Generalized Satisfiability Problems via Operator Assignments

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For the case of constraints given by a system of linear equations over the two-element field, this relaxation has received considerable attention in the foundations of quantum mechanics, where such constructions as the Mermin-Peres magic square show that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. We obtain a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments. To establish our main result, we adapt the notion of primitive-positive definability (pp-definability) to our setting, a notion that has been used extensively in the study of constraint satisfaction problems. Here, we show that pp-definability gives rise to gadget reductions that preserve satisfiability gaps. We also present several additional applications of this method. In particular and perhaps surprisingly, we show that the relaxed notion of pp-definability in which the quantified variables are allowed to range over operator assignments gives no additional expressive power in defining Boolean relations

Complexity of Generalized Satisfiability Counting Problems

AbstractThe class of generalized satisfiability problems, introduced in 1978 by Schaefer, presents a uniform way of studying the complexity of satisfiability problems with special conditions. The complexity of each decision and counting problem in this class depends on the involved logical relations. In 1979, Valiant defined the class #P, the class of functions definable as the number of accepting computations of a polynomial-time nondeterministic Turing machine. Clearly, all satisfiability counting problems belong to this class #P. We prove a Dichotomy Theorem for generalized satisfiability counting problems. That is, if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time, otherwise this function is #P-complete. This gives us a comparison between decision and counting generalized satisfiability problems. We can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already #P-complete. Moreover, taking advantage of a similar dichotomy result proved in 1978 by Schaefer for generalized satisfiability decision problems, we get as a corollary the implication that the counting counterpart of each NP-complete generalized satisfiability decision problem is #P-complete

Random subcubes as a toy model for constraint satisfaction problems

We present an exactly solvable random-subcube model inspired by the structure of hard constraint satisfaction and optimization problems. Our model reproduces the structure of the solution space of the random k-satisfiability and k-coloring problems, and undergoes the same phase transitions as these problems. The comparison becomes quantitative in the large-k limit. Distance properties, as well the x-satisfiability threshold, are studied. The model is also generalized to define a continuous energy landscape useful for studying several aspects of glassy dynamics.Comment: 21 pages, 4 figure

Computational Complexity and Phase Transitions

Phase transitions in combinatorial problems have recently been shown to be useful in locating "hard" instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been addressed in Statistical Mechanics and Artificial Intelligence, but not studied rigorously. We take a step in this direction by investigating the existence of sharp thresholds for the class of generalized satisfiability problems defined by Schaefer. In the case when all constraints are clauses we give a complete characterization of such problems that have a sharp threshold. While NP-completeness does not imply (even in this restricted case) the existence of a sharp threshold, it "almost implies" this, since clausal generalized satisfiability problems that lack a sharp threshold are either 1. polynomial time solvable, or 2. predicted, with success probability lower bounded by some positive constant by across all the probability range, by a single, trivial procedure.Comment: A (slightly) revised version of the paper submitted to the 15th IEEE Conference on Computational Complexit

Generalized satisfiability problems via operator assignments

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. This representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For constraints given by a system of linear equations over the two-element field, earlier work in the foundations of quantum mechanics has shown that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. Our main result is a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments.Peer ReviewedPostprint (published version

The Complexity of Reasoning with FODD and GFODD

Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a knowledge representation that is useful in mechanizing decision theoretic planning in relational domains. GFODDs generalize function-free first order logic and include numerical values and numerical generalizations of existential and universal quantification. Previous work presented heuristic inference algorithms for GFODDs and implemented these heuristics in systems for decision theoretic planning. In this paper, we study the complexity of the computational problems addressed by such implementations. In particular, we study the evaluation problem, the satisfiability problem, and the equivalence problem for GFODDs under the assumption that the size of the intended model is given with the problem, a restriction that guarantees decidability. Our results provide a complete characterization placing these problems within the polynomial hierarchy. The same characterization applies to the corresponding restriction of problems in first order logic, giving an interesting new avenue for efficient inference when the number of objects is bounded. Our results show that for $\Sigma_k$ formulas, and for corresponding GFODDs, evaluation and satisfiability are $\Sigma_k^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete. For $\Pi_k$ formulas evaluation is $\Pi_k^p$ complete, satisfiability is one level higher and is $\Sigma_{k+1}^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete.Comment: A short version of this paper appears in AAAI 2014. Version 2 includes a reorganization and some expanded proof

Decidability of predicate logics with team semantics

We study the complexity of predicate logics based on team semantics. We show that the satisfiability problems of two-variable independence logic and inclusion logic are both NEXPTIME-complete. Furthermore, we show that the validity problem of two-variable dependence logic is undecidable, thereby solving an open problem from the team semantics literature. We also briefly analyse the complexity of the Bernays-Sch\"onfinkel-Ramsey prefix classes of dependence logic.Comment: Extended version of a MFCS 2016 article. Changes on the earlier arXiv version: title changed, added the result on validity of two-variable dependence logic, restructurin