Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large size, but are still solved in minutes. Yet, small and difficult SAT instances must exist if P##NP. To this end, our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from Very Large Scale Integration (VLSI) routing problems. With an efficient implementation to solve the graph automorphism problem [39, 50, 51], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.

Research in algorithms for Boolean satisfiability and their efficient implementations [26, 8] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks from the early 1990s [12] can be solved in seconds on commodity PCs. More recent benchmarks take longer to solve primarily because of their large size, but are still solved in minutes [28]. However, small and difficult SAT instances must exist because Boolean satisfiability is NP-complete. Our work articulates a number of SAT instances that are unusually difficult for their size, including satisfiable instances from global routing and detailed routing for FPGAs [22]. Using an efficient implementation to solve the graph automorphism problem [21, 23, 25], we show that in structured SAT instances difficulty is sometimes associated with large numbers of symmetries. We propose a new, improved construction of symmetry-breaking clauses [11] and apply them to empirically demonstrate very significant speed-ups over current state of the art in Boolean satisfiability. Our techniques are formulated as pre-processing and can be applied to an arbitrary SAT solver without modifying its source code. We also show that considerations of symmetry may lead to more efficient reductions to SAT in the routing domain and potentially other applications.

Boolean Satisfiability solvers improved dramatically over the last seven years [14, 13] and are commonly used in applications such as bounded model checking, planning, and FPGA routing. However, a number of practical SAT instances remain difficult to solve. Recent work pointed out that symmetries in the search space are often to blame [1]. The framework of symmetry-breaking (SBPs) [5], together with further improvements [1], was then used to achieve empirical speed-ups. For symmetry-breaking to be successful in practice, its overhead must be less than the complexity reduction it brings. In this work we show how logic minimization helps to improve this trade-off and achieve much better empirical results. We also contribute detailed new studies of SBPs and their efficiency as well as new general constructions of SBPs.

Measure-adaptive state-space construction is the process of exploiting symmetry in high-level model and performance measure specifications to automatically construct reduced state-space Markov models that support the evaluation of the performance measure. This paper describes a new reward variable specification technique, which, combined with recently developed state-space construction techniques, will allow us to build tools capable of measure-adaptive state-space construction. That is, these tools will automatically adapt the size of the state space to constraints derived from the system model and the user-specified reward variables. The work described in this paper extends previous work in two directions. First, standard reward variable definitions are extended to allow symmetry in the reward variable to be identified and exploited. Then, symmetric reward variables are further extended to include the set of path-based reward variables described in earlier work. In addition to the theory, several examples are introduced to demonstrate these new techniques.

We present a fully automatic framework for identifying symmetries in structural descriptions of digital circuits and CTL* formulas and using them in a model checker. We show how the set of sub-formulas of a formula can be partitioned into equivalence classes so that truth values for only one sub-formula in any class need be evaluated for model checking. We unify and extend the theories developed by Clarke et al [CEFJ96] and Emerson and Sistla [ES96] for symmetries in Kripke structures. We formalize the notion of structural symmetries in net-list descriptions of digital circuits and CTL* formulas. We show how they relate to symmetries in the corresponding Kripke structures. We also show how such symmetries can automatically be extracted by constructing a suitable directed labeled graph and computing its automorphism group. We present a novel fast algorithm for solving the graph automorphism problem for directed labeled graphs.

Let k 0 and n 2 be integers. A SOMA, or more specifically a SOMA(k;n), is an n \Theta n array A, whose entries are k-subsets of a kn-set\Omega\Gamma such that each element of\Omega occurs exactly once in each row and exactly once in each column of A, and no 2-subset of\Omega is contained in more than one entry of A. A SOMA(k;n) can be constructed by superposing k mutually orthogonal Latin squares of order n with pairwise disjoint symbol-sets, and so a SOMA(k;n) can be seen as a generalization of k mutually orthogonal Latin squares of order n. In this paper we first study the structure of SOMAs, concentrating on how SOMAs can decompose. We then report on the use of computational group theory and graph theory in the discovery and classification of SOMAs. In particular, we discover and classify SOMA(3; 10)s with certain properties, and discover two SOMA(4; 14)s (SOMAs with these parameters were previously unknown to exist). Some of the newly discovered SOMA(3; 10)s come from superpos...

Algebra From a rectangular structure R, using the bijection d from equation (55) above and denoting by R(s; t) the unique rectangle on the pair (s; t) guaranteed by (52), define ffl : S \Theta S ! S (63) (s; t) 7! u where fug = (d \Gamma1 (s)) 2 " (d \Gamma1 (t)) 1 ffi : S \Theta S ! S (64) (s; t) 7! u where fug = d(R(s; t)) as binary operations on S.

We present 15 new partial difference sets over 4 non-abelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a by-product, a few new amorphic association schemes with 3 classes on 100 points are discovered.

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this paper we classify the graphs on which a Fischer group F i 22 , F i 22 : 2