1,313 research outputs found
Representation Invariant Genetic Operators
A genetic algorithm is invariant with respect to a set of representations if it runs the same no matter which of the representations is used. We formalize this concept mathematically, showing that the representations generate a group that acts upon the search space. Invariant genetic operators are those that commute with this group action. We then consider the problem of characterizing crossover and mutation operators that have such invariance properties. In the case where the corresponding group action acts transitively on the search space, we provide a complete characterization, including high-level representation-independent algorithms implementing these operators
Group Properties of Crossover and Mutation
It is supposed that the finite search space Ω has certain symmetries that can be described in terms of a group of permutations acting upon it. If crossover and mutation respect these symmetries, then these operators can be described in terms of a mixing matrix and a group of permutation matrices. Conditions under which certain subsets of Ω are invariant under crossover are investigated, leading to a generalization of the term schema. Finally, it is sometimes possible for the group acting on Ω to induce a group structure on Ω itself
On the Evolution of Boomerang Uniformity in Cryptographic S-boxes
S-boxes are an important primitive that help cryptographic algorithms to be
resilient against various attacks. The resilience against specific attacks can
be connected with a certain property of an S-box, and the better the property
value, the more secure the algorithm. One example of such a property is called
boomerang uniformity, which helps to be resilient against boomerang attacks.
How to construct S-boxes with good boomerang uniformity is not always clear.
There are algebraic techniques that can result in good boomerang uniformity,
but the results are still rare. In this work, we explore the evolution of
S-boxes with good values of boomerang uniformity. We consider three different
encodings and five S-box sizes. For sizes and , we
manage to obtain optimal solutions. For , we obtain optimal
boomerang uniformity for the non-APN function. For larger sizes, the results
indicate the problem to be very difficult (even more difficult than evolving
differential uniformity, which can be considered a well-researched problem).Comment: 15 pages, 3 figures, 4 table
Prescriptive formalism for constructing domain-specific evolutionary algorithms
It has been widely recognised in the computational intelligence and machine learning communities that the key to understanding the behaviour of learning algorithms is to understand what representation is employed to capture and manipulate knowledge acquired during the learning process. However, traditional evolutionary algorithms have tended to employ a fixed representation space (binary strings), in order to allow the use of standardised genetic operators. This approach leads to complications for many problem domains, as it forces a somewhat artificial mapping between the problem variables and the canonical binary representation, especially when there are dependencies between problem variables (e.g. problems naturally defined over permutations). This often obscures the relationship between genetic structure and problem features, making it difficult to understand the actions of the standard genetic operators with reference to problem-specific structures. This thesis instead advocates m..
Perfect Edge-Transmitting Recombination of Permutations
Crossover is the process of recombining the genetic features of two parents.
For many applications where crossover is applied to permutations, relevant
genetic features are pairs of adjacent elements, also called edges in the
permutation order. Recombination of edges without errors is thought to be an
NP-hard problem, typically approximated by heuristics that either introduce new
edges or are only able to produce a small variety of offspring. Here, we derive
an algorithm for crossover of permutations that achieves perfect transmission
of edges and produces a uniform sampling of all possible offspring, in
quadratic average computation time. The algorithm and its derivation reveal a
link between cycle crossover (CX) and edge assembly crossover (EAX), offering a
new perspective on these well-established algorithms. We also describe a
modification of the algorithm that generates the mathematically optimal
offspring for the asymmetric travelling salesman problem
Decay of Bogoliubov excitations in one-dimensional Bose gases
We study the decay of Bogoliubov quasiparticles in one-dimensional Bose
gases. Starting from the hydrodynamic Hamiltonian, we develop a microscopic
theory that enables one to systematically study both the excitations and their
decay. At zero temperature, the leading mechanism of decay of a quasiparticle
is disintegration into three others. We find that low-energy quasiparticles
(phonons) decay with the rate that scales with the seventh power of momentum,
whereas the rate of decay of the high-energy quasiparticles does not depend on
momentum. In addition, our approach allows us to study analytically the
quasiparticle decay in the whole crossover region between the two limiting
cases. When applied to integrable models, including the Lieb-Liniger model of
bosons with contact repulsion, our theory confirms the absence of the decay of
quasiparticle excitations. We account for two types of integrability-breaking
perturbations that enable finite decay: three-body interaction between the
bosons and two-body interaction of finite range.Comment: 17 page
Index theory of one dimensional quantum walks and cellular automata
If a one-dimensional quantum lattice system is subject to one step of a
reversible discrete-time dynamics, it is intuitive that as much "quantum
information" as moves into any given block of cells from the left, has to exit
that block to the right. For two types of such systems - namely quantum walks
and cellular automata - we make this intuition precise by defining an index, a
quantity that measures the "net flow of quantum information" through the
system. The index supplies a complete characterization of two properties of the
discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the
sense that there is a system S which locally acts like S_1 in one region and
like S_2 in some other region, if and only if S_1 and S_2 have the same index.
Second, the index labels connected components of such systems: equality of the
index is necessary and sufficient for the existence of a continuous deformation
of S_1 into S_2. In the case of quantum walks, the index is integer-valued,
whereas for cellular automata, it takes values in the group of positive
rationals. In both cases, the map S -> ind S is a group homomorphism if
composition of the discrete dynamics is taken as the group law of the quantum
systems. Systems with trivial index are precisely those which can be realized
by partitioned unitaries, and the prototypes of systems with non-trivial index
are shifts.Comment: 38 pages. v2: added examples, terminology clarifie
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