May 31, 2007 19:47 WSPC - Proceedings Trim Size: 9in x 6in ictcs 1 Definable sets in the weak Presburger arithmetic ∗ CHRISTIAN CHOFFRUT LIAFA, Universit´eParis 7 & CNRS, 2, pl. Jussieu – 75251 Paris Cedex – 05, France E-mail: [email protected] www. liafa. jussieu. fr/ ~cc ACHILLE FRIGERI Dipartimento di Matematica, Politecnico di Milano & LIAFA, Universit´eParis 7 via Bonardi, 9 – 20133 Milano, Italia E-mail: [email protected] We show the following: given a relation defined by a first order formula on the structure hZ; +, <i, it is recursively decidable whether or not it is first order definable in the structure hZ; +i. Keywords: Presburger arithmetic, arithmetical definability. Introduction Presburger arithmetic is the fragment of arithmetic concerning the integers with addition and order. Presburger’s supervisor considered the decidabil- ity of this fragment too modest a result to deserve a Ph.D. degree and he accepted it only as a Master’s Thesis in 1928. Looking at the number of citations, we may say that history revised this depreciative judgment long ago. There still remains, at least as far as we can see, some confusion concerning the domain of the structure: Z or N? with or without the or- der relation? (the main popular mathematical web sites disagree on that respect). The original paper deals with the additive group of positive and negative integers with no binary relation, but in a final remark of the origi- nal communication the author asserts that the same result, to wit quantifier elimination, holds on the structure of the “whole” integers, i.e., the natural ∗Partially supported by . ... ... May 31, 2007 19:47 WSPC - Proceedings Trim Size: 9in x 6in ictcs 2 numbers with the binary relation <. In ?, which is the main reference on the subject, Presburger arithmetic is defined as the elementary theory of integers with equality, addition, having 0 and 1 as constant symbols and < as binary predicate, see also ?. On the other hand, the majority of the “modern” papers referring to Presburger arithmetic is concerned with the natural numbers where order relation is unnecessary as it is first order ex- pressible. The origin of the present work is the simple remark that concerning the set of integers Z, the binary relation matters. Here we study the decidability of the definability in the structure hZ; +i for a given relation defined in hZ; +, <i. We show that it is indeed recursively decidable and we prove this result by revisiting the notion of linear subsets introduced by Ginsburg and Spanier? in the sixties. Despite of its simplicity, this arithmetic is central in many areas of theo- retical and applied computer science. From a theoretical point of view, it has many surprising properties: 1) it admits quantifiers elimination?,?,? and therefore it is decidable, 2) given a formula on the expansion of the structure obtained by adding the function which to each integer assigns the maximal power of 2 which divides it, it is decidable whether or not it is definable by a Presburger formula over N (Cobham-Sem¨enov theorem?, improved in ? with a polynomial time algorithm), and 3) Presburger arithmetic is self-definable (i.e. there is Presburger definable criterion for definability?). Moreover, there is a strong and old connection between language theory, Presburger definable sets and rational relations on Z and N dating back to the sixties?,?,?. The concept is also widely used in many application areas, such as program analysis and model-checking and more specifically timed automata: roughly speaking, the main idea is that we can describe an infi- nite system with unbounded integer variables using Presburger formulas as guards? (see also the introduction of ? for some historical remarks on the role of Presburger arithmetic in the development of theoretical and applied computer science). 1. Preliminaries 1.1. Variants of Presburger arithmetic As observed above, a source of confusion is the lack of agreement in the definition of Presburger arithmetic itself. We make the convention of calling weak Presburger arithmetic the structure ZW = hZ; +; 0, 1i origi- nally studied in ?, while with Z we mean the (standard) Presburger arith- May 31, 2007 19:47 WSPC - Proceedings Trim Size: 9in x 6in ictcs 3 metic hZ; =, <; +; 0, 1i. The positive Presburger arithmetic is the structure N = hN; =; +; 0, 1i and we observe that in this case the < predicate (as restriction of the order on Z to N) is already definable in N . All these three structures are decidable in the sense that given a closed formula, it is re- cursively decidable whether or not it holds. In particular ZW and Z admit quantifier elimination in the augmented languages with the additional unary functional symbol − and the (recursive) set of binary functional symbols (≡m)m∈N\{0,1}, having the usual meaning of opposite and modulo, while for N it suffices to add the binary functional symbols (<m)m∈N\{0}, where ?,? x <m y if and only if x < y ∧ x ≡m y . 1.2. Logical definability Here we are concerned with the definability issue. We recall that given a logical structure D with domain D and a first order formula on this structure, say φ(x1, . , xn) where x1, . , xn is the set of free variables, the n-ary relation R defined by φ is the set of n-tuples (a1, . , an) such that φ holds true when the variable xi is assigned the value ai, i.e., R = n {(a1, . , an) ∈ D | D |= φ(a1, . , an)}. Example 1.1. E.g., the formula (x1 + x2 = 0) ∨ (x1 = x2 + 1) defines, in the structure Z, the union of a point and of a line in the discrete plane. 2. N-linear and Z-linear sets 2.1. Some notations The free abelian monoid and the free abelian group on k generators are respectively identified with Nk and Zk with the usual additive structure. The addition is extended from elements to subsets: if X, Y ⊆ Nk (resp. X, Y ⊆ Zk), X + Y ⊆ Nk (resp. X + Y ⊆ Zk) is the set of all sums x + y where x ∈ X and y ∈ Y . It might be convenient to consider the elements of Nk and Zk as vectors of the Q-vector space Qk. Given v in Nk or in Zk, the expression Nv represents the subset of all vectors nv where n range over N. This expression can be extended to Zv in a natural way whenever v is in Zk. Thus Zu + Zv represents the subgroup generated by the vectors u and v. 2.2. Linear sets The following discussion requires (the adaptation of) few definitions. The symbol K stands either for N or for Z when concerning the free abelian May 31, 2007 19:47 WSPC - Proceedings Trim Size: 9in x 6in ictcs 4 group Zk or for N when concerning the free abelian monoid Nk. Definition 2.1. A subset of Zk (resp. Nk) is K-linear if it is of the form n X k k a + Kbi, a, bi ∈ Z (resp. N ), i = 1, . , n. (1) i=1 k It is K-simple if the bi’s are linearly independent as vectors of Q . It is K-semilinear if it is a finite union of K-linear sets and K-semisimple if it is a finite disjoint union of K-simple sets. A subset of Zk is Z-quasisimple Pn if it is of the form A + i=1 Zbi, where A is a finite set such that for all a, a0 ∈ A the vector (a − a0) belongs to the Q-vector space spanned by the bi’s. Example 2.1. The subset Z(1, 0) ∪ Z(0, 1) is Z-semilinear. It is also N- semilinear and N-semisimple (it is equal to the union of N(1, 0), (−1, 0) + N(−1, 0), N(0, 1) and (0, −1) + N(0, −1)). The subset {(0, 1), (1, 0)} + Z(2, 0)+Z(0, 2) is Z-quasilinear. The subset {(0, 1, 0), (1, 1, 1)}+Z(2, 0, 0)+ Z(0, 2, 0) is Z-semilinear but not Z-quasilinear. Ginsburg and Spanier proved? the following result for Nk, but it can readily be seen to hold for Zk. Theorem 2.1. Given a subset X of Nk (resp. Zk) the following assertions are equivalent: (i) X is first order definable in N (resp. Z); (ii) X is N-semilinear; (iii) X is N-semisimple. Example 2.2. The binary relation of Example 1.1 is the union of the two Z-linear subsets: {(1, 1)} ∪ Z(1, −1). Clearly, a finite union of Z-linear subsets is also a finite union of N-linear subsets but the converse does not hold, e.g., a moment’s reflection will convince the reader that the subset N is not expressible as a finite union of Z-linear subsets. Still every Z-linear set is ZW -definable. Indeed, given Pn X = a + i=1 Zbi, then x = (x1, . , xk) ∈ X if, and only if, the following May 31, 2007 19:47 WSPC - Proceedings Trim Size: 9in x 6in ictcs 5 (i) formula holds (where bj means the i-th component of the vector bj): k n ^ (i) X (i) P(x) = ∃z1 ... ∃zn (xi = a + zjbj ) . (2) i=1 j=1 Would the family of finite unions of Z-linear subsets by chance capture the notion of subsets which are definable in the structure ZW ? This is not the case, since this family is not closed under taking the complement. Actually we will see that the Boolean closure of the family of Z-linear subsets is precisely the class of ZW definable sets, see Theorem 4.1.