On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry

We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating DOUBLELEX completely is NP-hard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark problems.Comment: To appear in the Proceedings of the 16th International Conference on Principles and Practice of Constraint Programming (CP 2010

Symmetry Breaking Constraints: Recent Results

Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful cases: symmetry breaking constraints for row and column symmetry, and symmetry breaking constraints for eliminating value symmetryComment: To appear in Proceedings of Twenty-Sixth Conference on Artificial Intelligence (AAAI-12

Compatible abelian symmetries in N-Higgs-Doublet Models

We analyze the compatibility between abelian symmetries acting in two different sectors of a theory using the Smith Normal Form method. We focus on N-Higgs-doublet models (NHDMs) and on the compatibility between symmetries in the Higgs potential and in the Yukawa interactions, which were separately analyzed previous works. It is shown that two equal (isomorphic) symmetry groups that act in two separate sectors are not necessarily compatible in the whole theory and an upper bound is found for the size of the group that can be implemented in the entire NHDM. We also develop useful techniques to analyze compatibility and extend a symmetry from one sector to another. Consequences to the supersymmetric case are briefly discussed.Comment: v2: 40pp; some modifications in text, brief discussion on the supersymmetric case added; to appear in JHE

Hidden flavor symmetries of SO(10) GUT

The Yukawa interactions of the SO(10) GUT with fermions in 16-plets (as well as with singlets) have certain intrinsic ("built-in") symmetries which do not depend on the model parameters. Thus, the symmetric Yukawa interactions of the 10 and 126 dimensional Higgses have intrinsic discrete $Z_2\times Z_2$ symmetries, while the antisymmetric Yukawa interactions of the 120 dimensional Higgs have a continuous SU(2) symmetry. The couplings of SO(10) singlet fermions with fermionic 16-plets have $U(1)^3$ symmetry. We consider a possibility that some elements of these intrinsic symmetries are the residual symmetries, which originate from the (spontaneous) breaking of a larger symmetry group $G_f$. Such an embedding leads to the determination of certain elements of the relative mixing matrix $U$ between the matrices of Yukawa couplings $Y_{10}$, $Y_{126}$, $Y_{120}$, and consequently, to restrictions of masses and mixings of quarks and leptons. We explore the consequences of such embedding using the symmetry group conditions. We show how unitarity emerges from group properties and obtain the conditions it imposes on the parameters of embedding. We find that in some cases the predicted values of elements of $U$ are compatible with the existing data fits. In the supersymmetric version of SO(10) such results are renormalization group invariant.Comment: 28 pages, a reference added, typos corrected, to be published in NP

U(1)U(1) flavour symmetries as Peccei-Quinn symmetries

We investigate to what extent a generic, generation-dependent $U(1)$ symmetry acting on the quark Yukawa operators can reduce the number of free parameters by forcing some entries in the Yukawa matrices to vanish. The maximal reduction compatible with CP violation yields nine real parameters and one phase, which matches the number of physical observables, implying that such models have no free parameters. We derive a set of results: (i) the only possible structures have the form $M_4 \oplus M_5$, where the subscripts indicate the number of real parameters in the Yukawa matrices, (ii) there are only two inequivalent Yukawa structures, each one giving rise to six different models depending on quark flavour assignments, (iii) the $U(1)$ symmetries that generate these textures all have a QCD anomaly, and hence are Peccei-Quinn symmetries, reinforcing the idea of a possible connection between the quark flavour puzzle and the axion solution to the strong CP problem, (iv) in some cases the contributions to the QCD anomaly of two generations cancels out, and this opens the possibility that the axion coupling to nucleons could be strongly suppressed. Flavour-violating axion couplings to quarks are completely fixed, up to the axion decay constant, providing a non-trivial complementarity between low-energy flavour-violating processes and standard axion searches.Comment: v2: version accepted for publication in JHEP; figure 1 updated; minor additions; 23 pages, 1 figure. v1: 20 pages, 1 figur

Origin of Constrained Maximal CP Violation in Flavor Symmetry

Current data from neutrino oscillation experiments are in good agreement with $\delta =-\frac{\pi}{2}$ and $\theta_{23}^{} = \frac{\pi}{4}$. We define the notion of "constrained maximal CP violation" (CMCPV) for predicting these features and study their origin in flavor symmetry. We derive the parametrization-independent solution of CMCPV and give a set of equivalent definitions for it. We further present a theorem on how the CMCPV can be realized. This theorem takes advantage of residual symmetries in the neutrino and charged lepton mass matrices, and states that, up to a few minor exceptions, $(|\delta |, \theta_{23}) =(\frac{\pi}{2}, \frac{\pi}{4})$ is generated when those symmetries are real. The often considered $\mu$-$\tau$ reflection symmetry, as well as specific discrete subgroups of O(3), are special cases of our theorem.Comment: Phys.Lett.B Final Version, 13pp. All conclusions unchanged, only minor improvement to stress the parametrization-independence of our CMCP

Symmetry Breaking Using Value Precedence

We present a comprehensive study of the use of value precedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is both efficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreath value and partial interchangeability. We also show that value precedence is closely related to lexicographical ordering. Finally, we consider the interaction between value precedence and symmetry breaking constraints for variable symmetries.Comment: 17th European Conference on Artificial Intelligenc