A Sound and Complete Axiomatization of Majority-n Logic

Manipulating logic functions via majority operators recently drew the attention of researchers in computer science. For example, circuit optimization based on majority operators enables superior results as compared to traditional logic systems. Also, the Boolean satisfiability problem finds new solving approaches when described in terms of majority decisions. To support computer logic applications based on majority a sound and complete set of axioms is required. Most of the recent advances in majority logic deal only with ternary majority (MAJ- 3) operators because the axiomatization with solely MAJ-3 and complementation operators is well understood. However, it is of interest extending such axiomatization to n-ary majority operators (MAJ-n) from both the theoretical and practical perspective. In this work, we address this issue by introducing a sound and complete axiomatization of MAJ-n logic. Our axiomatization naturally includes existing majority logic systems. Based on this general set of axioms, computer applications can now fully exploit the expressive power of majority logic.Comment: Accepted by the IEEE Transactions on Computer

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Unconditional Lower Bounds in Complexity Theory

This work investigates the hardness of solving natural computational problems according to different complexity measures. Our results and techniques span several areas in theoretical computer science and discrete mathematics. They have in common the following aspects: (i) the results are unconditional, i.e., they rely on no unproven hardness assumption from complexity theory; (ii) the corresponding lower bounds are essentially optimal. Among our contributions, we highlight the following results. Constraint Satisfaction Problems and Monotone Complexity. We introduce a natural formulation of the satisfiability problem as a monotone function, and prove a near-optimal 2^{Ω (n/log n)} lower bound on the size of monotone formulas solving k-SAT on n-variable instances (for a large enough k ∈ ℕ). More generally, we investigate constraint satisfaction problems according to the geometry of their constraints, i.e., as a function of the hypergraph describing which variables appear in each constraint. Our results show in a certain technical sense that the monotone circuit depth complexity of the satisfiability problem is polynomially related to the tree-width of the corresponding graphs. Interactive Protocols and Communication Complexity. We investigate interactive compression protocols, a hybrid model between computational complexity and communication complexity. We prove that the communication complexity of the Majority function on n-bit inputs with respect to Boolean circuits of size s and depth d extended with modulo p gates is precisely n/log^{ϴ(d)} s, where p is a fixed prime number, and d ∈ ℕ. Further, we establish a strong round-separation theorem for bounded-depth circuits, showing that (r+1)-round protocols can be substantially more efficient than r-round protocols, for every r ∈ ℕ. Negations in Computational Learning Theory. We study the learnability of circuits containing a given number of negation gates, a measure that interpolates between monotone functions, and the class of all functions. Let C^t_n be the class of Boolean functions on n input variables that can be computed by Boolean circuits with at most t negations. We prove that any algorithm that learns every f ∈ C^t_n with membership queries according to the uniform distribution to accuracy ε has query complexity 2^{Ω (2^t sqrt(n)/ε)} (for a large range of these parameters). Moreover, we give an algorithm that learns C^t_n from random examples only, and with a running time that essentially matches this information-theoretic lower bound. Negations in Theory of Cryptography. We investigate the power of negation gates in cryptography and related areas, and prove that many basic cryptographic primitives require essentially the maximum number of negations among all Boolean functions. In other words, cryptography is highly non-monotone. Our results rely on a variety of techniques, and give near-optimal lower bounds for pseudorandom functions, error-correcting codes, hardcore predicates, randomness extractors, and small-bias generators. Algorithms versus Circuit Lower Bounds. We strengthen a few connections between algorithms and circuit lower bounds. We show that the design of faster algorithms in some widely investigated learning models would imply new unconditional lower bounds in complexity theory. In addition, we prove that the existence of non-trivial satisfiability algorithms for certain classes of Boolean circuits of depth d+2 leads to lower bounds for the corresponding class of circuits of depth d. These results show that either there are no faster algorithms for some computational tasks, or certain circuit lower bounds hold

Generalized asset integrity games

Generalized assets represent a class of multi-scale adaptive state-transition systems with domain-oblivious performance criteria. The governance of such assets must proceed without exact specifications, objectives, or constraints. Decision making must rapidly scale in the presence of uncertainty, complexity, and intelligent adversaries. This thesis formulates an architecture for generalized asset planning. Assets are modelled as dynamical graph structures which admit topological performance indicators, such as dependability, resilience, and efficiency. These metrics are used to construct robust model configurations. A normalized compression distance (NCD) is computed between a given active/live asset model and a reference configuration to produce an integrity score. The utility derived from the asset is monotonically proportional to this integrity score, which represents the proximity to ideal conditions. The present work considers the situation between an asset manager and an intelligent adversary, who act within a stochastic environment to control the integrity state of the asset. A generalized asset integrity game engine (GAIGE) is developed, which implements anytime algorithms to solve a stochastically perturbed two-player zero-sum game. The resulting planning strategies seek to stabilize deviations from minimax trajectories of the integrity score. Results demonstrate the performance and scalability of the GAIGE. This approach represents a first-step towards domain-oblivious architectures for complex asset governance and anytime planning