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Algorithmic MetaTheorems
 In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
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Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P”. A particularly well known example of a metatheorem is Courcelle’s theorem that every decision problem definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth [1]. The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded treewidth and for which approximate solutions can be computed efficiently from solutions of certain subinstances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many
Logic, Graphs, and Algorithms
, 2007
"... Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all ..."
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Cited by 10 (0 self)
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Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all graph properties definable in monadic secondorder logic can be decided in linear time on graphs of bounded tree width. This article is an introduction into the theory underlying such meta theorems and a survey of the most important results in this area.
Concurrency Makes Simple Theories Hard
 in Proceedings of STACS’12, ser.LeibnizInternationalProceedingsin Informatics (LIPIcs
"... A standard way of building concurrent systems is by composing several individual processes by product operators. We show that even the simplest notion of product operators (i.e. asynchronous products) suffices to increase the complexity of model checking simple logics like HennessyMilner (HM) logic ..."
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A standard way of building concurrent systems is by composing several individual processes by product operators. We show that even the simplest notion of product operators (i.e. asynchronous products) suffices to increase the complexity of model checking simple logics like HennessyMilner (HM) logic and its extension with the reachability operator (EFlogic) from PSPACE to nonelementary. In particular, this nonelementary jump happens for EFlogic when we consider individual processes represented by pushdown systems (indeed, even with only one control state). Using this result, we prove nonelementary lower bounds on the size of formula decompositions provided by FefermanVaught (de)compositional methods for HM and EF logics, which reduce theories of asynchronous products to theories of the components. Finally, we show that the same nonelementary lower bounds also hold when we consider the relativization of such compositional methods to finite systems.
Logical complexity of graphs: a survey
 CONTEMPORARY MATHEMATICS
, 2004
"... We discuss the definability of finite graphs in firstorder logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W(G) is the minimum number of ..."
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Cited by 2 (0 self)
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We discuss the definability of finite graphs in firstorder logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W(G) is the minimum number of variables occurring in such a sentence. The logical length L(G) is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the WeisfeilerLehman algorithm in isomorphism testing, the evolution of a random graph, quantitative characteristics of the zeroone law, or the contribution of Frank Ramsey to the research on Hilbert’s Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible
The FixedParameter Tractability of Model Checking Concurrent Systems ∗
"... We study the fixedparameter complexity of model checking temporal logics on concurrent systems that are modeled as the product of finite systems and where the size of the formula is the parameter. We distinguish between asynchronous product and synchronous product. Sometimes it is possible to show ..."
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We study the fixedparameter complexity of model checking temporal logics on concurrent systems that are modeled as the product of finite systems and where the size of the formula is the parameter. We distinguish between asynchronous product and synchronous product. Sometimes it is possible to show that there is an algorithm for this with running time ( ∑ i Ti) O(1) · f(ϕ), where the Ti are the component systems and ϕ is the formula and f is computable function, thus model checking is fixedparameter tractable when the size of the formula is the parameter. In this paper we concern ourselves with the question, provided fixedparameter tractability is known, whether it holds for an elementary function f. Negative answers to this question are provided for modal logic and EF logic: Depending on the mode of synchronization we show the nonexistence of such an elementary function f under different assumptions from (parameterized) complexity theory.
Acknowledgements xi
, 2014
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Universität Bremen
"... Abstract—We show that the satisfiability problem for the twodimensional extension K ×K of unimodal K is nonelementary, hereby confirming a conjecture of Marx and Mikulás from 2001. Our lower bound technique allows us to derive further lower bounds for manydimensional modal logics for which only e ..."
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Abstract—We show that the satisfiability problem for the twodimensional extension K ×K of unimodal K is nonelementary, hereby confirming a conjecture of Marx and Mikulás from 2001. Our lower bound technique allows us to derive further lower bounds for manydimensional modal logics for which only elementary lower bounds were previously known. We also derive nonelementary lower bounds on the sizes of FefermanVaught decompositions w.r.t. product for any decomposable logic that is at least as expressive as unimodal K. Finally, we study the sizes of FefermanVaught decompositions and formulas in Gaifman normal form for fixedvariable fragments of firstorder logic. I.
The Complexity of Decomposing Modal and FirstOrder Theories
"... We study the satisfiability problem of the logic K2 = K ×K, i.e., the twodimensional variant of unimodal logic, where models are restricted to asynchronous products of two Kripke frames. Gabbay and Shehtman proved in 1998 that this problem is decidable in a tower of exponentials. So far the best kn ..."
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We study the satisfiability problem of the logic K2 = K ×K, i.e., the twodimensional variant of unimodal logic, where models are restricted to asynchronous products of two Kripke frames. Gabbay and Shehtman proved in 1998 that this problem is decidable in a tower of exponentials. So far the best known lower bound is NEXPhardness shown by Marx and Mikulás in 2001. Our first main result closes this complexity gap: We show that satisfiability inK2 is nonelementary. More precisely, we prove that it is kNEXPcomplete, where k is the switching depth (the minimal modal rank among the two dimensions) of the input formula, hereby solving a conjecture of Marx and Mikulás. Using our lowerbound technique allows us to derive also nonelementary lower bounds for the twodimensional modal logics K4 ×K and S52 ×K for which only elementary lower bounds were previously known. Moreover, we apply our technique to prove nonelementary lower bounds for the sizes of FefermanVaught decompositions with respect to product for any decomposable logic that is at least as expressive as unimodal K, generalizing a recent result by the first author and Lin. For the threevariable fragment FO3 of firstorder logic, we obtain the following immediate corollaries: (i) the size of FefermanVaught decompositions with respect to disjoint sum are inherently nonelementary and (ii) equivalent formulas in Gaifman normal form are inherently nonelementary.