Decidability frontier for fragments of first-order logic with transitivity? Lidia Tendera[0000−0003−0681−4040] University of Opole, Poland [email protected] Abstract. Several decidable fragments of first-order logic have been identified in the past as a generalisation of the standard translation of modal logic. These include: the fluted fragment, the two-variable frag- ment, the guarded fragment and the unary negation fragment; some of them have been recently generalised or combined to yield even more expressive decidable logics (guarded negation fragment or uniform one- dimensional fragment). None of the fragments allows one to express tran- sitivity of a binary relation or related properties like being an equivalence, a linear or a partial order, that naturally appear in specifications or in verification. The question therefore arises what is the impact of adding transitivity to these fragments and, where the cost is too high, how can these languages be tamed. In this talk we survey results concerning the decidability frontier of the above-mentioned fragments extended with transitivity. We discuss both, general and finite satisfiability, as presence of transitivity axioms often allows one to express axioms of infinity. Simultaneously, we locate in the picture known description logics, discuss relevant technics, admire a few exotic results and state some open questions. Keywords: First-Order logic · Decidability · (Finite) Satisfiability · Transitivity · Complexity. Introduction and Preliminaries It is well-known that formulas of propositional modal logics under Kripke se- mantics can be naturally encoded in first-order logic, using the so-called stan- dard translation (cf. [2]); the same applies to the basic description logic ALC and many of its variations. Since first-order logic is not so well-behaved as modal or description logics, in particular the (finite) satisfiability problems are undecid- able, it was natural to ask what the right image of the standard translation is and 'Why is modal logic so robustly decidable?'. This question opened an in- teresting and fruitful research path in theoretical computer science, and in this talk we explore part of the path. The formulas obtained under the standard translation evince some patterns: (i) variables appear in some fixed order and no rescoping of variables occurs, (ii) ? The talk is supported by the European Association for Artificial Intelligence. quantifiers are relativized by atomic formulas, (iii) negation is applied only to subformulas with a single free variable. Moreover, by properly reusing variables: (iv) the number of variables can be restricted to two. The four syntactic restrictions considered separately give rise to four frag- ments of first order logic: (i) the fluted fragment FL, (ii) the guarded fragment GF, (iii) the unary negation fragment UNF, and (iv) the two-variable fragment FO2. All of the four languages have the finite model property, and hence are decidable. As already mentioned in the abstract, none of them can express tran- sitivity of a binary relation or related properties like being a partial order or an equivalence relation. Hence, their extensions are studied when it is assumed that some of the binary symbols from the signature are interpreted in a special way (as transitive relations, orderings, equivalences etc.). In the talk we review main properties of the above mentioned fragments and describe the decidability frontier for their extensions with various transitive relations. The presentation includes joint work with Georg Gottlob, Emanuel Kiero´nski,Jakub Michaliszyn, Andreas Pieris, Ian Pratt-Hartmann, and Wies law Szwast. In the next section we recall formal definitions of the four fragments mentioned above; more surveying material with several references can be found e.g. in [7, 4]. In Section 2 we present supplementary material that is featured in the talk but has not yet been published elsewhere. 1 Definitions We assume relational signatures not containing constants or function symbols. Definition 1. The fluted fragment [5]: Let x¯! = x1; x2;::: be a fixed sequence of variables. We define the sets of formulas FL[k] (for k ≥ 0) by structural induction as follows: (i) any atom α(x`; : : : ; xk), where x`; : : : ; xk is a contiguous [k] [k] subsequence of x¯!, is in FL ; (ii) FL is closed under boolean combinations; [k+1] [k] (iii) if ' is in FL , then 9xk+1' and 8xk+1' are in FL . The set of fluted S [k] formulas is defined as FL = k≥0 FL . A fluted sentence is a fluted formula over an empty set of variables, i.e. an element of FL[0]. Thus, when forming Boolean combinations in the fluted fragment, all the combined formulas must have as their free variables some suffix of some prefix x1; : : : ; xk of x¯!; and when quantifying, only the last variable in this sequence may be bound. Definition 2. The guarded fragment [1], GF, is defined as the least set of for- mulas such that: (i) every atomic formula belongs to GF; (ii) GF is closed under logical connectives :; _; ^; !; and (iii) quantifiers are appropriately relativised by atoms. More specifically, in GF, condition (iii) is understood as follows: if ' is a formula of GF, α is an atomic formula featuring all the free variables of ', and x¯ is any sequence of variables in α, then the formulas 8x¯(α ! ') and 9x¯(α ^ ') belong to GF. In this context, the atom α is called a guard. The equality symbol when present in the signature is also allowed in guards. Definition 3. The unary negation fragment [6], UNF, consists of formulas in which the use of negation is restricted only to subformulas with at most one free variable. More precisely, UNF is defined as the least set of formulas such that: (i) every atomic formula of the form R(¯x) or x = y belongs to UNF; (ii) UNF is closed under logical connectives _, ^ and under existential quantification; (iii) if '(x) is a formula of UNF featuring no free variables besides (possibly) x, then :'(x) belongs to UNF. Definition 4. The two variable fragment: By the k-variable fragment of a logic L, denoted Lk, we mean the set of formulas of L featuring at most k distinct variables. In particular FOk denotes the set of all first-order formulas with at most k variables. All of the above languages are incomparable in terms of expressive power, have the finite model property and the satisfiability problem (=finite satisfiabil- ity problem) is decidable. 2 Fluted Fragment with Transitive Relations We show two undecidability results for the fluted fragment with two variables, FL2, extended with transitive relations. We employ the apparatus of tiling sys- tems (cf. e.g. [3]). A tiling system is a tuple C = (C; CH ; CV ), where C is a finite set of tiles, and CH , CV ⊆ C × C are the horizontal and vertical constraints. Let S be any of the spaces N × N, Z × Z or Zt × Zt. A tiling system C tiles S, if there exists a function ρ : S !C such that for all (p; q) 2 S:(ρ(p; q); ρ(p + 1; q)) 2 CH and (ρ(p; q); ρ(p; q + 1)) 2 CV . It is known that the problem whether a given tiling system tiles any of the spaces N×N, Z×Z, Zt ×Zt is undecidable. To reduce one of the problems to the (finite) satisfiability problem for some logic L we proceed as follows. Let C = (C; CH ; CV ) be a tiling system. We write an L-formula ηC such that the following reduction properties hold (i) ηC is satisfiable iff C tiles N × N or Z × Z, and (ii) ηC is finitely satisfiable iff C tiles Zt × Zt, for some t ≥ 1. Properties (i) and (ii) above yield undecidability of the satisfiability, respectively, finite satisfiability, problem for L. In order to encode tilings using two-variable logics it is useful to define struc- tures that are grid-like. A structure G = (G; h; v) with two binary relation h and v is grid-like, if one of the standard grids N × N, Z × Z or Zt × Zt can be homomorphically embedded into G. To show that a structure G is grid-like it suffices to require that G j= 8x(9yh(x; y) ^ 9yv(x; y)) (note that this is an FL2-formula) and the following confluence property holds G j= 8x; x0; y; y0((h(x; y) ^ v(x; x0) ^ v(y; y0) ! h(x0; y0)): (*) The confluence property above uses four variables and is not fluted. We will enforce it (or a variation of it) in FL2 using transitive relations. 2.1 Undecidability of FL2 with two transitive relations and equality Suppose the signature contains two transitive relations b (blue) and r (red), and additional unary predictes ci;j (0 ≤ i ≤ 3, 0 ≤ j ≤ 3) called colours. We write a formula 'grid capturing several properties of the intended expansion of the Z×Z grid as shown03 in Figure13 1.23 There,33 each element03 13 with23 coordinates33 (k; l) satisfies ci;j, where i = k mod 4 and j = l mod 4 and the transitive relations connect only some elements that are close in the grid. The formula 'grid is a conjunction 02 12 22 32 02 12 22 32 of the statements (1)-(7) described below; in the formulas in this subsection we assume that addition in subscripts of the cijs is always understood modulo 4. 01 11 21 31 01 11 21 31 00 10 20 30 00 10 20 30 03 13 23 33 03 13 23 33 02 12 22 32 02 12 22 32 01 11 21 31 01 11 21 31 00 10 20 30 00 10 20 30 Fig. 1. Intended expansion of the Z × Z grid with transitive relations b and r.