Algebraic Structure Helps in Finding and Using Almost-Symmetries Igor L. Markov Department of EECS, The University of Michigan, Ann Arbor, MI 48109-2122 [email protected] Abstract other optimization problems in natural sciences. A rich literature exists on semantic symmetries of Boolean Many successful uses of symmetries in dis- functions and describes many uses in synthesis and op- crete computational problems rely on group- timization of digital circuits [21, ibid]. For example, in theoretical properties. For example, large a given VLSI layout one may permute the input pins sets of permutational symmetries can be of a single AND gate or a whole 64-bit multiplier to compactly represented by small sets of gen- improve wiring congestion by uncrossing wires [6]. erators and manipulated using stabilizer- Identifying and using a greater variety of symme- chain algorithms. While almost-symmetries tries often improves computational efficiency. Hence, may be more numerous than symmetries, it is natural to relax the notion of symmetry and no group-like properties or efficient algo- deal with almost-symmetries which retain useful prop- rithms are currently known for them. In erties, occur more often and have greater impact. this work, we identify an algebraic structure Such extensions have already been studied for tilings formed by almost-symmetries and, using it, and partial differential equations in Mathematics [22; develop a complete “life-cycle” for almost- 12], quasicrystals and approximate conservation laws symmetries, including their discovery, com- in Physics [16; 22], stars and planets in Astrophysics pact representation and symmetry-breaking. [23], molecular configurations in Chemistry [17; 9] and computational problems in Artificial Intelligence [15]. To be useful, the effort to exploit symmetries or 1 Introduction almost-symmetries should be justified by tangible ben- A structure-preserving reversible transformation of a efits, especially when symmetry-agnostic processing is structured object is often called a symmetry (or auto- possible [14; 8]. Such trade-offs are poorly understood morphism), with prime examples being (1) a permu- for almost-symmetries. In Section 2 we show that for tation of vertices of a given graph, that maps edges regular symmetries, computational improvements can to edges and preserves vertex labels, (2) a permuta- only be achieved by using the algebraic structure of tion of variables in a system of constraints, that pre- symmetries. The latter observation is driving much of serves the system, (3) a permutation of possible values our work on almost-symmetries as outlined below. of some variables, such as the simultaneous negation Existing techniques for using symmetries tend to be of two Boolean (or two integer) variables, (4) a permu- undermined by the freedom of choice. Namely, all tation of input and output variables of a Boolean func- computational techniques cited above assume simulta- tion that leaves the function invariant. Computational neous constraints, as in Boolean satisfiability, but are applications often involve finite domains. For exam- not sensitive to alternative constraints involving multi- ple, identifying a pair of symmetric variables x and y in ple variables [11]. In another example, the behavior an equation allows one to introduce the additional con- of digital circuits is often left underspecified on in- straint x ≤ y so as to reduce the amount of searching by valid inputs (resolved later so as to simplify the de- up to 25%, or closer to 50% for non-Boolean variables. sign), which leads to Boolean functions with don’t- Combining many such symmetries sometimes reduces cares. In Section 3 we attempt to generalize symme- the complexity of search, proofs or refutations from ex- tries to these cases, but discover that natural ”almost- ponential to polynomial, both in provable lower bounds symmetries” cannot be composed as freely as regular [18] and empirical performance [1]. Recent progress symmetries. Therefore almost-symmetries do not form in understanding and manipulating symmetries accel- groups, cosets, semi-groups, monoids, groupoids, or erates the solution of large, practical Boolean equa- other named algebraic structures. However, in Section tions [1], instances of Integer Linear Programming [3; 4 we identify another algebraic structure that character- 4], and other discrete problems. This is often accom- izes almost-symmetries and helps to compactly repre- plished through the use of fast symmetry detection [20; sent them. This facilitates symmetry-finding in Section 8] and symmetry-breaking techniques [7; 13; 2]. A 5 and almost-symmetry-breaking in Section 7. Section broad range of affected applications include (i) formal 6 discusses generalizations and applications of almost- verification of microprocessors and software, (ii) opti- symmetries. We articulate open problem in Section 8 mal scheduling and planning, (iii) protein folding and and discuss our ongoing work in the Appendix B. 2 Necessary Background 3 Possible Notions of Almost-symmetries Below we review a representative modern paradigm for Semantic symmetries include syntactic ones and form exploiting syntactic symmetries in search and discrete the same algebraic structure. However, using all se- optimization [1; 2], and then contrast it with seman- mantic symmetries is often computationally impracti- tic (functional) symmetries and related notions of in- cal. The same can be expected for almost-symmetries. terchangeability and substitutability [24]. In order to relax the notion of symmetry, one con- Finding and representing symmetries efficiently. siders transformations that do not necessarily preserve Consider an instance of constraint-programming or the structure in question. One known possibility is combinatorial optimization where all constraints must that of conditional symmetries that arise in the course be simultaneously satisfied. To identify its syntactic of constraint-solving algorithms based on backtrack- symmetries, all variables, constraints and optimization ing [13; 24]. Such algorithms work by assigning trial objectives are represented by labeled vertices in a graph values to variables, which simplifies the original prob- G where occurrences of variables are represented by lem instance and potentially facilitates new symme- edges, e.g., a forest of parse trees (uses of hypergraphs tries. Given that conditional symmetries may exist in or graph gadgets are allowed and do not affect our dis- an exponential number of different contexts, it may be cussion). The graph is constructed in such a way that difficult to batch symmetry-detection and symmetry- the group Hsym of symmetries of the original problem breaking with as much efficiency as for unconditional is isomorphic to the graph’s group of automorphisms symmetries. A typical conditional symmetry is appli- Aut(G), which can be found by advanced software cable in a considerably smaller scope than a regular tools such as NAUTY [20] or SAUCY [8]. Aut(G) symmetry and therefore holds less promise to acceler- is captured in a compact form by an unordered list of ate constraint-solving.1 Yet, much of what we propose group generators, which cannot be much larger than in this paper works for conditional symmetries as well. jGj. Having a group isomorphism, rather than an ar- Almost-symmetries are commonly defined as sym- bitrary one-to-one mapping between two sets, ensures metries of slightly modified objects [15]. For example, that sets of generators of Aut(G) map onto those of adding or removing one constraint can make the overall Hsym. Also, NAUTY and SAUCY would not have been set of constraints more symmetric. This may seem like applicable if symmetries did not form groups. a generalization of conditional symmetries, whose con- Using symmetries in search. For a given symmetry, ditions can be viewed as constraints of a special kind. a symmetry-breaking predicate (SBP) is a set of addi- However, conditional symmetries rely on the interpre- tional constraints added to the original constraints to tation of their conditions, i.e., assigned values are used prevent redundant search through symmetric branches. to simplify constraints, whereas almost-symmetries are For multiple symmetries, compatibility is ensured by often formulated entirely in terms of syntactic manip- lex-leader SBP constructions [7] which have recently ulations on the set of constraints and promise greater been improved [2]. It is often best to generate SBPs computational efficiency. We note that such differences for group generators only [1], thus adding very few are only meaningful in the context of specific search constraints (breaking fewer symmetries does not affect procedures and can be ignored at first. Therefore we π correctness). In practice, the SBPs for symmetries 1 focus on labeled graphs, except in Section 7. and π2 make the addition of an SBP for products of π π A popular example of an almost-symmetry of a 1 and 2 (and products of their powers) essentially graph is a symmetry of the graph derived by adding [ ] unnecessary 2 . Group properties are crucial in other or removing a small number of edges [19]. However, paradigms for using symmetries, where efficiency calls we are first going to warm up by studying a different [ ] for algorithms based on stabilizer chains 14 . relaxation of graph symmetries that deals with vertex Functional symmetries & don’t cares. A Boolean labels instead of edges, motivated by functions with function f with n input bits can be captured by a truth n don’t-cares (see Section 2). From now on we are go- table with 2 lines and one output column filled with ing to perceive vertex labels as colors — recall that a 0s and 1s. Permutational and negational symmetries of graph symmetry is a permutation of vertices that maps f can be captured as symmetries of its truth table, or each vertex to a vertex of the same color and maps ev- as automorphisms of the n-dimensional Boolean cube ery pair of vertices connected by an edge to another whose vertices represent the lines of the truth table and such pair. The color-related limitation can be relaxed are labeled 0 or 1 (see further optimizations in [6]).