D. Seese. The structure of models of decidable monadic theories of graphs. Annals of Pure and Applied Logic, 53:169-195, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for the veri cation of in nite state systems, recently such graph classes have received increasing interest, see [28] for an overview. Courcelle showed that the class of graphs of tree width at most b has a decidable MSO theory (for any b 2 N) 5] A partial converse was proved by Seese [24] This work was done while the rst author worked at University of Leicester, parts of it were done while the second author was on leave at IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France and supported by the INRIA cooperative research action FISC. in conjunction with another result by ....

....cation over sets of edges is not possible. On the other hand, for graphs of bounded degree, Courcelle [6] has shown that the extension of MSO by quanti cation over sets of edges, which is known as MSO 2 , can be de ned within MSO. Thus the MSO 2 theory of G is decidable. A result of Seese [24] implies that undir(G) has nite tree width. Thus, assume that H = undir(G) has tree width at most b for some b 2 N. Then also any nite subgraph of H has tree width at most b. Since the degree of H is bounded by some constant d, the same holds for its nite subgraphs. Hence, by a result from ....

D. Seese. The structure of models of decidable monadic theories of graphs. Annals of Pure and Applied Logic, 53:169-195, 1991.

....it is in some sense hard to find properties of graphs that correspond to NP complete problems which are not MSO. One example is Bandwidth which is W [t] hard for all t even for trees [BFH94] Also in terms of MSO, bounded treewidth is the boundary of tractability. Theorem 4. 6 (Seese [Se91]) Suppose that C is any family of graphs with a decidable monadic second order (MS 2 ) theory. Then there is a number n such that for all G 2 F , the treewidth of G is less than n. Finally treewdith is of interest to us also because its recognition, despite being NP complete, is FPT. The ....

D. Seese, "The structure of Models of Decidable Monadic Theories of Graphs," Ann. Pure and Appl. Logic, Vol. 53, (1991) 169-195.

....GSO model checking is decidable for A if and only if A is of finite tree width. The same holds for GSO(# ) Proof. If A is of finite tree width then GSO(# ) collapses to MSO(# ) which is decidable (see Courcelle [10] 13] The other direction is a special case of a result of Seese [23]. If A has infinite tree width then its Gaifman graph contains subgraphs K n,n for all n 0 . Note that the grid [n] n] is bipartite since we can partition the vertices (i, k) n] depending on whether i k is even or odd. Thus, also contains subgraphs [n] n] for all n 0 . ....

D. Seese, The structure of the models of decidable monadic theories of graphs, Annals of Pure and Applied Logic, 53 (1991), pp. 169--195.

....(ii) However, on graphs of treewidth at most 1 (undirected forests) monadic PH collapses to Sigma 1 MSOL. i) is proven by Fagin [Fag75] for graphs of treewidth at most 2. ii) follows from the padding lemma for MSOL of Grohe and Mari no, GM99] A predecessor to theorem 11 may be found in [See91] who showed a similar theorem, were 9MSOL polynomial is replaced by decidability of the MSOL theory of K, and no closure condition is needed. Not so similar but related to theorem 11 are the characterization of classes of formulas for which the evaluation of first order conjunctive queries is ....

D. Seese. The structure of the models of decidable monadic theories of graphs. Annals of Pure Applied Logic, 53:169--195, 1991.

....of L by [55] 5) 8) by a result of Courcelle and S enizergues [27] 2 7. 2 The structure of sets of graphs having decidable monadic theories By combining techniques from logic (de nable graph transductions) and from graph theory (minor inclusion) one obtains the following result of Seese [58]. Theorem 7.7 : If a set of graphs has a decidable MS 2 theory, then it is a subset of some HR set of graphs. Proof sketch : The mapping from a graph to the set of its minors is (2,2) de nable. Hence, if a set L of graphs has a decidable MS 2 theory, then so has the set of its minors. But this ....

....one (Corollary 6.7) saying that the MS 2 theory of a HR set is decidable. We make the following conjecture. Conjecture 7.8 : If a set of graphs has a decidable MS 1 theory then it is a subset of some V R set of graphs. 70 Special cases of this conjecture are proved in Courcelle [16] Seese [58] has made a conjecture which is equivalent to Conjecture 7.8 by the result of [34] in the generalized form given in [23] Acknowledgements Many thanks to Mrs. A. Dupont and to A. Pari es for the Latex typing, and to E. Grandjean and the referees for comments. ....

: SEESE D., The structure of the models of decidable monadic theories of graphs, Annals Pure Applied Logic 53 (1991) 169-195.

....in favor of the standard simpler operations de ning clique width. We can prove that edge contractions are not compatible with the other operations. Edge contraction is a fusion operation de ned from a binary relation. Our fusions are de ned from unary relations. Seese has conjectured in [See91] that if a set of graphs has a decidable monadic theory then it is tree like. If this conjecture is true, then this means that one cannot nd sets of operations generating more that classes of bounded clique width and that would satisfy Q5. ....

D. Seese. The structure of the models of decidable monadic theories of graphs. Annals of Pure Applied Logic, 53:169-195, 1991.

....MSformulas that are true on each element of G. This theory of G is decidable if it is recursive (i.e. there is an algorithm that decides whether any given MS formula holds for all the elements of G) Courcelle showed in [10] that the MS theory of the class of partial k trees is decidable. Seese [20] proved that if the MS theory of a class of finite graphs M is decidable, then the graphs in M have uniformly bounded tree width. Thus, tree width characterizes classes of finite graphs having decidable MS theories. Courcelle [11] showed that a recognizable set of partial k trees is ....

Seese, D. The structure of the models of decidable monadic theories of graphs. Ann. Pure Appl. Logic 53 (1991), 169--195.

....in [2] that the MS theory of the class of partial k trees This research was done while the author was at Simon Fraser University [8] The author s present address is Department of Computer Science, University of Toronto, Toronto, ON, Canada M5S 3G4; kabanets cs.utoronto.ca. is decidable. Seese [11] proved that if the MS theory of a class of finite graphs G is decidable, then the graphs in G have uniformly bounded tree width. Thus, tree width characterizes classes of finite graphs having decidable MS theories. Strictly speaking, the above results hold for so called MS 2 logic, where MS 2 ....

D. Seese. The structure of the models of decidable monadic theories of graphs. Ann. Pure Appl. Logic, 53:169--195, 1991. This article was processed using the L a T E X macro package with LLNCS style

....Only local tests are necessary to verify that successive rows in the grid correspond to successive tapes in a successful computation of the machine. In its infinitary version (i.e. Z Theta Z) the grid has been used, for instance, to obtain undecidability results for monadic second order theories [29, 24, 11, 18, 19]. The reader is referred to Borger et al. book [4] for further reading on tiling, dominoes, grids and (un)decidability. In this paper we prove undecidability results for computational mechanisms over finite terms. Turing computations are encoded in so called grid terms. Roughly speaking, a ....

Seese, D. The structure of the models of decidable monadic theories of graphs. Annals of Pure and Applied Logic 53 (1991), 169--195.

....Courcelle [Cou89] and later Arnborg, Lagergren, Seese [ALS91] Se92] showed that relative to the class of graphs of bounded tree width the satisfiability of monadic secondorder logic is decidable. The decidability of monadic second order theories of graphs in general was investigated by Seese [Se91]. It remained open whether existential monadic second order logic is closed under complement relative to the class of graphs of bounded tree width. In the present paper we show that this does not hold. Moreover, we introduce within the domain of graphs of bounded tree width a whole hierarchy of ....

D. Seese, The structure of the models of decidable monadic theories of graphs, Annals of Pure and Applied Logic 53 (1991), 169-195.

....a class T of trees such that the monadic second order theory of K is interpretable into the monadic second order theory of T . For the class of planar graphs, graphs of bounded genus and any class of graphs excluding an arbitrary fixed graph as a minor one can show that this conjecture holds (see [38] and [9] 7.2. Digraphs and Decidability (Dirk Vertigan) For a directed graph G, let G denote the corresponding undirected graph. Let C be a minor closed class of directed graphs. Let C be the corresponding minor closed class of undirected graphs, that is, C = fG : G 2 Cg. Note that C is not ....

D. Seese, The structure of the models of decidable monadic theories of graphs, Annals of Pure and Applied Logic 53 (1991) 169-195.

....may consist of vertices or of vertices and edges. In the latter case, quantified variables of MS formulas may denote sets of edges. We shall refer to MS logic with vertex and edge quantifications by the notation MS 2 and to MS logic with vertex quantifications only by the notation MS 1 . Seese [16] proved that if a set of finite graphs has a decidable MS 2 theory, then it has bounded tree width. He conjectured that if a set of graphs L has a decidable MS 1 theory (which is a weaker condition) then it is interpretable in a set of trees . This condition is equivalent by results in [10, 11, ....

....of Theorem 2.3.4 that the sets TWD(k) and TWD 0 (k) are HR. The corresponding fact for CWD and CWD is an open question (Question 2.2.4) 3. Assertions (3) and (4) of Theorem 2.3.4 correspond to Propositions 2.3.1 and 2.3.2, respectively. 4. The references are Courcelle [5] for (3) and Seese [16] for (4) Seese conjectured that every set of graphs the MS 1 theory of which is decidable is interpretable in a set of trees . This is equivalent to Conjecture 2.3.3 by the results of Engelfriet [13] and Courcelle and Engelfriet [10] 5. The proof of (4) of Theorem 2.3.4 sketched in Courcelle ....

D. Seese, The structure of the models of decidable monadic theories of graphs, Annals of Pure Applied Logic , 53, (1991), 169--195.

.... the means provided by (1) 2) and (3) 2 The following proposition is a corollary of a theorem of Trakhtenbrot [Tr50] on the undecidability of the first order logic of graphs (see [Co90b] for a discussion) We remark that Seese has shown that undecidability still holds even for planar graphs [Se75, Se91]. Proposition 1 ( Tr50, Se91] Given an MSO formula OE, there is no algorithm to decide if there is a finite graph G such that G j= OE. 2 We can now prove our main result. Theorem 1. There is no effective procedure to compute the obstruction set for a minor ideal F from a monadic second order ....

.... and (3) 2 The following proposition is a corollary of a theorem of Trakhtenbrot [Tr50] on the undecidability of the first order logic of graphs (see [Co90b] for a discussion) We remark that Seese has shown that undecidability still holds even for planar graphs [Se75, Se91] Proposition 1 ([Tr50, Se91]) Given an MSO formula OE, there is no algorithm to decide if there is a finite graph G such that G j= OE. 2 We can now prove our main result. Theorem 1. There is no effective procedure to compute the obstruction set for a minor ideal F from a monadic second order description of F . Proof. We ....

D. Seese. The structure of the models of decidable monadic theories of graphs. Annals of Pure and Applied Logic 53 (1991), 169--195.

....(see also [Se92] showed that relative to the class of graphs of bounded tree width the satisfiability of monadic second order logic is decidable. The importance of the notion of bounded treewidth for the decidability of theories of monadic second order logic in general was investigated by Seese [Se91]. It remained open whether closure under complement holds. In the present paper we show that this complementation fails; and we introduce within the domain of bounded tree width a whole hierarchy of graph classes where the same is true. These graph classes are introduced by means of a ....

D. Seese, The structure of the models of decidable monadic theories of graphs, Annals of Pure and Applied Logic 53 (1991), 169-195.

....of (S, i.e. of the structure S augmented with a binary relation which is a linear order of D S , and such that for any two linear orders and on D S it holds that Q(S, iff Q(S, Every CMS property is MS( 5.1) Conjecture ( 9] There exists a MS( property which is not CMS. Seese [24] has proved that if a set of graphs is such that its MS 2 theory is decidable then its elements have uniformly bounded tree width. See the beginning of Section 4 for the definition of tree width. 5.2) Conjecture ( 24] If a set of graphs L has a decidable MS 1 theory, then L def D (K) ....

....Conjecture ( 9] There exists a MS( property which is not CMS. Seese [24] has proved that if a set of graphs is such that its MS 2 theory is decidable then its elements have uniformly bounded tree width. See the beginning of Section 4 for the definition of tree width. 5. 2) Conjecture ([24]) If a set of graphs L has a decidable MS 1 theory, then L def D (K) for some MS definition scheme D where K is the set of binary trees. An equivalent form of this conjecture is discussed in [7] and proved for certain sets of chordal graphs and for any class of directed graphs L closed ....

D. Seese, The structure of the models of decidable monadic theories of graphs, Ann. Pure Applied Logic 53 (1991) 169-195.

.... 2) k 1) 2k 2 2. 26 Hence G contains Kk 1 as a subgraph. There exist in G two adjacent vertices x, y having the same color g n (x) g n (y) Hence g n is not a semistrong coloring of K n . We now discuss some consequences of Theorem (4. 1) We recall the conjecture made by Seese [Se] and already considered in [Cou8] saying that if a set of finite graphs L has a decidable MS1 theory then L t(B) for some (1,1) definable transduction t. We recall from section (1.1) that L denotes the set of edge complements of the graphs in L. 4.4) Proposition : Let L be a set of finite ....

....either few edges, or on the contrary are dense . It is still open for intermediate cases. Proof : Let L be uniformly k sparse. Since the identity on a superset of L is (1,2) definable, since L has a decidable MS1 theory, then it has also a decidable MS 2 theory, hence L has bounded tree width ([Se], Cou8] Hence L 2 t (B) for some (1,2) definable transduction t and L 1 i o t (B) t(B) where i maps G 2 to G 1 , and t = i o t is a (1,1) definable transduction. If L is uniformly k sparse and L has a decidable MS 1 theory, then, L also has a decidable MS 1 theory ....

Seese D., The structure of the models of decidable monadic theories of graphs. Annals of Pure and Applied Logic, 53 (1991) 169-195.

....are well known tools to prove undecidability results. The grid structure provides convenient means for encoding computation sequences of Turing machines. In its infinitary version (i.e. Z ThetaZ) it has been used for instance to obtain undecidability results for monadic second order theories [21,19,6,14,15]. A classical encoding of the computation of a Turing Machine can be done only with a local matching on a grid, where, roughly speaking, row i contains a description of the tape at time i, and column j contains the values of cell j of the tape during the computation. Only local tests are ....

D. Seese. The structure of the models of decidable monadic theories of graphs. Annals of Pure and Applied Logic, 53:169--195, 1991.

No context found.

Seese D., The structure of the models of decidable monadic theories of graphs. Annals of Pure and Applied Logic, 53 (1991) 169-195.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC