{"title": "Lifted Symmetry Detection and Breaking for MAP Inference", "book": "Advances in Neural Information Processing Systems", "page_first": 1315, "page_last": 1323, "abstract": "Symmetry breaking is a technique for speeding up propositional satisfiability testing by adding constraints to the theory that restrict the search space while preserving satisfiability. In this work, we extend symmetry breaking to the problem of model finding in weighted and unweighted relational theories, a class of problems that includes MAP inference in Markov Logic and similar statistical-relational languages. We introduce term symmetries, which are induced by an evidence set and extend to symmetries over a relational theory. We provide the important special case of term equivalent symmetries, showing that such symmetries can be found in low-degree polynomial time. We show how to break an exponential number of these symmetries with added constraints whose number is linear in the size of the domain. We demonstrate the effectiveness of these techniques through experiments in two relational domains. We also discuss the connections between relational symmetry breaking and work on lifted inference in statistical-relational reasoning.", "full_text": "Lifted Symmetry Detection and\nBreaking for MAP Inference\n\nTim Kopp\n\nUniversity of Rochester\n\nRochester, NY\n\nParag Singla\nI.I.T. Delhi\n\nHauz Khas, New Delhi\n\nHenry Kautz\n\nUniversity of Rochester\n\nRochester, NY\n\ntkopp@cs.rochester.edu\n\nparags@cse.iitd.ac.in\n\nkautz@cs.rochester.edu\n\nAbstract\n\nSymmetry breaking is a technique for speeding up propositional satis\ufb01ability test-\ning by adding constraints to the theory that restrict the search space while preserv-\ning satis\ufb01ability. In this work, we extend symmetry breaking to the problem of\nmodel \ufb01nding in weighted and unweighted relational theories, a class of problems\nthat includes MAP inference in Markov Logic and similar statistical-relational\nlanguages. We introduce term symmetries, which are induced by an evidence\nset and extend to symmetries over a relational theory. We provide the important\nspecial case of term equivalent symmetries, showing that such symmetries can\nbe found in low-degree polynomial time. We show how to break an exponential\nnumber of these symmetries with added constraints whose number is linear in the\nsize of the domain. We demonstrate the effectiveness of these techniques through\nexperiments in two relational domains. We also discuss the connections between\nrelational symmetry breaking and work on lifted inference in statistical-relational\nreasoning.\n\nIntroduction\n\n1\nSymmetry-breaking is an approach to speeding up satis\ufb01ability testing by adding constraints, called\nsymmetry-breaking predicates (SBPs), to a theory [7, 1, 16]. Symmetries in the theory de\ufb01ne a\npartitioning over the space of truth assignments, where the assignments in a partition either all satisfy\nor all fail to satisfy the theory. The added SBPs rule out some but not all of the truth assignments in\nthe partitions, thus reducing the size of the search space while preserving satis\ufb01ability.\nWe extend the notion of symmetry-breaking to model-\ufb01nding in relational theories. A relational\ntheory is speci\ufb01ed by a set of \ufb01rst-order axioms over \ufb01nite domains, optional weights on the axioms\nor predicates of the theory, and a set of ground literals representing evidence. By model \ufb01nding\nwe mean satis\ufb01ability testing (unweighted theories), weighted MaxSAT (weights on axioms), or\nmaximum weighted model \ufb01nding (weights on predicates). The weighted versions of model \ufb01nding\nencompass MAP inference in Markov Logic and similar statistical-relational languages.\nWe introduce methods for \ufb01nding symmetries in a relational theory that do not depend upon solving\ngraph isomorphism over its full propositional grounding. We show how graph isomorphism can be\napplied to just the evidence portion of a relational theory in order to \ufb01nd the set of what we call term\nsymmetries. We go on to de\ufb01ne the important subclass of term equivalent symmetries, and show that\nthey can be found in O(nM log M ) time where n is the number of constants and M is the size of\nthe evidence.\nNext we provide the formulation for breaking term and term equivalent symmetries. An inherent\nproblem in symmetry-breaking is that a propositional theory may have an exponential number of\nsymmetries, so breaking them individually would increase the size of the theory exponentially. This\nis typically handled by breaking only a portion of the symmetries. We show that term equivalent\nsymmetries provide a compact representation of exponentially many symmetries, and an exponen-\n\n1\n\n\ftially large subset of these can be broken by a small (linear) number of SBPs. We demonstrate these\nideas on two relational domains and compare our approach to other methods for MAP inference in\nMarkov Logic.\n2 Background\nSymmetry Breaking for SAT Symmetry-breaking for satis\ufb01ability testing, introduced by Craw-\nford et. al.[7], is based upon concepts from group theory. A permutation \u03b8 is a mapping from a set L\nto itself. A permutation group is a set of permutations that is closed under composition and contains\nthe identity and a unique inverse for every element. A literal is an atom or its negation. A clause\nis a disjunction over literals. A CNF theory T is a set (conjunction) of clauses. Let L be the set of\nliterals of T . We consider only permutations that respect negation, that is \u03b8(\u00acl) = \u00ac\u03b8(l) (l \u2208 L).\nThe action of a permutation on a theory, written \u03b8(T ), is the CNF formula created by applying \u03b8 to\neach literal in T . We say \u03b8 is a symmetry of T if it results in the same theory i.e. \u03b8(T ) = T .\nA model M is a truth assignment to the atoms of a theory. The action of \u03b8 on M, written \u03b8(M ),\nis the model where \u03b8(M )(P ) = M (\u03b8(P )). The key property of \u03b8 being a symmetry of T is that\nM |= T iff \u03b8(M ) |= T . The orbit of a model M under a symmetry group \u0398 is the set of models\nthat can be obtained by applying any of the symmetries in \u0398. A symmetry group divides the space\nof models into disjoint sets, where the models in an orbit either all satisfy or all do not satisfy the\ntheory. The idea of symmetry-breaking is to add clauses to T rule out many of the models, but are\nguaranteed to not rule out at least one model in each orbit. Note that symmetry-breaking preserves\nsatis\ufb01ability of a theory.\nSymmetries can be found in CNF theories using a reduction to graph isomorphism, a problem that is\nthought to require super-polynomial time in the worst case, but which can often be ef\ufb01ciently solved\nin practice [18]. The added clauses are called symmetry-breaking predicates (SBPs). If we place\na \ufb01xed order on the atoms of theory, then a model can be associated with a binary number, where\nthe i-th digit, 0 or 1, speci\ufb01es the value of the i-th atom, false or true. Lex-leader SBPs rule out\nmodels that are not the lexicographically-smallest members of their orbits. The formulation below\nis equivalent to the lex-leader SBP given by Crawford et. al. in [7]:\n\n(cid:94)\n\n(cid:16) (cid:94)\n\nSBP (\u03b8) =\n\n1\u2264i\u2264n\n\n1\u2264j