Monadic second-order definable graph transductions

Vertex-minors, monadic second-order logic, and a conjecture by Seese

It is known that a language is context-free iff it is the set of borders of the trees of recognizable set, where the border of a (labelled) tree is the word consisting of its leaf labels read from left to right.We give a generalization of this result in terms of planar graphs of bounded tree-width. Here the border of a planar graph is the word of edge labels of a path which borders a face for some planar embedding. We prove that a language is context-free iff it is the set of borders of the graphs of a set of (labelled) planar graphs of bounded tree-width which is definable by a formula of monadic second-order logic.

Hyperedge replacement has been studied for more than twenty years as a rulebased device for the generation of sets of graphs, hypergraphs and other graphlike structures. By now, it is one of the best understood and developed branches of the theory of graph grammars providing mainly a graph-grammatical counterpart to context-free Chomsky grammars. But there is more to it than just context-freeness. Hyperedge replacement has got some other facets to overcome — at least to a certain degree — the restricted generative power and applicability: 1. parallel hyperedge replacement with a mode of rewriting like OL- and TOL-systems, 2. hyperedge replacement with rendezvous, 3. hyperedge replacement with required or forbidden context, 4. hyperedge replacement with rule-selection specifications like path expressions, 5. infinite hyperedge replacement generating potentially infinite graphs.

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.

Model-checking is one of the most powerful techniques for verifying systems and programs, which since the pioneering results by Knapik et al., Ong, and Kobayashi, is known to be applicable to functional programs with higher-order types against properties expressed by formulas of monadic second-order logic. What happens when the program in question, in addition to higher-order functions, also exhibits algebraic effects such as probabilistic choice or global store? The results in the literature range from those, mostly positive, about nondeterministic effects, to those about probabilistic effects, in the presence of which even mere reachability becomes undecidable. This work takes a fresh and general look at the problem, first of all showing that there is an elegant and natural way of viewing higher-order programs producing algebraic effects as ordinary higher-order recursion schemes. We then move on to consider effect handlers, showing that in their presence the model checking problem is bound to be undecidable in the general case, while it stays decidable when handlers have a simple syntactic form, still sufficient to capture so-called generic effects . Along the way, we hint at how a general specification language could look like, this way justifying some of the results in the literature, and deriving new ones.

The monadic second-order logic of graphs VII: graphs as relational structures

We establish that every equational graph can be characterized, up to isomorphism, by a formula of monadic second-order logic. It follows that the isomorphism of two equational graphs is decidable. Equational graphs can be used to describe the behaviour of recursive applicative program schemes. We obtain a sufficient and decidable condition for the equivalence of these program schemes.

This work presents an algebraic method based on rational transductions to study the sequential and parallel complexity of counting problems for regular and context-free languages. This approach allows to obtain old and new results on the complexity of ranking and unranking as well as on other problems concerning the number of prefixes, suffixes, subwords and factors of a word which belong to a fixed language. Other results concern a suboptimal compression of finitely ambiguous c.f. languages, the complexity of the value problem for rational and algebraic formal series in noncommuting variables and a characterization of regular and Z-algebraic languages by means of rank functions.KeywordsFormal SeriesRegular LanguageRegulate RepresentationCounting ProblemConstant SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recent results concerning efficient solvability of graph problems on graphs with bounded tree-width and decidability of graph properties for hyperedge-replacement graph grammars are systematised by showing how they can be derived from recognisability of corresponding tree classes by finite tree automata, using only well-known techniques from tree-automata theory.

Every graph can be represented uniquely in a hierarchical way by means of its modular decomposition. We establish that the modular decomposition of a linearly ordered graph is definable in monadic second-order (MS) logic in the graph itself. The modular decomposition does not depend on the linear order of the given graph. A set of graphs is recognizable w.r.t. the operations associated with graph substitution iff it is definable by a formula of an extension of MS logic based on the use of auxilliary linear orderings. This paper is an extended abstract: complete proofs can be found in [7].

Computing with graph rewriting systems with priorities

Handle-rewriting hypergraph grammars

We present a method for generating random hypergraphs in context-free hypergraph languages. It is obtained by adapting Mairson's generation algorithm for context-free string grammars to the setting of hyperedge replacement grammars. Our main results are that for non-ambiguous hyperedge replacement grammars, the method generates hypergraphs uniformly at random and in quadratic time. We illustrate our approach by a running example of a hyperedge replacement grammar generating term graphs.

Borel hierarchy and omega context free languages

This tutorial will present an overview of the use of Monadic Second-Order Logic to describe sets of finite graphs and graph transformations, in relation with the notions of tree-width and clique-width. It will review applications to the construction of algorithms, to Graph Theory and to the extension to graphs of Formal Language Theory concepts.