Two-Variable Descriptions of Regularity

We prove that the class of all languages that are definable in 11(FO2), that is, in (non-monadic) existential second-order logic with only two first-order variables, coincides with the regular languages. This provides an alternative logical description of regularity to both the traditional one in terms of monadic second-order logic, due to Büchi and Trakhtenbrot, and the more recent ones in terms of prefix fragments of 11, due to Eiter, Gottlob and Gurevich. Our result extends to more general settings than words. Indeed, definability in 11(FO2) coincides with recognizability by appropriate notions of automata on a large class of objects, including !-words, trees, pictures and, more generally, all weakly deterministic, triangle-free transition systems. 1 Logical characterizations of the regular languages An important connection between automata theory and mathematical logic was established in the early 1960’s by Büchi and Trakhtenbrot [5, 24], who showed that a language is regular if and only if it is definable in monadic secondorder logic MSO. This result has been extended in many directions. For example, McNaughton and Papert [16] characterized the regular languages that admit a first-order definition, and Büchi [6], McNaughton [15], and Rabin [18] established equivalences between monadic second-order logic and finite automata also on !-words and on trees. Recently, Thomas and others [9, 10, 20, 23, 21, 22] have developed an automata theory on more general objects (for instance two-dimensional pictures, partial orders, or arbitrary graphs of bounded degree) and established a close relationship between recognizability by automata and definability in existential monadic second-order logic. Let us briefly explain the connection between regularity and monadic second-order logic. A word w0 wn 1 2 A (for some finite alphabet A) can be viewed as a structure W = (n; S;min;max; (Pa)a2A) where the universe n = f0; : : : ; n 1g is the set of letter positions in the word, S is the usual successor relation on n, min and max are constants for the first and the last element of n, and the Pa are monadic predicates indicating the set of positions at which the letter a occurs, i.e. Pa = fi < n : wi = ag. We write A for the vocabulary fS;min;max; (Pa)a2Ag of word structures for the alphabet A. A sentence of vocabulary A defines the language L( ) A consisting of all words W such that W j= . The Büchi-Trakhtenbrot Theorem says that a language is regular if and only if it is definable by a sentence in monadic second-order logic. We write FO for first-order logic and 11 := f9R1 9Rk' : ' 2 FOg for existential second-order logic. The proof of the Büchi-Trakhtenbrot Theorem shows actually that the existential fragment mon 11 of MSO suffices to define all regular languages and that therefore, over word structures, MSO collapses to mon 11. Recall that, by Fagin’s Theorem, existential secondorder logic captures the complexity class NP and is hence much more powerful than finite automata. Recently, Eiter, Gottlob and Gurevich gave new characterizations of the regular languages by fragments of non-monadic existential second-order logic. Instead of restricting secondorder quantification to monadic predicates (as in the BüchiTrakhtenbrot Theorem), they admit quantification over arbitrary relations but restrict the quantifier prefix of the firstorder part of the formulae. Definition 1.1. Let X FO be a fragment of first-order logic. We denote by 11(X) the set of all sentences of the form 9R1 9Rk' such that ' is a sentence in X . A prefix class in FO is given by a set of words Q over the alphabet f9;8g that is closed under taking subwords (i.e. if w 2 Q and w0 is obtained by deleting some of the letters in w, then also w0 2 Q). We identify such a set of prefixes with the set of first-order formulae of form Q1x1 Qrxr' such that ' is quantifier-free and the word Q1 Qr belongs to Q. Fragments of first-order logic defined by quantifier prefixes have been studied very intensively in the context of the classical decision problem (see [4]). Eiter, Gottlob and Gurevich gave a complete classification of the prefix classes Q such that 11(Q) over words characterizes the regular languages. Theorem 1.2 (Eiter, Gottlob, Gurevich). Let Q FO be any prefix class in first-order logic. Then, over word structures, either (i) Q 9 89 [ 9 82 = 9 8(8 [ 9 ) and 11(Q) defines only regular languages, or (ii) Q contains at least one of the prefix classes 888, 889, 898 and 11(Q) defines some NP-complete language. Hence there is an interesting gap phenomenon. For any logic 11(Q) (whereQ is a prefix class in FO) the data complexity of model checking on words is either very simple (it can be done by a finite automaton) or NP-complete. There are no intermediate levels of complexity. Further, in the case of (i) the reduction from a given sentence 2 11(Q) to an automaton recognizing the language defined by is effective. Hence the satisfiability problem for 11(Q) over strings (or equivalently, the satisfiability problem forQ over finite successor structures) is decidable. It was observed by the present authors that in the case of (ii) the satisfiability problem for 11(Q) over finite successor structures is undecidable (see [7, p. 24-25] for a sketch of the proof). As a consequence one obtains another dichotomy theorem. Corollary 1.3. Let Q be any prefix class in first-order logic. Then, over words, either (i) the satisfiability problem for Q is decidable and 11(Q) defines only regular languages, or (ii) the satisfiability problem for Q is undecidable and 11(Q) defines some NP-complete language. There are of course other fragments of first-order logic which are known to be ‘well-behaved’ in a certain sense, and which may possibly lead to interesting fragments of 11. The two-variable fragment of first-order logic is one example which has recently been studied quite intensively ([8, 12, 13]). Definition 1.4. Two-variable first-order logic FO2 is the set of all first-order formulae whose vocabulary contains only relation symbols and constants (but no function symbols of positive arity), and which contain only two variables x and y. It is known that FO2 has the finite model property [17] and the satisfiability problem for FO2 is complete for NEXPTIME [12]. Etessami, Vardi and Wilke [8] establish an interesting relationship between FO2 and regular languages that are definable by a fragment of temporal logic. Recently, the expressive power of 11(FO2) has been investigated by Le Bars [2, 3]. Our main result is the following new characterisation of the regular languages. Theorem 1.5. 11(FO2) characterizes the regular languages. The fact that every regular language can be defined by a sentence in 11(FO2) is not difficult to prove and wellknown. Indeed, the straightforward description of the behaviour of a finite automaton in monadic second-order logic only requires the use of monadic 11-sentences whose firstorder parts have prefix 88 or 89 (see e.g. [7]). Hence REG = mon 11 = mon 11(88) = mon 11(89) = mon 11(FO2) 11(FO2): The other direction is not obvious, and the proof will be given in Sect. 2. The result may be useful for verification purposes since interesting properties of transition systems can sometimes be described more naturally by nonmonadic existential second-order formulae than by monadic ones. If such a description requires only two variables, then our main result immediately implies that the property can be checked by a finite automaton. In Sect. 5 we extend our main result to more general settings than words. Indeed, we can show that definability in 11(FO2) coincides with recognizability by appropriate notions of automata on a large class of objects, including !words, trees, pictures and, more generally, all weakly deterministic, triangle-free transition systems. In Sect. 6 we discuss the power of 11(GF) on words, where GF is the guarded fragment, another ‘well-behaved’ portion of first-order logic that has received a lot of attention recently. However, it will turn out that 11(GF) has the same expressive power as 11 on words and hence captures all of NP. 2 11(FO2) characterizes the regular languages In this section we prove our main result, that every sentence in 11(FO2) defines a regular language. To this end we will show that, over words, every 11(FO2)-sentence is equivalent to an 11(9 88)-sentence. We can then use the result of Eiter, Gottlob and Gurevich that, on words, every sentence in 11(9 88) is equivalent to a monadic 11-sentence and hence defines a regular language. Theorem 2.1. Over words, every 11(FO2)-sentence is equivalent to a sentence in 11(9 88).