Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

Abstract We study interactions between Skolem Arithmetic and certain classes of Circuit Satisfiability and Constraint Satisfaction Problems (CSPs). We revisit results of Glaser et al. [1] in the context of CSPs and settle the major open question from that paper, finding a certain satisfiability problem on circuits—involving complement, intersection, union and multiplication—to be decidable. This we prove using the decidability of Skolem Arithmetic. Then we solve a second question left open in [1] by proving a tight upper bound for the similar circuit satisfiability problem involving just intersection, union and multiplication. We continue by studying first-order expansions of Skolem Arithmetic without constants, ( N ; × ) , as CSPs. We find already here a rich landscape of problems with non-trivial instances that are in P as well as those that are NP-complete.

[1]  Michael Pinsker,et al.  Schaefer's Theorem for Graphs , 2015, J. ACM.

[2]  Libor Barto,et al.  Constraint Satisfaction Problems of Bounded Width , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[3]  Artur Jez,et al.  Computational completeness of equations over sets of natural numbers , 2014, Inf. Comput..

[4]  Andrei A. Bulatov,et al.  A Dichotomy Theorem for Nonuniform CSPs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[5]  Manuel Bodirsky.  Cores of Countably Categorical Structures , 2007, Log. Methods Comput. Sci..

[6]  K. Wagner,et al.  The Complexity of Problems Concerning Graphs with Regularities (Extended Abstract) , 1984, MFCS.

[7]  Ke Yang.  Integer circuit evaluation is PSPACE-complete , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[8]  Artur Jez,et al.  Complexity of Equations over Sets of Natural Numbers , 2009, Theory of Computing Systems.

[9]  Christian Glaßer,et al.  Equivalence Problems for Circuits over Sets of Natural Numbers , 2007, Theory of Computing Systems.

[10]  Hans-Georg Breunig.  The Complexity of Membership Problems for Circuits over Sets of Positive Numbers , 2007, FCT.

[11]  Peter Jonsson,et al.  Computational complexity of linear constraints over the integers , 2013, Artif. Intell..

[12]  Iain A. Stewart,et al.  Dichotomies for classes of homomorphism problems involving unary functions , 2004, Theor. Comput. Sci..

[13]  H. Putnam,et al.  The Decision Problem for Exponential Diophantine Equations , 1961 .

[14]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[15]  Dejan Deli'c.  Constraint Satisfaction Problem Dichotomy for Finite Templates: a Proof Via Consistency Checks , 2017 .

[16]  Erich Grädel,et al.  Dominoes and the Complexity of Subclasses of Logical Theories , 1989, Ann. Pure Appl. Log..

[17]  B. Scarpellini.  Complexity of subcases of Presburger arithmetic , 1984 .

[18]  Albert R. Meyer,et al.  Word problems requiring exponential time(Preliminary Report) , 1973, STOC.

[19]  Ju. V. Matijasevic,et al.  ENUMERABLE SETS ARE DIOPHANTINE , 2003 .

[20]  Dmitriy Zhuk,et al.  A Proof of CSP Dichotomy Conjecture , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[21]  Peter Jeavons,et al.  On the Algebraic Structure of Combinatorial Problems , 1998, Theor. Comput. Sci..

[22]  Peter Jonsson,et al.  Essential Convexity and Complexity of Semi-Algebraic Constraints , 2012, Log. Methods Comput. Sci..

[23]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraint satisfaction problems on a 3-element set , 2006, JACM.

[24]  Saeed Salehi,et al.  On the Multiplicative Theory of Numbers , 2017, ArXiv.

[25]  J. Ferrante,et al.  The computational complexity of logical theories , 1979 .

[26]  I. Borosh,et al.  Bounds on positive integral solutions of linear Diophantine equations , 1976 .

[27]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[28]  Jeanne Ferrante,et al.  A Decision Procedure for the First Order Theory of Real Addition with Order , 1975, SIAM J. Comput..

[29]  Titus Dose,et al.  Complexity of Constraint Satisfaction Problems over Finite Subsets of Natural Numbers , 2016, Electron. Colloquium Comput. Complex..

[30]  Ivo Düntsch,et al.  Functions Definable by Arithmetic Circuits , 2009, CiE.

[31]  Christian Glaßer,et al.  Emptiness Problems for Integer Circuits , 2020, MFCS.

[32]  Christos H. Papadimitriou,et al.  Computational complexity , 1993 .

[33]  Manuel Bodirsky,et al.  Non-dichotomies in Constraint Satisfaction Complexity , 2008, ICALP.

[34]  I. L. Novak.  Review: Andrzej Mostowski, On Direct Products of Theories , 1952 .

[35]  Barnaby Martin,et al.  Constraint Satisfaction Problems over the Integers with Successor , 2015, ICALP.

[36]  Pierre McKenzie,et al.  The Complexity of Membership Problems for Circuits Over Sets of Natural Numbers , 2007, computational complexity.

[37]  Christian Glaßer,et al.  Satisfiability of Algebraic Circuits over Sets of Natural Numbers , 2007, FSTTCS.

[38]  Eugene C. Freuder.  Complexity of K-Tree Structured Constraint Satisfaction Problems , 1990, AAAI.

[39]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[40]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[41]  Stephen D. Travers.  The Complexity of Membership Problems for Circuits over Sets of Integers , 2004, MFCS.

[42]  Libor Barto,et al.  The CSP Dichotomy Holds for Digraphs with No Sources and No Sinks (A Positive Answer to a Conjecture of Bang-Jensen and Hell) , 2008, SIAM J. Comput..

[43]  Christian Glaßer,et al.  Satisfiability of algebraic circuits over sets of natural numbers , 2010, Discret. Appl. Math..

[44]  Christos H. Papadimitriou,et al.  On the complexity of integer programming , 1981, JACM.

[45]  C. Smorynski.  The Incompleteness Theorems , 1977 .

[46]  Martin Grohe.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2007, JACM.

[47]  Andrzej Mostowski,et al.  On direct products of theories , 1952, Journal of Symbolic Logic.

[48]  Manuel Bodirsky,et al.  The complexity of temporal constraint satisfaction problems , 2008, STOC.

[49]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[50]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .