K-sat on groups and undecidability

The general Boolean formula can be quickly converted into a normal form, A$, the conjunction of triple disjunctions of literals, which is satisfiable iff the original is. On the other hand, formulae in Ns, the conjunction of 2-fold disjunctions of liter&s, can be checked for a satisfaction in polynomial time. Thus these hvo satisfaction problems, ‘Y-sat” and “3-sat”, have been considered as an interesting boundary point between IP and NP. We define an infinite generalization of 2-sat and 3-sat which are respectively algorithmic and tmdecidable. As a corollary it is noted that the 3colorability of doubly-periodic planar graphs is undecidable. It was suggested in m that a general approach to proving P # PIP would be to construct some infinitary limit of decision problems with the property that those admitting polynomial time algorithms would be decidable in this limit. The hope here is to exploit the strong connection behseen polynomial growth and finite dimensionality. Since logic has a method-selfreference-for establishing problems as undecidable, this technique applied to the limit could potentially show that the finite-decision problem lies outside of P. This paper supplies one way of extending k-sat to an infinite context in which decidability distinguishes 2-sat from 3-sat. Consider a finitely-generated iscrete group G and subgroup of finite index H C G. The elements of G are our alphabet, and a literal is a symbol g or 3, g E G, read “g” or “not g”. Let g’ be ambiguous, representing (precisely) one of g or 3, so that me can write an arbitrary k-fold disjunction as gi\f. Vgi. We let Nk be the finite Boolean formulae which are finite conjunctions of k-foldjisjunctions as above. A typical element of A& is written: A (gi,j V . . V gizlj). Now let HNI, be the set of (generally :$nite) Boolean formulae: The set of disjunctive clauses of a formula T in HIV/: is H-invariant with respect o the obvious left action; this is the defining property. Notice that if H is an infinite group, Twill have infinitely-many clauses. We will study this k-sat problem where the clauses are H-equivariant, but there is no requirement hat a solution be H-equivariant. That is, a solution might specify hrg and (G) in the satisfaction, hl, hs E H. (If the solution were required to be equivnriant, we would only be discussing the ordinary b-sat problem with alphabet he elements of the coset space H \ 6’. We have sought a convenient level of generality in which to phrase the discussion. A broader definition would allow H to act co-compactly on any discrete space. In this paper the interesting assertions will be mainly for the special cases G = 2, the integers, and G = 2 $ 2, the Gaussian integers. When G = 2 we are considering periodic formulae with period p, that is H = pZ and Jr E pZ& Here we find that for any k > 2, that if 3 has a satisfaction, it also has a periodic satisfaction, although the period P of the satisfaction may have to be much larger, P >> p. However P is bounded by an explicit function of p, enabling an exhaustive search for the satisfaction. Thus for G = 2 no interesting difference between k = 2 and 1; = 3 is seen: both problems are decidable. when G = 2 @I 2, we are in effect considering doubly-periodic formulae with periods (pl,pz); so, for the remainder of this paragraph, fix H = {(nlpr, nepe) 1 nl, n2 E 2). In this case we construct an (exponential-time) decision procedure for T E HA5 and show that there is no algorithm for deciding if 3 E HAra has a satisfaction. Consequently there are specific formulae & E HIV3 whose satisfaction is independent of the axioms of set theory (ZFC). The negative results follow from translating Berger’s theorem (see [B] and [Rl], [R2]) on the undecidability of tiling problems into the setting of doubly-periodic Boolean formulae, As a byproduct of our argument we will see that formulae in HN2 which admit a satisfaction admit doubly-periodic satisfactions (of estimatable periods); whereas there are formulae in HNs which are satisfiable, but no satisfaction admits even a single period (or other symmetry), Beyond the two groups studied in detail, we conjecture that HiVz is also decidable when G is word-hyperbolic. On seeing an early draft of this paper, Fan Chung directed my attention to Theorem 2 [Bu] which establishes the undecidability of the vertex coloring problem (k 2 3) on doublyperiodic graphs on which the coloring is fixed in advance on a finite subset of vertices. Our Corollary 3 improves this statemcnt by removing the relativization. The broader program, to which we hope these examples might contribute, is to find some infinite setting or limit for finite decision problems in which problems in Itp are decidable, Experience with taking limits of groups m, suggests that the structure of P should be exploited to yield “finitedimensional” limits for which the decision problem can be solved by continuous earch. Then those finite-decision problems which arc undecidable in the infinite setting will be retognizcd as lying outside l? The topic of this paper is quite new to me, and I would like to express my thanks to many mathematical friends who contributed to these ideas, These include Ian Agol, Sam Buss, Jcong Han Kim, and Van Vu. I am particularly indebted to David Meyer and Peter Doyle for drawing my attention to aperlodic tilings.