Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Robust Constraint Satisfaction and Local Hidden Variables in Quantum Mechanics Samson Abramsky1, Georg Gottlob1, Phokion G. Kolaitis2 1. Department of Computer Science, University of Oxford 2. University of California, Santa Cruz and IBM Research - Almaden Abstract hidden-variable models, while [Abramsky, 2013] developed a purely relational framework. In the latter framework, the Motivated by considerations in quantum mechan- focus is on n-dimensional relational models, n ≥ 1, of the ics, we introduce the class of robust constraint sat- form (M; O; e), where M is the cartesian product of n sets isfaction problems in which the question is whether M ;:::;M of measurements, O is the cartesian product of every partial assignment of a certain length can 1 n n sets O1;:::;On of outcomes, and e ⊆ M ×O. [Abramsky, be extended to a solution, provided the partial as- 2013] explored the connections between the key properties signment does not violate any of the constraints of relational hidden-variable models and, in particular, char- of the given instance. We explore the complex- acterized when such a model belongs to the class HV(n) of ity of specific robust colorability and robust sat- local hidden-variable models. A consequence of this char- isfiability problems, and show that they are NP- acterization is that the membership problem for HV(n) is in complete. We then use these results to establish NP. However, the exact complexity of this decision problem, the computational intractability of detecting local which is also denoted by HV(n), was left open. This was the hidden-variable models in quantum mechanics. original motivation for the work reported here. In this paper, we introduce the class of robust constraint 1 Introduction and Summary of Results satisfaction problems1 in which the question is whether every Since the 1930’s, the study of hidden variables and locality partial assignment of a certain length can be extended to a so- has occupied a prominent place in the foundations of quantum lution, provided the partial assignment does not violate any of mechanics. The hidden-variable program in quantum me- the constraints of the given instance. Special cases of robust chanics was developed in an attempt to explain the celebrated constraint satisfaction problems were studied earlier in totally Einstein-Podolsky-Rosen paradox [Einstein et al., 1935]. The different contexts. In particular, [Beacham, 2000] studied a main idea behind this program is that the original formulation robust version of 3-HYPERGRAPH 2-COLORABILITY in an of quantum mechanics is “incomplete”, but it can be “com- investigation of problems with no frozen variables (no back- pleted” via the introduction of hidden variables, that is, quan- bone), while [Gottlob, 2012] studied a robust version of SAT- tities that cannot be seen or measured, but which control the ISFIABILITY in the context of minimal constraint networks. observable behaviour. A hidden-variable model is supposed However, the general concept of a robust constraint satisfac- to be consistent with the predictions of quantum mechanics, tion problem had not been formulated earlier. while at the same time possessing certain desirable proper- We focus on two robust constraint satisfaction problems ties of classical systems, such as locality and the observer- that turn out to be just the right tool needed to settle the independence of properties of physical systems. Later on, the computational complexity of local hidden-variable models. hidden-variable program was dealt a blow by the no-go theo- Specifically, we show that ROBUST 3-COLORABILITY and rems, which are results establishing that certain quantum me- ROBUST 3SAT are NP-complete problems. The former prob- chanics predictions cannot be explained via hidden-variable lem asks: given a graph G, is it 3-colorable and also is it true models. Bell’s theorem [Bell, 1964], the Kochen-Specker that every partial assignment of one of three colors to two in- theorem [Kochen and Specker, 1967], and Hardy’s paradox dependent nodes u and v can be extended to a 3-coloring of [Hardy, 1993] are among the best-known no-go theorems. G? The latter problem asks: given a 3CNF formula ', is it More recently, researchers have embarked on the devel- true that every partial assignment to three variables can be ex- opment of unifying mathematical frameworks in which the tended to a satisfying truth assignment of ', provided it does key properties (such as locality, determinism, and indepen- not directly contradict any clause of '? dence) of hidden-variable models can be expressed, the rela- Armed with these two new NP-completeness results, we tions between these properties can be studied, and existence show that for every n ≥ 2, testing for membership in HV(n) and non-existence (no-go) theorems can be stated in precise terms and proved rigorously. In particular, [Brandenburger 1This notion is unrelated to the notion of robust constraint satis- and Yanofsky, 2008] developed a probabilistic framework for faction algorithm recently introduced by [Barto and Kozik, 2012] 440 is a NP-complete problem, thus resolving the problem left certain form. For example, 3SAT can be viewed as a 3CSP open in [Abramsky, 2013]. In fact, we obtain a complete pic- whose constraints represent the satisfying assignments of the ture for the computational complexity of the parameterized different types of clauses with 3 literals. Similarly, GRAPH subclasses of HV(n) that also take into account the size of 3-COLORABILITY is a 2CSP, where a given graph G = the domains Oi, 1 ≤ i ≤ n, of possible outcomes. More (V; E) is viewed as a CSP-instance I having var(I) = V , precisely, for every n ≥ 2 and every k ≥ 2, let HV(n)=k be dom(I) = f1; 2; 3g (the three colors), and the same con- the subclass of HV(n) consisting of all hidden-variable mod- straint c = f(1; 2); (1; 3); (2; 1); (2; 3); (3; 1); 3; 2)g, for each els (M; O; e) in which each domain Oi of outcomes has at edge (u; v) 2 E. most k elements. We show that HV(2)=2 is in PTIME, while Here, we will also be interested in 3-HYPERGRAPH 2- HV(n)=k) is NP-complete, for all other values of n and k. In COLORABILITY, another well known NP-complete problem. particular, both HV(2)=3 and HV(3)=2 are NP-complete. An instance of this problem is a 3-hypergraph, i.e., a pair Roadmap. In Section 2, we define robust constraint satis- H = (V; R), where R is a ternary relation on V . The ques- faction problem and present several examples. The complex- tion is whether there is an assignment of one of two colors to ity of robust colorability and robust satisfiability problems is each node of V such that for every triple (3−hyperedge) in R, studied in Section 3. In Section 4, we give some background two of its three nodes are assigned different color. It is easy material about hidden-variable models and then show that de- to see that 3-HYPERGRAPH 2-COLORABILITY is a 3CSP. tecting hidden variables is NP complete. Section 5 concludes In what follows, we introduce the notion of robust con- the paper with a brief summary and outlook on future work. straint satisfaction and present a number of examples. Let P be a CSP and r a non-negative integer. Intuitively, 2 Robust Constraint Satisfaction the r-robust version of P , denoted by rROBUST P , is the de- cision problem that, given an instance I of P , asks whether I We first review some basic notions from database theory. A is satisfiable under every partial assignment to r variables that relation schema is a finite set of variables (a.k.a. attributes), does not directly violate any constraint of I. The following where each variable x has an associated domain dom(x) definition makes this precise. of values. A relation r over a schema S (also denoted by Definition 2.1 Let P be a CSP and let r ≥ 0 an integer. schema(r)) is a set of tuples, where each tuple t is a map- • Let I be an instance of P , V a set of variables from I, ping that assigns a value t(x) 2 dom(x) to each variable and t : V −! dom(I) a partial assignment for I. We say x 2 S. If jSj = s, then we say that r is s-ary. If that t is compatible 2 with I if for every constraint c of I such 0 0 S = fx1; : : : ; xng ⊆ S and t is a tuple over S, then t[S ] that V \ scope(c) 6= ;, we have that t[V \ scope(c)] belongs (and also t[x1; : : : ; xn]) denotes the restriction of the func- to the projection π (c) of c on V \ scope(c). 0 V \scope(c) tion t to the variables of S. If S ⊆ S, then the projection • The r-robust version of P , denoted by rROBUST P , is the 0 0 πS0 (r) is the relation ft[S ]jt 2 rg over the schema S . following decision problem. The instances of rROBUST P Next we recall the standard definition of a constraint sat- are exactly those of P . Given such an instance I, the question isfaction problem (see also [Tsang, 1993; Dechter, 2003; is whether I is satisfiable under every partial assignment t : Rossi et al., 2006]). V ! dom(I) such that jV j = r and t is compatible with I. ACONSTRAINT SATISFACTION PROBLEM (CSP) is a de- • If P is a kCSP, for some k ≥ 1, then we will write RO- cision problem P that has the following characteristics.