A Continuous-Discontinuous Second-Order Transition in the Satisfiability of Random Horn-SAT Formulas

Moore, Cristopher; Istrate, Gabriel; Δημόπουλος, Δημήτριος; Vardi, Moshe Y.

doi:10.1007/11538462_35

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…Additionally, we observe the chaotic phase transition in the XORSAT as well. Moreover, HornSAT, which is solvable in linear time (thus it is also in P ) [49,50], was mathematically proven [51] to also present several phase transitions. The phase transition picture appears too coarse in describing the essential nature of the algorithmic barrier; one needs tools that can attack this question at the level of single formulas.…”

Section: Summary and Discussionmentioning

confidence: 99%

Order-to-chaos transition in the hardness of random Boolean satisfiability problems

et al. 2016

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Transient chaos is a ubiquitous phenomenon characterizing the dynamics of phase-space trajectories evolving towards a steady-state attractor in physical systems as diverse as fluids, chemical reactions, and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems, and even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability (k-SAT), a canonical class of constraint satisfaction problems, by using an analog deterministic algorithm based on a system of ordinary differential equations. Problem hardness is defined through the escape rate κ, an invariant measure of transient chaos of the dynamical system corresponding to the analog algorithm, and it expresses the rate at which the trajectory approaches a solution. We show that for a given density of constraints and fixed number of Boolean variables N, the hardness of formulas in random k-SAT ensembles has a wide variation, approximable by a lognormal distribution. We also show that when increasing the density of constraints α, hardness appears through a second-order phase transition at α_{χ} in the random 3-SAT ensemble where dynamical trajectories become transiently chaotic. A similar behavior is found in 4-SAT as well, however, such a transition does not occur for 2-SAT. This behavior also implies a novel type of transient chaos in which the escape rate has an exponential-algebraic dependence on the critical parameter κ∼N^{B|α-α_{χ}|^{1-γ}} with 0<γ<1. We demonstrate that the transition is generated by the appearance of metastable basins in the solution space as the density of constraints α is increased.

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Section: Summary and Discussionmentioning

confidence: 99%

Order-to-chaos transition in the hardness of random Boolean satisfiability problems

et al. 2016

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“…While the study of phase transitions is a well established topic in Complex Systems and A.I. [78,79], with phase transitions apearing even in settings relevant to model checking [80], the logical study of such "phase transitions" is still a relatively underdeveloped area. An exception is the topic of "zero-one laws" in the theory of random graphs [81].…”

Section: Parameterized Logics Of Stylized Facts: "Continuous Statemen...mentioning

confidence: 99%

Models we Can Trust: Toward a Systematic Discipline of (Agent-Based) Model Interpretation and Validation

Istrate¹

2021

Preprint

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We advocate the development of a discipline of interacting with and extracting information from models, both mathematical (e.g. game-theoretic ones) and computational (e.g. agent-based models). We outline some directions for the development of a such a discipline: -the development of logical frameworks for the systematic formal specification of stylized facts and social mechanisms in (mathematical and computational) social science. Such frameworks would bring to attention new issues, such as phase transitions, i.e. dramatical changes in the validity of the stylized facts beyond some critical values in parameter space. We argue that such statements are useful for those logical frameworks describing properties of ABM.the adaptation of tools from the theory of reactive systems (such as bisimulation) to obtain practically relevant notions of two systems "having the same behavior".the systematic development of an adversarial theory of model perturbations, that investigates the robustness of conclusions derived from models of social behavior to variations in several features of the social dynamics. These may include: activation order, the underlying social network, individual agent behavior.

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“…The occurence of phase transitions is not limited to hard problems; they occur in provably easy problems as well, including graph properties such as connectivity (Erdős and Rényi 1960), 2-SAT (Chvatal and Reed 1992;Goerdt 1996), XOR-SAT (Creignou and Daude 1999) and Horn-SAT (Moore et al 2005). For hard problems, the existence of a solvability threshold does not necessarily imply an easyhard-easy pattern, though counterexamples are rare.…”

Section: Related Workmentioning

confidence: 99%

An Investigation of Phase Transitions in Single-Machine Scheduling Problems

Wang

O’Gorman

Tran

et al. 2017

ICAPS

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We investigate solvable-unsolvable phase transitions in the single-machine scheduling (SMS) problem. SMS is at the core of practical problems such as telescope and satellite scheduling and manufacturing. To study the solvability phase transition, we construct a variety of instance families parameterized by the set of the processing times, the window size (deadline minus release time), and the horizon. We empirically establish the phase transition and look for an easy-hard-easy pattern for this family using several common solvers. While in many combinatorial problems a phase transition coincides with typically hard instances, whether or not that is the case with SMS remains an open question, and merits further study.

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A Continuous-Discontinuous Second-Order Transition in the Satisfiability of Random Horn-SAT Formulas

Cited by 3 publications

References 31 publications

Order-to-chaos transition in the hardness of random Boolean satisfiability problems

Order-to-chaos transition in the hardness of random Boolean satisfiability problems

Models we Can Trust: Toward a Systematic Discipline of (Agent-Based) Model Interpretation and Validation

An Investigation of Phase Transitions in Single-Machine Scheduling Problems

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