Complexity classifications for different equivalence and audit problems for Boolean circuits

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.Comment: 25 pages, 1 figur

Necessary and sufficient conditions for non-perturbative equivalences of large N orbifold gauge theories

Large N coherent state methods are used to study the relation between U(N) gauge theories containing adjoint representation matter fields and their orbifold projections. The classical dynamical systems which reproduce the large N limits of the quantum dynamics in parent and daughter orbifold theories are compared. We demonstrate that the large N dynamics of the parent theory, restricted to the subspace invariant under the orbifold projection symmetry, and the large N dynamics of the daughter theory, restricted to the untwisted sector invariant under "theory space'' permutations, coincide. This implies equality, in the large N limit, between appropriately identified connected correlation functions in parent and daughter theories, provided the orbifold projection symmetry is not spontaneously broken in the parent theory and the theory space permutation symmetry is not spontaneously broken in the daughter. The necessity of these symmetry realization conditions for the validity of the large N equivalence is unsurprising, but demonstrating the sufficiency of these conditions is new. This work extends an earlier proof of non-perturbative large N equivalence which was only valid in the phase of the (lattice regularized) theories continuously connected to large mass and strong coupling.Comment: 21 page, JHEP styl

An efficient implementation of a test for EA-equivalence

We implement an algorithm for testing EA-equivalence between vectorial Boolean functions proposed by Kaleyski in the C programming language, and observe that it reduces the running time (as opposed to the original Magma implementation of the algorithm) necessary to decide equivalence up to 300 times in many cases. Our implementation also significantly reduces the memory usage, and makes it possible to run the algorithms for dimensions from 10 onwards, which was impossible using the original implementation due to its memory consumption. Our approach allows us to reconstruct the exact form of the equivalence and to prove that two given functions are equivalent (for comparison, computing invariants for the functions, which is the approach typically used in practice, only allows us to show that two functions are not equivalent). Furthermore, our approach works for functions of any algebraic degree, while most existing approaches (such as invariants and other algorithms for EA-equivalence) are restricted to the quadratic case. We then adapt Kaleyski’s algorithm to test for linear and affine equivalence instead of EA-equivalence. We supply an implementation in C of this procedure as well. As an application, we show how this method can be used to test quadratic APN functions for EA-equivalence through the linear equivalence of their orthoderivatives. We observe that by taking this approach, we can reduce the time necessary for deciding EA-equivalence up to 20 times (as compared with our efficient C implementation from the previous paragraph). The downside compared to Kaleyski’s original algorithm is that this faster method makes it difficult to recover the exact form of the EA-equivalence between the tested APN functions. We confirm this by running some computational experiments in dimension 6, and observing that only one out of all possible linear equivalences between the orthoderivatives corresponds to the EA-equivalence between the APN functions in question. To the best of our knowledge, this is the first investigation into the exact relationship between the EA-equivalence of quadratic APN functions and the affine equivalence of their orthoderivatives given in the literature.Masteroppgave i informatikkINF399MAMN-INFMAMN-PRO