Relating Boolean Gate Truth Tables to One-Way Functions

In this paper, the authors present a schema to build one way functions from a family of Boolean gates. Moreover, the authors relate characteristics of these Boolean gate truth tables to properties of the derived one-way functions

Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis

A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n-arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is derived. The common control gates can be easily obtained by considering the logical correspondence between the control logic operator and the binary propositional logic operator. A new polynomial and exponential formulation of the Toffoli gate is presented. The method has parallels to quantum gate-T optimization methods using powers of multilinear operator polynomials. The method is then applied naturally to alphabets greater than two for multi-valued logical gates used for quantum Fourier transform, min-max decision circuits and multivalued adders

Effective Theories for Circuits and Automata

Abstracting an effective theory from a complicated process is central to the study of complexity. Even when the underlying mechanisms are understood, or at least measurable, the presence of dissipation and irreversibility in biological, computational and social systems makes the problem harder. Here we demonstrate the construction of effective theories in the presence of both irreversibility and noise, in a dynamical model with underlying feedback. We use the Krohn-Rhodes theorem to show how the composition of underlying mechanisms can lead to innovations in the emergent effective theory. We show how dissipation and irreversibility fundamentally limit the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.Comment: 11 pages, 9 figure

Pure Nash Equilibria: Hard and Easy Games

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems

Verified AIG Algorithms in ACL2

And-Inverter Graphs (AIGs) are a popular way to represent Boolean functions (like circuits). AIG simplification algorithms can dramatically reduce an AIG, and play an important role in modern hardware verification tools like equivalence checkers. In practice, these tricky algorithms are implemented with optimized C or C++ routines with no guarantee of correctness. Meanwhile, many interactive theorem provers can now employ SAT or SMT solvers to automatically solve finite goals, but no theorem prover makes use of these advanced, AIG-based approaches. We have developed two ways to represent AIGs within the ACL2 theorem prover. One representation, Hons-AIGs, is especially convenient to use and reason about. The other, Aignet, is the opposite; it is styled after modern AIG packages and allows for efficient algorithms. We have implemented functions for converting between these representations, random vector simulation, conversion to CNF, etc., and developed reasoning strategies for verifying these algorithms. Aside from these contributions towards verifying AIG algorithms, this work has an immediate, practical benefit for ACL2 users who are using GL to bit-blast finite ACL2 theorems: they can now optionally trust an off-the-shelf SAT solver to carry out the proof, instead of using the built-in BDD package. Looking to the future, it is a first step toward implementing verified AIG simplification algorithms that might further improve GL performance.Comment: In Proceedings ACL2 2013, arXiv:1304.712