A characterization of context-free NCE graph languages by monadic second-order logic on trees

2005 ◽  
pp. 311-327 ◽  
Author(s):  
Joost Engelfriet
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Context Free ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Graph Languages

Recognizable languages of arrows and cospans

2018 ◽  
Vol 28 (8) ◽  
pp. 1290-1332
Author(s):  
H. J. SANDER BRUGGINK ◽  
BARBARA KÖNIG
Keyword(s):  
Finite Automata ◽  
General Setting ◽  
Second Order ◽  
Order Logic ◽  
Straightforward Generalization ◽  
Simple Logic ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Graph Languages

In this article, we generalize Courcelle's recognizable graph languages and results on monadic second-order logic to more general structures.First, we give a category-theoretical characterization of recognizability. A recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets. This can be seen as a straightforward generalization of finite automata. We show that our notion corresponds to recognizable graph languages if we apply the theory to the category of cospans of graphs.In the second part of the paper, we introduce a simple logic that allows to quantify over the subobjects of a categorical object. Again, we show that, for the category of graphs, this logic is equally expressive as monadic second-order graph logic (msogl). Furthermore, we show that in the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle's result that everymsogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into automaton functors.

On spectra of sentences of monadic second order logic with counting

2004 ◽  
Vol 69 (3) ◽  
pp. 617-640 ◽  
Author(s):  
E. Fischer ◽  
J. A. Makowsky
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Graph Grammars ◽  
Free Graph ◽  
Unary Relation ◽  
Tree Width ◽  
Binary Relation Symbol ◽  
Context Free ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Abstract.We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle.Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result.

scholarly journals Graphs and Decidable Transductions based on Edge Constraints

1994 ◽  
Vol 1 (4) ◽  
Author(s):  
Nils Klarlund ◽  
Michael I. Schwartzbach
Keyword(s):  
Spanning Trees ◽  
Second Order ◽  
Order Logic ◽  
Graph Grammar ◽  
Hoare Logic ◽  
Free Graph ◽  
Graph Families ◽  
Context Free ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We give examples to show that not even <strong> c-edNCE</strong>, the most general known notion of context-free graph grammar, is suited for the specification of some common data structures.<br /> <br />To overcome this problem, we use monadic second-order logic and introduce <em> edge constraints</em> as a new means of specifying a large class of graph families. Our notion stems from a natural dichotomy found in programming practice between ordinary pointers forming spanning trees and auxiliary pointers cutting across.<br /> <br />Our main result is that for certain transformations of graphs definable in monadic second-order logic, the question of whether a graph family given by a specification A is mapped to a family given by a specification B is decidable. Thus a decidable Hoare logic arises.

scholarly journals Graphs and Decidable Transductions based on Edge Constraints: Extended abstract

1994 ◽  
Vol 23 (469) ◽  
Author(s):  
Nils Klarlund ◽  
Michael I. Schwartzbach
Keyword(s):  
Spanning Trees ◽  
Second Order ◽  
Order Logic ◽  
Graph Grammar ◽  
Hoare Logic ◽  
Free Graph ◽  
Graph Families ◽  
Context Free ◽  
Second Order Logic ◽  
Monadic Second Order Logic

<p>We give examples to show that not even <strong> c-edNCE </strong>, the most general known notion of context-free graph grammar, is suited for the specification of some common data structures.</p><p> </p><p>To overcome this problem, we use monadic second-order logic and introduce <em> edge constraints</em> as a new means of specifying a large class of graph families. Our notion stems from a natural dichotomy found in programming practice between ordinary pointers forming spanning trees and auxiliary pointers cutting across.</p><p> </p><p>Our main result is that for certain transformations of graphs definable in monadic second-order logic, the question of whether a graph family given by a specification <strong> A </strong> is mapped to a family given by a specification <strong> B </strong> is decidable. Thus a decidable Hoare logic arises.</p>

scholarly journals Map Genus, Forbidden Maps, and Monadic Second-Order Logic

10.37236/1656 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
B. Courcelle ◽  
V. Dussaux
Keyword(s):  
Orientable Surface ◽  
Second Order ◽  
Order Logic ◽  
Minimal Genus ◽  
Planar Embeddings ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Circular Order

A map is a graph equipped with a circular order of edges around each vertex. These circular orders represent local planar embeddings. The genus of a map is the minimal genus of an orientable surface in which it can be embedded. The maps of genus at most $g$ are characterized by finitely many forbidden maps, relatively to an appropriate ordering related to the minor ordering of graphs. This yields a "noninformative" characterization of these maps, that is expressible in monadic second-order logic. We give another one, which is more informative in the sense that it specifies the relevant surface embedding, in addition to stating its existence.

Sign in / Sign up

Export Citation Format

Share Document