides, operator representation allows for the efficient application of the axiomatic technique for the investigation of computation.