Certificate complexity and symmetry of nested canalizing functions

Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs.

SHAPE METAPHORS IN ENGLISH AND SPANISH: SYMMETRY, TOTAL ASYMMETRY AND PARTIAL SYMMETRY

The objective of this article is to measure the symmetry between Spanish and English in the application of technical metaphors of shape. The reason for this is double: they have not received too much attention and Forceville (2007, 2008, 2012) recommends studying how they interact with cultural/linguistic factors. The method has measured percentage of symmetry / asymmetry in different metaphors. Research shows that symmetry between English/Spanish metaphors rises to 60%, whereas partial metaphoric asymmetry is caused by excessive metaphorization (30%) or by metonymy (10%) as a relation of cause-effect between two concepts.Keywords: Shape Metaphor, Lexicography, Translation, Overmetaphorization, Metonymy

Enhancing Static Symmetry Breaking with Dynamic Symmetry Handling in CDCL SAT Solvers

Symmetry exploitation techniques in SAT can be classified into two main categories: static symmetry breaking and dynamic symmetry handling. It is currently recognized that the most effective approach is the static one, which proceeds by preprocessing the formula to be solved in order to add symmetry-breaking predicates (SBPs) that guarantee the preservation of equisatisfiability but not equivalence. Due to the large size of the CNF symmetry groups, only a subset of the SBPs is generated resulting in a partial symmetry breaking. It is at this level that dynamic symmetry handling can intervene to help the solver avoid as much as possible, the exploration of isomorphic symmetrical subspaces for symmetries not (entirely) broken by the static approach. However, the use of both techniques within the same solver could compromise the result. We propose in this paper in addition to an optimization of the dynamic scheme SLS, an approach that allows the two methods of symmetry exploitation to coexist within the same solver in order to increase its efficiency while preserving its correctness. This is to the best of our knowledge, the first time such an integration is envisaged within the framework of SAT solving. Experimental results conducted on hard combinatorial instances drawn from SAT competitions show that symmetrical learning can indeed improve static symmetry breaking.