Static Symmetry Breaking Research Articles

SAT Modulo Symmetries for Graph Generation and Enumeration

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality with respect to a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator performs a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of forbidden subgraphs and diameter. In particular, we could confirm the Murty-Simon Conjecture (1979) on diameter-2-critical graphs for graphs up to 19 vertices and prove the exact number of Ramsey graphs \(\mathcal{R}(3,5,n)\) and \(\mathcal{R}(4,4,n)\) .

High-order harmonic generation from a thin film crystal perturbed by a quasi-static terahertz field

Studies of laser-driven strong field processes subjected to a (quasi-)static field have been mainly confined to theory. Here we provide an experimental realization by introducing a bichromatic approach for high harmonic generation (HHG) in a dielectric that combines an intense 70 femtosecond duration mid-infrared driving field with a weak 2 picosecond period terahertz (THz) dressing field. We address the physics underlying the THz field induced static symmetry breaking and its consequences on the efficient production/suppression of even-/odd-order harmonics, and demonstrate the ability to probe the HHG dynamics via the modulation of the harmonic distribution. Moreover, we report a delay-dependent even-order harmonic frequency shift that is proportional to the time derivative of the THz field. This suggests a limitation of the static symmetry breaking interpretation and implies that the resultant attosecond bursts are aperiodic, thus providing a frequency domain probe of attosecond transients while opening opportunities in precise attosecond pulse shaping.

New Techniques for Static Symmetry Breaking in Many-Sorted Finite Model Finding

Symmetry in finite model finding problems of many-sorted first-order logic (MSFOL) can be exploited to reduce the number of interpretations considered during search, thereby improving solver performance for tools such as the Alloy Analyzer. We present a framework to soundly compose static symmetry breaking schemes for many-sorted finite model finding. Then, we introduce and prove the correctness of three static symmetry breaking schemes for MSFOL: 1) one for functions with distinct sorts in the domain and range; 2) one for functions where the range sort appears in the domain; and 3) one for predicates. We provide a novel presentation of sort inference in the context of symmetry breaking that yields a new mathematical link between sorts and symmetries. We empirically investigate how our symmetry breaking approaches affect solving performance.

Experimental analysis of the effect of local base blowing on three-dimensional wake modes

Wake modes of a three-dimensional blunt-based body near a wall are investigated at a Reynolds number . The targeted modes are the static symmetry-breaking mode and two antisymmetric periodic modes. The static mode orientation is aligned with the horizontal major -axis of the base and randomly switches between a positive and a negative state leading to long-time bistable dynamics of the turbulent wake. The modifications of these modes are studied when continuous blowing is applied at different locations through four slits along the base edges (denoted L for left, R for right, T for top and B for bottom) in either four single asymmetric configurations or two double symmetric configurations (denoted LR and TB). Two regimes, referred to as mass and momentum, are clearly identifiable for all configurations. The mass regime, which is fairly insensitive to blowing momentum and location, is characterized by the growth of the recirculating bubble as the total injected flow rate is increased, and is associated with a base drag reduction and interpreted as resulting from the equilibrium between mass fluxes feeding and emptying the recirculating region. A simple budget model is shown to be in agreement with entrainment velocities measured for isolated turbulent mixing layers. The strength of the static mode is reduced up to 20 % when the bubble length is maximum, whereas no change in the periodic mode frequencies is found. On the other hand, the momentum regime is characterized by the deflating of the recirculating bubble, leading to base drag increase, and it is interpreted by the free shear layer forcing, which increases the entrainment velocity, thus emptying the recirculating bubble. In this regime the static mode orientation is imposed by the blowing symmetry. Lateral L and R (respectively top/bottom T and B) blowing configurations select or states in the horizontal (respectively vertical) direction, while bistable dynamics persists for the symmetric LR and TB configurations. The shape of periodic modes follows the changes in wake static orientation. The transition between the two regimes is governed by both the total injected flow rate and the location of the injection.

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.

A numerical investigation of the asymmetric wake mode of a squareback Ahmed body – effect of a base cavity

Numerical simulations of the turbulent flow over the flat backed Ahmed model at Reynolds number $Re\simeq 4\times 10^{5}$ are conducted using a lattice Boltzmann solver to clarify the mean topology of the static symmetry-breaking mode of the wake. It is shown that the recirculation region is occupied by a skewed low pressure torus, whose part closest to the body is responsible for an extra low pressure imprint on the base. Shedding of one-sided vortex loops is also reported, indicating global quasi-periodic dynamics in conformity with the seminal work of Grandemange et al. (J. Fluid Mech., vol. 722, 2013, pp. 51–84). Despite the limited low frequency resolution of the simulation, power spectra of the lateral velocity fluctuations at different locations corroborate the presence of this quasi-periodic mode at a Strouhal number of $St=0.16\pm 0.03$. A shallow base cavity of $5\,\%$ of the body height reduces the drag coefficient by $3\,\%$ but keeps the recirculating torus and its interaction with the base mostly unchanged. The drag reduction lies in a global constant positive shift of the base pressure distribution. For a deep base cavity of $33\,\%$ of the body height, a drag reduction of $9.5\,\%$ is obtained. It is accompanied by a large elongation of the recirculation inside the cavity that considerably attenuates the low pressure sources therein together with a symmetrization of the low pressure torus. The global quasi-periodic mode is found to be inhibited by the cavity.

A New Routing Method Based on Hybrid Algorithm

A new routing algorithm is proposed for FPGA to improve the increasing transformation cost of pseudo-Boolean Satisfiability algorithm in the routing process. The approach combines advantages of pseudo-Boolean Satisfiability and geometric routing algorithm. First, Frontier-one of geometric routing algorithm is chosen for FPGA routing. Then pseudo-Boolean Satisfiability algorithm is used when the process of Frontier is not successful. Technique of static symmetry-breaking is also adding to carry out pretreatment of pseudo-Boolean constraints, detecting and breaking the symmetries in the routing flow. The advantage is that the search path is pruned, and the cost is consequently reduced. Preliminary experiments results show that the hybrid method have no adverse affect on overall program. It has also reduced the runtime observably, and sped up the solving process.

The Improvement of Pseudo-Boolean Satisfiability Algorithm for FPGA Routing

A new routing algorithm is proposed for FPGA to improve the increasing transformation cost of pseudo-Boolean Satisfiability algorithm in the routing process, which combined advantages of pseudo-Boolean Satisfiability and geometric routing algorithm. In the routing process, one of geometric routing algorithm-VPR5.0 was chosen firstly for FPGA routing. If not successful, then use pseudo-Boolean Satisfiability algorithm. Technique of static symmetry-breaking is also adding to carry out pretreatment of pseudo-Boolean constraints, detecting and breaking the symmetries in the routing flow. The purpose was to prune search path, and the cost was consequently reduced. Preliminary experiments results show that the hybrid approach can reduce the runtime observably, speed up the solving process, and have no adverse affect on overall program.

A New Hybrid Algorithm for FPGA Routing

A new routing algorithm is proposed for FPGA to improve the increasing transformation cost of pseudo-Boolean Satisfiability algorithm in the routing process. The approach combines advantages of pseudo-Boolean Satisfiability and geometric routing algorithm. First, Frontier- one of geometric routing algorithm is chosen for FPGA routing. Then pseudo-Boolean Satisfiability algorithm is used when the process of Frontier is not successful. Technique of static symmetry-breaking is also adding to carry out pretreatment of pseudo-Boolean constraints, detecting and breaking the symmetries in the routing flow. The advantage is that the search path is pruned, and the cost is consequently reduced. Preliminary experiments results show that the hybrid method have no adverse affect on overall program. It has also reduced the runtime observably, and sped up the solving process.

Reformulations in mathematical programming: automatic symmetry detection and exploitation

If a mathematical program has many symmetric optima, solving it via Branch-and-Bound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for automatically finding the formulation group of any given Mixed-Integer Nonlinear Program, and for reformulating the problem by means of static symmetry breaking constraints. The reformulated problem—which is likely to have fewer symmetric optima—can then be solved via standard Branch-and-Bound codes such as CPLEX (for linear programs) and Couenne (for nonlinear programs). Our computational results include formulation group tables for the MIPLib3, MIPLib2003, GlobalLib and MINLPLib instance libraries and solution tables for some instances in the aforementioned libraries.

Dynamic symmetry-breaking for Boolean satisfiability

With impressive progress in Boolean Satisfiability (SAT) solving and several extensions to pseudo-Boolean (PB) constraints, many applications that use SAT, such as high-performance formal verification techniques are still restricted to checking satisfiability of certain conditions. However, there is also frequently a need to express a preference for certain solutions. Extending SAT-solving to Boolean optimization allows the use of objective functions to describe a desirable solution. Although recent work in 0---1 Integer Linear Programming (ILP) offers extensions that can optimize a linear objective function, this is often achieved by solving a series of SAT or ILP decision problems. Our work articulates some pitfalls of this approach. An objective function may complicate the use of any symmetry that might be present in the given constraints, even when the constraints are unsatisfiable and the objective function is irrelevant. We propose several new techniques that treat objective functions differently from CNF/PB constraints and accelerate Boolean optimization in many practical cases. We also develop an adaptive flow that analyzes a given Boolean optimization problem and picks the symmetry-breaking technique that is best suited to the problem characteristics. Empirically, we show that for non-trivial objective functions that destroy constraint symmetries, the benefit of static symmetry-breaking is lost but dynamic symmetry-breaking accelerates problem-solving in many cases. We also introduce a new objective function, Localized Bit Selection (LBS), that can be used to specify a preference for bit values in formal verification applications.

Solving necklace constraint problems

Some constraint problems have a combinatorial structure where the constraints allow the sequence of variables to be rotated ( necklaces), if not also the domain values to be permuted ( unlabelled necklaces), without getting an essentially different solution. We bring together the fields of combinatorial enumeration, where efficient algorithms have been designed for (special cases of) some of these combinatorial objects, and constraint programming, where the requisite symmetry breaking has at best been done statically so far. We design the first search procedure and identify the first symmetry-breaking constraints for the general case of unlabelled necklaces. Further, we compare dynamic and static symmetry breaking on real-life scheduling problems featuring (unlabelled) necklaces.