B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an algorithm to find such a total coloring in time O(sn ) where n is the number of the vertices. It is known that many NP hard problems, including the ordinary vertex and edge coloring problems, can be efficiently solved, mostly in linear time, for partial k trees, graphs with treewidth k [ACPS93, AL91, BPT92, Cou90]: for example, the vertex coloring problem can be solved in linear time for partial k trees by a standard dynamic programming algorithm# and the edge coloring problem can be solved in linear time for partial k trees [ZNN96] However, there has been no known efficient algorithm to solve the total ....

B. Courcelle, The monadic second-order logic of graphs I: Recognizable sets of finite graphs, Inform. Comput., 85, pp. 12--75, 1990.

....to the subset relation on the defined sets of feature trees. We are curious to extend the developed theory in the following ways. First, we would like to find a logical characterization of the class of recognizable feature trees, extending the results of Doner, Thatcher Wright and Courcelle [Don70, TW67, Cou90]. It will be interesting to combine second order logic and the counting constraints introduced here, in order to account for the flexibility in the depth as well as in the out degree of the nodes of feature trees. Also, in order to account for circular data structures, like, e.g. circular lists, ....

Bruno Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, pages 12-75, 1990.

....been investigated in [23] including another kind of local control mechanism) 3] deals with the expressibility of backtracking graph algorithms by means of PGRS s. In [19] the computational power of PGRS s is studied and some links with classes of graphs definable by logic formulae (see e.g. [7, 8]) are established. The use of PGRS s as a tool for encoding and proving graph algorithms or algorithms on networks is discussed in [21] Finally, the power and limitations of local computations on graphs (by means of PGRS s) is discussed in [24] 9 2 Expanding Graph Relabeling Systems and Graph ....

B. Courcelle, The monadic second-order logic of graphs I : recognizable sets of finite graphs, Inform. and Comput. 85 (1990), 12-75.

....Note that all these steps can be done in constant time, and we can actually do them if the constant time algorithms to compute S[H i ] and S S[H] are known. This completes the proof. 2 13 As an important special case, we consider the graph properties that are MS definable (see e.g. [9] or [3] Let k 1. Suppose we have a graph property P which can be written as P (G) 9 S2D(G) Q(G;S) where D(G) D 1 (G) Theta D 2 (G) Theta Delta Delta Delta Theta D t (G) for some t 1, each D i (G) is either equal to V (G) to E(G) to P(V (G) or to P(E(G) and we have a definition ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990. 42

....of Bodlaender [6] we can determine that no such decomposition of width b(k) exists or be given a decomposition of G. In either case, the running time for this procedure is linear in the size of G but exponential only in k. By means of one of several general algorithmic design methodologies (see [1, 4, 5, 15, 20, 54]) we may then answer the original question in time linear in the size of G. Hence, for small values of k, this procedure may lead to algorithms that are practical even for very large graphs G. Examples where these methods have been successful include Treewidth, Pathwidth, Min Cut Linear ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990. 21

....Thomassen [19] and using a new graph coloring method, it is shown that if a E 1 ae sentence gives rise to a pure pattern graph, then a fixed integer k can be found such that every undirected graph without self loops and having tree width bigger than k satisfies . Second, Courcelle s theorem [5] is used by which model checking for MSO sentences is polynomial on graphs of bounded tree width. Third, Bodlaender s result [1] is used that, for each fixed k, there is a polynomial time algorithm to check if a given graph has tree width at most k. The polynomial time algorithm for mixed pattern ....

B. Courcelle. The monadic second-order logic of graphs I: recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

.... notice, that for fixed v; w; a 1 ; a k 1 ; b 1 ; b k 1 , the existence of T fulfilling the given properties can be formulated in Monadic Second Order Logic, and hence be decided (with an algorithm that can be constructed) in linear time for graphs of bounded treewidth (see [1, 10, 12]) As we must make in total less than jEj Delta n Delta k = O(n ) checks, the time bound follows. The algorithms for the cases of k IRS, k LIRS, and k SLIRS are similar: because b k 1 is not used, the time bounds for k LIRS and k SLIRS are a linear factor smaller. ut Corollary 13 One can ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....relation of the vertices of the graph. Although we don t have the equality operator in the definition of MS, we used it freely, since it can be written as (Z = Y ) j 8x(x 2 Z ( x 2 Y ) We say a graphic problem has the MS property if it can be formulated in the MS formulation. Courcelle [9, 10] introduced the use of MS to solve problems restricted to partial k trees. He proved that any problem that can be formulated by a MS formula has a linear time algorithm, when restricted to graphs with bounded tree width and if a tree decomposition of the graph is given. Later, Arnborg et al. 1] ....

B.Courcelle "The monadic Second order Logic of graphs I: Recognizable sets of finite graphs", Information and Computation, 85, 12-75 (1990).

....the constrained subproblem restricted to G u , the sourced subgraph of G represented by the subtree of T rooted at u. The table of a leaf is initialized according to the base case, usually by a brute force strategy. The 4 G[9 8 7 6] 987(654321) 9876(54321) 9876(541) 876(32) 987(4) 976(51) G[10 9 8 7] G[8 7 6 3] G[9 8 7 4] G[8 7 6 2] G[9 6 5 1] G[9 7 6 5] 965(1) 9765(1) 9876(4) 876(2) 8763(2) JOIN REDUCE PRIMITIVE sources(non sources) 10 9 8(7654321) 10 9 8 7(654321) FORGET Fig. 2. The binary parse tree T of the partial 3 tree G based on the peo tree P (see Figure 1) Nodes u 2 ....

B.Courcelle, The monadic second-order logic of graphs I: Recognizable sets of finite graphs, Information and Computation 85: (1990) 12-75.

....and introducing a new graph coloring method, we show that if a E 1 ae formula gives rise to a pure pattern graph, then we can nd a xed integer k such that the formula is satis ed by every undirected graph without self loops having tree width bigger than k. Second, we use Courcelle s theorem [4] to the e ect that the model checking problem for formulas of monadic second order logic on graphs of bounded tree width is solvable on polynomial time. Third, we use Bodlaender s result [1] to the e ect that, for each xed k, there is a polynomial time algorithm to check if a given graph has ....

....a graph has tree width k can be done in linear time. The second result, by Thomassen [18] states that for every constant c there is an integer r(c) such that every undirected graph having tree width r(c) contains a cycle whose length is a multiple of c. The third result is Courcelle s Theorem [4], which states that for each constant b and each xed formula of monadic secondorder logic (MSO) the problem of deciding whether G j= for graphs of treewidth b is solvable in linear time. In particular, since SATU(P ) is in MSO, for each constant b there exists a linear algorithm ....

B. Courcelle. The Monadic Second-Order Logic of Graphs I: Recognizable Sets of Finite Graphs. Information and Computation, 85:12-75, 1990.

.... and sometimes even linear, time on graphs of bounded treewidth, using the covering hyperforest s tree decomposition [AP89] More generally, any predicate that can be expressed in (generalized) monadic second order logic over the graph can be solved in linear time for graphs of bounded tree width [Cou90]. The dependence on the treewidth is, of course, exponential. 54 Chapter 4 Hypertrees and the Maximum Hypertree Problem We are now ready to state the central combinatorial optimization problem this work is concerned with the maximum hypertree problem. In this chapter we present the problem ....

B. Courcelle. The monadic second-order logic of graphs i: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....Irred R 5 (G; has no I labelled vertex, and thus satis es , if and only if G has no cycle. Hence, the pair (R 5 ; K( is a deterministic recognizer for the class of trees. In [10, 13] the recognizable classes of graphs are compared to the classes of graphs de nable by logic formulas (see [4] for the notion of de nability by logic formulas) In particular, it is proved that (deterministically or not) recognizable classes of graphs are not comparable with classes of graphs de nable by logic formulas expressed in rst order logic (FOL) monadic second order logic (MSOL) or second order ....

Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of nite graphs. Inform. and Comput. 85:12-75, 1990.

....y x 3 = y) 2 It follows that one can express in CMS that a set X is nite and has a cardinality of the form q p for some 2 N where 1 q p : it suces to write that for some Y and Z, X = Y [ Z, Y Z = Card(Y ) q and Card(Z) is a multiple of p. We now recall a result from Courcelle [11]. Proposition 3.2 Every formula 2 MS( fX 1 ; Xn g) is equivalent to a nite disjunction of conjunctions of conditions of the forms Card(Y 1 Y 2 Yn ) m or Card(Y 1 Y 2 Yn ) m where m 2 N and, for each i = 1; n; Y i is either X i or D X i , where D is ....

....this variant, one could construct a second order formula 0 as in Proposition 4. 3 (with quanti cations on binary relations) but not a monadic second order one (at least in general) We wish to avoid non monadic second order logic because most constructions and decidability results (like those of [11], 12] break down. In the third example given before Fact 4.1.1, we have shown that the transduction associating the automaton A B with an automaton A is de nable (via the chosen representation of nite state automata by relational structures) for xed B. Here are some other closure properties ....

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: COURCELLE B., The monadic second-order logic of graphs I : Recognizable sets of nite graphs. Information and Computation 85 (1990) 12-75. 71

....of these important problems on graph grammars. 1 Introduction Graph grammars can be seen as the extension of context free grammars from strings to graphs. They are based on node replacement (NR graph grammars) Nag80,JR80a,JR80b,Cou87,Eng89,EKR91] or on edge replacement (HR graph grammars) [Cou90,Hab92]. A graph grammar consists of a finite set of productions, which are used to expand graphs or hypergraphs by repeated replacements of nodes or hyperedges. The language of a graph grammar is the set of all terminal labeled graphs derivable from some axiom. The most powerful NR graph grammars are ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Inform. and Comput., 85:12--75, 1990.

....is presented in [Eng94] Hyperedge Replacement In the theory of HR graph grammars and languages essentially three different types of graph properties with nice decidability and structural results have been found: the compatible, recognizable (or finite) and inductive graph properties. In [Cou90b, HKL93] it is shown that these notions are essentially equivalent. In [Cou90b] it was proved that every MSOL definable set of graphs is recognizable. This innovating paper was the start of a long series of powerful applications of MSOL to context free graph grammars. In [Cou91] sets of graphs ....

....grammars and languages essentially three different types of graph properties with nice decidability and structural results have been found: the compatible, recognizable (or finite) and inductive graph properties. In [Cou90b, HKL93] it is shown that these notions are essentially equivalent. In [Cou90b] it was proved that every MSOL definable set of graphs is recognizable. This innovating paper was the start of a long series of powerful applications of MSOL to context free graph grammars. In [Cou91] sets of graphs are investigated for which the reverse of the above implication also holds. In ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Inform. Comput., 85:12--75, 1990.

....automaton. This was generalized in [Don,ThaWri] to sets of node labeled ordered trees, with an appropriate generalization of the nite state automaton to the bottom up nite state tree automaton. The trees considered are the usual representations of terms over a nite set of operators. In [Cou1] one direction of the result was generalized to The present address of the rst author is: Department of Computer Science, University of Colorado at Boulder, P.O.Box 430, Boulder, CO 80303, email: Roderick. Bloem colorado.edu The second author was supported by ESPRIT BRWG No.7183 ....

....logic is recognizable in the algebraic sense (see [MezWri] for a speci c algebra of graphs. This was used to show that the class of context free graph languages, generated by context free graph grammars, is closed under intersection with mso de nable sets of graphs. Inspired by the ideas of [Cou1], a characterization of the context free graph languages in terms of monadic second order logic was shown in [Oos,Eng4] published in [EngOos] based on the following natural logical way to specify graph transductions, i.e. functions from graphs to graphs. The idea is that the output graph is ....

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B. Courcelle; The monadic second-order logic of graphs I: Recognizable sets of nite graphs, Inform. and Comput. 85 (1990), 12-75

....that PrExt with fixed color bound is solvable in polynomial time for graphs of bounded treewidth. Actually a slightly more general result is proved in [8] We note here that the result on bounded treewidth follows from the method of monadic second order logic for graphs, developed by Courcelle [4]. The metaresult is that every graph property expressible in the monadic second order logic is decidable in polynomial time on graphs of bounded treewidth. In the case of PrExt with color bound k, we consider unary predicates lab i (v) ae TRUE if v is precolored by color i FALSE otherwise, ....

Courcelle B., The monadic second order logic of graphs I: Recognizable sets of finite graphs, Information and Comput. 85 (1990), 12--75.

....in polynomial time is the class of tree width bounded graphs, see Bodlaender [Bod98] for a survey. All graph properties expressible in monadic second order logic with quanti cations over vertex sets and edge sets (MSO 2 logic) can be solved in linear time for tree width bounded graphs, see [Cou90] The MSO 2 logic has also been extended by counting mechanisms to express optimization problems which can then be solved in polynomial time for tree width bounded graphs, see [ALS91] It is already known that each graph of tree width at most k has clique width at most 2 k 1 1, see [CO00] ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of nite graphs. Information and Computation, 85:12-75, 1990.

....in proofs of properties of graph generating tree grammars (see, e.g. 8, 14, 12] Of particular interest are the (usual) topdown and bottom up tree transducers (see, e.g. 31] because they are syntaxdirected, i.e. compute recursively on the structure of the input tree. Moreover, implicitly in [6, 5, 35, 36, 16, 45] and explicitly by Drewes in [17] the topdown tree transducer was formulated as a model for the computation of boolean Supported by the ESPRIT BRWG No.7183 COMPUGRAPH II and numerical functions on graphs, i.e. the translation of graphs into booleans or natural numbers, using boolean ....

....operations (plus) Delta (times) max (maximum) and all constants in N. For this reason we will say that f is F transducible if it is F PTM transducible (for a set of graph operations F ) Taking F 0 to be the set of all boolean operations, transducible boolean functions were introduced in [6, 5], where they are called inductive predicates. In [35] they are called compatible predicates (for a different, but equivalent, F ) Transducible numerical functions (called compatible functions) were introduced in [36] and also studied in [16] where they are called inductively computable) ....

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B.Courcelle; The monadic second-order logic of graphs I: Recognizable sets of finite graphs, Inform. and Comput. 85 (1990), 12-75

....Note that these characterizations differ from the one of C edNCE: by definition, a language L is MSO definable if there exists a closed MSO formula OE such that L consists of all strings (trees, graphs) that satisfy OE. The class of MSO definable graph languages is incomparable with C edNCE (cf. [Cou2]) The proof of our MSO characterization is heavily based on the classical results of [Buc, Elg, Don, TW] for instance, the domain of an MSO definable function is MSO definable (by OE dom ) and corresponds directly to the regular tree language of the regular path description. A relationship ....

....of [Buc, Elg, Don, TW] for instance, the domain of an MSO definable function is MSO definable (by OE dom ) and corresponds directly to the regular tree language of the regular path description. A relationship between context free graph languages and MSO logic was first established by Courcelle in [Cou2] (see also Section 4 of [Cou3] where he showed that the class of HR context free graph languages is closed under intersection with MSO definable graph languages (generalizing the corresponding result for strings) For C edNCE and several of its subclasses similar results are shown in [Cou1, ....

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B.Courcelle; The monadic second-order logic of graphs I: Recognizable sets of finite graphs, Inform. and Comput. 85 (1990), 12-75

....lineartime algorithms for subgraph problems on certain families of graphs, that include the graphs with bounded treewidth. The set of problems solvable in linear time includes all problems that can be described using monadic second order logic (with some extension) as shown by Courcelle [4] and others [1, 3] and in particular it includes the edge deletion problem on trees for every fixed H. Using the approach of [2] the maximum subgraph problem can be solved in 2 p O(k) n time when the graph G has treewidth p. Throughout, n denotes the number of vertices in G, and k is the ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....of Bodlaender [7] we can find a decomposition of width b(k) for G or determine that no such decomposition exists. In either case, the running time for this procedure is linear in the size of G but exponential only in k. By means of one of several general algorithmic design methodologies (see [1, 5, 6, 15, 19, 52]) we may then answer the original question in time linear in the size of G. Hence, for small values of k, this procedure may lead to algorithms that are practical even for very large instances. Examples where these methods have been successful include Treewidth, Pathwidth, Min Cut Linear ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....see also [Cou87, Lemma 2.14] Note that this does not hold for the usual substitution. The idea of our intended transformation is based on the definition of a finite congruence on extended graphs with respect to the extended substitution mechanism. Similar concepts are considered, for example, in [Cou90, LW93, Wan91] for the hyperedge replacement embedding and the NLC embedding. Here, we restrict the definition of the congruence to extended graphs whose nodes are all terminal. In the following, we assume that each graph over Sigma; Gamma is also associated with the sets Delta; Omega of terminal labels. ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....are equal by the assumption and the associativity of the extended substitution. 2 The idea of our intended transformation is based on the definition of a finite congruence on extended graphs with respect to the extended substitution mechanism. Similar concepts are considered, for example, in [4, 16, 17] for the hyperedge replacement embedding and the NLC embedding. Here, we restrict the definition of the congruence to extended graphs whose nodes are all terminal. In the following, we assume that each graph over Sigma ; Gamma is also associated with the sets Delta; Omega of terminal labels. ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....been investigated in [23] including another kind of local control mechanism) 3] deals with the expressibility of backtracking graph algorithms by means of PGRS s. In [19] the computational power of PGRS s is studied and some links with classes of graphs definable by logic formulae (see e.g. [7, 8]) are established. The use of PGRS s as a tool for encoding and proving graph algorithms or algorithms on networks is discussed in [21] Finally, the power and limitations of local computations on graphs (by means of PGRS s) is discussed in [24] 2 Expanding Graph Relabeling Systems and Graph ....

B. Courcelle, The monadic second-order logic of graphs I : recognizable sets of finite graphs, Inform. and Comput. 85 (1990), 12-75.

....second order logic and its extensions, all problems that are finite state , etc. A large number of interesting and important graph problems can be dealt in this way, including CHROMATIC NUMBER, MAX IMUM CLIQUE, MAXIMUM INDEPENDENT SET, HAMILTONIAN CIRCUIT, STEINER TREE, LONGEST PATH, etc. See [3, 10, 9]. With the results of this paper, this implies that we can solve the problems described above on graphs of treewidth at most two with the same resource bounds as the algorithm for finding a tree decomposition of width two. One of the problems which can be solved if a tree decomposition of bounded ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

.... notice, that for fixed v; w; a 1 ; a k 1 ; b 1 ; b k 1 , the existence of T fulfilling the given properties can be formulated in Monadic Second Order Logic, and hence be decided (with an algorithm that can be constructed) in linear time for graphs of bounded treewidth (see [1, 10, 12]) As we must make in total less than jEj Delta n 2k 1 Delta k = O(n 2k 2 ) checks, the time bound follows. The algorithms for the cases of k IRS, k LIRS, and k SLIRS are similar: because b k 1 is not used, the time bounds for k LIRS and k SLIRS are a linear factor smaller. ut Corollary 13 ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....papers in this series) See also [20] Also, many graph problems, including a very large number of well known NP hard problems, have been shown to be linear time solvable on graphs that are given together with a tree decomposition of treewidth at most k, for constant k. See, amongst others [2, 5, 6, 7, 11, 9, 13, 14, 15, 31, 33]. The first step of algorithms that exploit small treewidth of input graphs is to find a tree decomposition with treewidth bounded by a constant, although possible This work was partially supported by the ESPRIT Basic Research Actions of the EC under contract 7141 (project ALCOM II) y ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....0 l of P k ;l which is effectively decidable. Finite index corresponds to finite state : there exists a linear time algorithm that decides finite index properties on graphs, given their tree decomposition of bounded treewidth. Moreover, this algorithm is of a special, well described structure [10, 9, 1]. The disadvantage of this algorithm is that a tree decomposition of the input graph is needed. Although for each fixed k, there is a linear time sequential algorithm which, given a graph G, checks if tw(G) k, and if so, computes a minimum width tree decomposition of G [6] this algorithm is ....

....let l be a refinement of P;l . Let k 1. If l is finite for each l 0, then there is a special reduction system (R ; I ) for P k , such that for each (H;H 0 ) 2 R , H l H 0 . Moreover, if l and P are effectively decidable, then such a system can effectively be constructed. Courcelle [10] has given a large class of graph properties which are of finite index, namely the class of properties that are definable in Monadic Second Order Logic or MSOL for graphs. MSOL for graphs G = V;E) consists of a language in which predicates can be built with ffl the logic connectives , and ....

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B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....Note that all these steps can be done in constant time, and we can actually do them if the constant time algorithms to compute S[H 0 i ] and S 00 Phi S[H] are known. This completes the proof. 2 As an important special case, we consider the graph properties that are MS definable (see e.g. [9] or [3] Let k 1. Suppose we have a graph property P which can be written as P (G) 9 S2D(G) Q(G;S) where D(G) D 1 (G) Theta D 2 (G) Theta Delta Delta Delta Theta D t (G) for some t 1, each D i (G) is either equal to V (G) to E(G) to P(V (G) or to P(E(G) and we have a ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....9 ; x 3 ) where again the R s depend on the rules of the grammar. 2 From these facts we conclude that Proposition15. The notion of well formed formulas of FOL and SOL are not definable in SAA Sigma(1; 1) but definable in SAA Sigma(3; 1) For similar results, cf. Courcelle and Engelfriet [Cou90, Eng91] 4 Self Satisfying Sentences Let F be a subset of SOL formulas. In this section we discuss in detail the class of structures AUTOSAT (F ) Let SOL be a finite vocabulary rich enough to describe SOL( formulas for arbitrary . SOL consists of one binary relation symbol denoted by and ....

B. Courcelle. The monadic second--order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....such as infinite Mazurkiewicz traces and asynchronous automata) we confine ourselves to the case of finite partial orders in the present paper. Many approaches have been developed to obtain natural generalizations of finite automata theory to cover partial orders and graphs, among them [KS81] [Cou90], and [Th91] We concentrate here on the last mentioned proposal, which is closely related to the tiling systems of [GR96] over labelled rectangular grids (pictures) This approach is based on the view that a finite automaton is a finite system which checks (by its finitely many transitions) ....

....in the introduction we mentioned the subject of monadic second order properties of infinite partial orders. Another track is the description of properties by calculi of regular expressions (as pursued in [BDW95] or by algebraic notions of recognizability (as developed in Courcelle s work [Cou90], Cou96] Finally, for applications in decision problems of logic or in program verification the complexity of the transformation procedures from logical formulas to finite state acceptors need to be analyzed. ....

B. Courcelle, The monadic second-order logic of graphs I: recognizable sets of finite graphs Inform. and Comput. 85 (1990), 12-75.

.... that are NP hard in general are polynomially solvable for restricted classes of graphs [GaJo79] Recently, a long line of work has culminated in the development of various methodologies for devising polynomial time (and, indeed, often linear time) algorithms for graphs of bounded tree width [AbFe92, ALS91, ArPr89, Bod87, BPT88, BLW87, Cou90, Wim87] 3 A preliminary version of this paper will appear in the proceedings of the 3rd Scandinavian Conference on Algorithm Theory, 1992. y Supported in part by the National Science Foundation under grant No. CCR 8909626. for a definition of tree width, see [RoSe86] or section 2 of this paper) ....

....in section 6. 2.2 Regular Graph Properties The various subgraph problems that are amenable to dynamic programming on graphs of bounded tree width share two key properties. First, the space of potential solutions to these problems can be partitioned into a finite number of equivalence classes [ALS91, ArPr89, Bod87, BPT88, BLW87, Cou90]. Second, there is a finite set of rules whereby partial solutions, computed for portions of the input graph, can be combined into larger partial solutions. Rules are expressed in tables which are fixed for each problem and each family of graphs. These two facts are essential in the proof of our ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....of Bodlaender [6] we can determine that no such decomposition of width b(k) exists or be given a decomposition of G. In either case, the running time for this procedure is linear in the size of G but exponential only in k. By means of one of several general algorithmic design methodologies (see [1, 4, 5, 15, 20, 54]) we may then answer the original question in time linear in the size of G. Hence, for small values of k, this procedure may lead to algorithms that are practical even for very large graphs G. Examples where these methods have been successful include Treewidth, Pathwidth, Min Cut Linear ....

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

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B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

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B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of nite graphs. Information and Computation, 85:12-75, 1990.

....aG (e) holds iff e 2 E a (or e = #G ) and incG = f(e; x; y) j e 2 EG ; e links x to yg. Hence, sets of graphs can be defined by formulas in Monadic Second Order logic or in Counting MS logic, a refinement of MS logic using special predicate expressing cardinality of sets modulo fixed integers; see [2 4]. Such sets are said MS definable (resp. CMS definable) for short. MS logic is the extension of First Order logic with set variables. For words and binary trees, MS definability equals recognizability. Definition 6 (Tree width) A tree decomposition is a pair (T ; g) where T is a tree and where ....

....are even not special or define graphs that are not special) Condition (1) is CMS definable (because val Gamma1 preserves CMS definability, see Courcelle [3] condition (2) is easy to express in MS logic. It follows that K is CMS definable, hence, recognizable by the theorem of Doner et al. [5, 2]. As a consequence of Lemma 14, each compact (T ; g) is represented by some term t 2 T(F k ) which verifies ff(t) bd(fl(G) It follows that bd(L 0 ) bd(val(t) ff(K) hence is a context free language by Proposition 10. ut Our aim is now to extend Proposition 15 to sets of graphs having ....

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B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....aG (e) holds iff e 2 E a (or e = #G ) and incG = f(e; x; y) j e 2 EG ; e links x to yg. Hence, sets of graphs can be defined by formulas in Monadic Second Order logic or in Counting MS logic, a refinement of MS logic using special predicate expressing cardinality of sets modulo fixed integers; see [2 4]. Such sets are said MS definable (resp. CMS definable) for short. MS logic is the extension of First Order logic with set variables. For words and binary trees, MS definability equals recognizability. Definition 6 (Tree width) A tree decomposition is a pair (T ; g) where T is a tree and where ....

....are even not special or define graphs that are not special) Condition (1) is CMS definable (because val Gamma1 preserves CMS definability, see Courcelle [3] condition (2) is easy to express in MS logic. It follows that K is CMS definable, hence, recognizable by the theorem of Doner et al. [5, 2]. As a consequence of Lemma 14, each compact (T ; g) is represented by some term t 2 T(F k ) which verifies ff(t) bd(fl(G) It follows that bd(L 0 ) bd(val(t) ff(K) hence is a context free language by Proposition 10. ut Our aim is now to extend Proposition 15 to sets of graphs having ....

[Article contains additional citation context not shown here]

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....is given as input. These problems are characterized by their expressibility in certain variations of Monadic Second Order Logic, MSOL( 1;p ) for decision problems) or LinEMSOL( 1;p ) for optimization problems) the study of which was initiated by B. Courcelle and others in a sequence of papers [Cou90, Cou91, Cou94b, Cou95, Cou96, CM93, ALS91]. Roughly speaking, MSOL( 1 ) is Monadic Second Order Logic with quantification over subsets of vertices, but not of edges; MSOL( 1;p ) is the extension of MSOL( 1 ) by unary predicates representing labels attached to the vertices. LinEMSOL( 1;p ) is the extension of MSOL( 1;p ) which allows ....

....23 it is easy to see that 2 tree(G) can be constructed from T (G) in time linear in the number of nodes of T (G) But since the number of nodes of T (G) is O(jV j) as proved in [Spi92] we get that the total construction of 2 tree(G) takes O(jV j jEj) time. 2 The following theorem is from [Cou90, CM93, ALS91] using the linear time algorithm (cf. Bod96] for constructing tree decompositions of partial k trees. Theorem 27. Let p and k be fixed integers. Every LinEMSOL( 1;p ) optimization problem on the class of partial k trees can be solved in O(jV j) time and the corresponding algorithm can be ....

[Article contains additional citation context not shown here]

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

....properties. Monadic second order logic (MS logic for short) is in this respect an interesting logical language because the graph properties it can express are decidable in linear time on many families of graphs, whereas they form a significantly large class, containing many basic graph properties [2, 9]. It is also possible to use monadic second order logic to express graph transformations [8] and functions on graphs [10] Certain graphs have a unique hierarchical decomposition : for an example the so called cographs [1] have a unique algebraic expression in terms of two binary operations. In ....

COURCELLE B., The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85 (1990) 12-75.

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B. Courcelle. The monadic second-order logic of graphs i: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

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B. Courcelle. The monadic second-order logic of graphs i: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

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B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

No context found.

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

No context found.

B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

No context found.

B. Courcelle. The monadic second order logic of graphs I: recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

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B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

No context found.

B. Courcelle. The monadic second-order logic of graphs i: Recognizable sets of finite graphs. Information and Computation, 85:12--75, 1990.

No context found.

B. Courcelle, The monadic second-order logic of graphs I: Recognizable sets of nite graphs, Information and Computation 85 (1991) 12-75 21

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B. Courcelle, The monadic second-order logic of graphs I: Recognizable sets of nite graphs, Information and Computation 85 (1991) 12-75

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