The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic second-order logic of graphs and of sets of graphs, using techniques from universal algebra, and the theory of formal languages. (By a graph, we mean a finite directed hyperedge-labelled hypergraph, equipped with a sequence of distinguished vertices.) A survey of this research can be found in Courcelle [ 111. An algebraic structure on the set of graphs (in the above sense) has been proposed by Bauderon and Courcelle [2,7]. The notion of a recognizable set of finite graphs follows, as an instance of the general notion of recognizability introduced by Mezei and Wright in [25]. A graph can also be considered as a logical structure of a certain type. Hence, properties of graphs can be written in first-order logic or in secondorder logic. It turns out that monadic second-order logic, where quantifications over sets of vertices and sets of edges are used, is a reasonably powerful logical language (in which many usual graph properties can be written), for which one can obtain decidability results. These decidability results do not hold for second-order logic, where quantifications over binary relations can also be used. Our main theorem states that every definable set of finite graphs (i.e., every set that is the set of finite graphs satisfying a graph property expressible in monadic second-order logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.”

By considering graphs as logical structures, one...

We define matchings, and show that they capture the essence of context-freeness. More precisely, we show that the class of context-free languages coincides with the class of those sets of strings which can be defined by sentences of the form 9 b', where ' is first order, b is a binary predicate symbol, and the range of the second order quantifier is restricted to the class of matchings. Several variations and extensions are discussed.

this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSL--TR--95--664, Stanford University

. Two well-known formalisms for the specication and computation of tree transductions are compared: the mso graph transducer and the attributed tree transducer with look-ahead, respectively. The mso graph transducer, restricted to trees, uses monadic second order logic to dene the output tree in terms of the input tree. The attributed tree transducer is an attribute grammar in which all attributes are trees; it is preceded by a look-ahead phase in which all attributes have nitely many values. The main result is that these formalisms are equivalent, i.e., that the attributed tree transducer with look-ahead is an appropriate implementation model for the tree transductions that are speciable in mso logic. This result holds for mso graph transducers that produce trees with shared subtrees. If no sharing is allowed, the attributed tree transducer satises the single use restriction. 1 Introduction Formulas of monadic second order (mso) logic can be used to express properti...

. The finiteness of ranges of tree transductions is shown to be decidable for TBY + , the composition closure of macro tree transductions. Furthermore, TBY + definable sets and TBY + computable relations are considered, which are obtained by viewing a tree as an expression that denotes an element of a given algebra. A sufficient condition on the considered algebra is formulated under which the finiteness problem is decidable for TBY + definable sets and for the ranges of TBY + computable relations. The obtained result applies in particular to the class of string languages that can be defined by TBY + transductions via the yield mapping. This is a large class which is proved to form a substitution-closed full AFL. 1 Introduction The finiteness problem is one of the classical decidability problems in formal language theory. For a given language of interest, one usually does not wish to know whether that language is finite (because it usually is not), but rather whether the l...

. The concept of tree-based picture generation is introduced. It is shown that there are equivalent tree-based definitions of four picture-generating devices known from the literature, namely collage grammars, mutually recursive function systems, context-free chain-code grammars, and 0L-systems with turtle interpretation. Furthermore, generalisations of each of these systems are discussed. 1 Introduction During the last two decades picture generation has become a large field whose manyfold aspects are studied in mathematics as well as in practical and theoretical computer science. It attracts the interest of numerous researchers from diverse directions, which is no surprise because one can find in this area a great number of intellectually appealing mathematical and computational problems, interesting applications like the modelling of plant development and, one should not forget to mention this, an astonishing variety of beautiful pictures. In this paper picture generation is studied...

We show that simultaneous rigid E-unification, or SREU for short, is decidable and in fact EXPTIME-complete in the case of one variable. This result implies that the ... fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the 99-fragment, we obtain a complete classification of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. It is also proved that SREU with one variable and a constant bound on the number of rigid equations is Pcomplete.

We prove that for each k, there exists a MSO-transduction that associates with every graph of tree-width at most k one of its treedecompositions of width at most k. Courcelle proves in (The Monadic secondorder logic of graphs, I: Recognizable sets of finite graphs) that every set of graphs is recognizable if it is definable in Counting Monadic Second-Order logic. It follows that every set of graphs of bounded tree-width is CMSOdefinable if and only if it is recognizable. A fundamental theorem by Buchi [2] states that a language of words is recognizable iff it is definable by some formula in a monadic second-order logic (MSOL). This result is extended to finite ranked ordered trees by Doner [7], and to sets of finite unranked unordered trees by Courcelle [3]. This last result uses an extension of MSOL, called counting monadic second-order logic (CMSOL), that allows counting of cardinality of sets modulo fixed integers. These results relate an algebraic aspect, namely recognizab...

. This paper considers languages in a free algebra which has a binary associative operation called the series product. We define automata operating in these algebras and rational expressions, and we show that their expressive powers coincide (a Kleene theorem). We also show that this expressive power equals that of algebraic recognizability (a Myhill-Nerode theorem). This generalizes the work of Thatcher and Wright. The first equivalence continues to hold when conditions such as associativity and commutativity are imposed on the term operations, but recognizability is weaker when one of the term operations (other than the series product) is associative. We also consider languages which have a bound on the number of nested occurrences of certain designated term operations and get both the equivalences mentioned above. This generalizes our earlier work and answers a question left open therein. 1 Automata form one of the most commonly used computing device, from their histor...