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Finite Model Theory and Descriptive Complexity
, 2002
"... This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the ..."
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Cited by 24 (7 self)
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This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the
XPath, transitive closure logic, and nested tree walking automata
 In Proceedings PODS 2008
, 2008
"... We consider the navigational core of XPath, extended with two operators: the Kleene star for taking the transitive closure of path expressions, and a subtree relativisation operator, allowing one to restrict attention to a specific subtree while evaluating a subexpression. We show that the expressiv ..."
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Cited by 22 (0 self)
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We consider the navigational core of XPath, extended with two operators: the Kleene star for taking the transitive closure of path expressions, and a subtree relativisation operator, allowing one to restrict attention to a specific subtree while evaluating a subexpression. We show that the expressive power of this XPath dialect equals that of FO(MTC), first order logic extended with monadic transitive closure. We also give a characterization in terms of nested treewalking automata. Using the latter we then proceed to show that the language is strictly less expressive than MSO. This solves an open question about the relative expressive power of FO(MTC) and MSO on trees. We also investigate the complexity for our XPath dialect. We show that query evaluation be done in polynomial time (combined complexity), but that satisfiability and query containment (as well as emptiness for our automaton model) are 2ExpTimecomplete (it is ExpTimecomplete for Core XPath).
Hierarchies in Transitive Closure Logic, Stratified Datalog and Infinitary Logic
 ANNALS OF PURE AND APPLIED LOGIC
, 1994
"... We establish a general hierarchy theorem for quantifier classes in the infinitary logic L ! 1! on finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure. This implies the solution of se ..."
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Cited by 12 (4 self)
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We establish a general hierarchy theorem for quantifier classes in the infinitary logic L ! 1! on finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure. This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of Immerman. We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog.
On the Power of Deterministic Transitive Closures
 INFORMATION AND COMPUTATION
, 1995
"... We show that transitive closure logic (FO + TC) is strictly more powerful than deterministic transitive closure logic (FO + DTC) on finite (unordered) structures. In fact, on certain classes of graphs, such as hypercubes or regular graphs of large degree and girth, every DTCquery is bounded and the ..."
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Cited by 10 (1 self)
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We show that transitive closure logic (FO + TC) is strictly more powerful than deterministic transitive closure logic (FO + DTC) on finite (unordered) structures. In fact, on certain classes of graphs, such as hypercubes or regular graphs of large degree and girth, every DTCquery is bounded and therefore first order expressible. On the other hand there are simple (FO + pos TC) queries on these classes that cannot be defined by first order formulae.
Incremental Model Checking for Fixed Point Properties on Decomposable Structures
, 1995
"... , April 1995 Abstract. Assume we are given a transition system which is composed from several well identified components. We propose a method which allows us to reduce the model checking of Safety, Reachability and Liveness Properties (expressible in fixed point logic) in the complex system to model ..."
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Cited by 5 (1 self)
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, April 1995 Abstract. Assume we are given a transition system which is composed from several well identified components. We propose a method which allows us to reduce the model checking of Safety, Reachability and Liveness Properties (expressible in fixed point logic) in the complex system to model checking of derived formulas in Transitive Closure Logic in the components, provided the complex systems is a \Phisum of its components. The basic idea goes back to a method proposed first by Feferman and Vaught in 1959, which we generalize to Transitive Closure Logic TC 1 . The generalization for Fixed Point Logic LFP 1 is due to U. Bosse. We adapt the method to the specific context of model checking of transition systems. Our method allows for a precise definition of incremental model checking. We give also estimates on when our incremental method starts to be better than traditional methods. 1 Introduction In hardware verification we find the following situation: We are given a mat...
Arity Hierarchies
 ANNALS OF PURE AND APPLIED LOGIC
, 1996
"... Many logics considered in finite model theory have a natural notion of an arity. The purpose of this article is to study the hierarchies which are formed by the fragments of such logics whose formulae are of bounded arity. Based on a construction of finite graphs with a certain property of homogenei ..."
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Cited by 3 (1 self)
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Many logics considered in finite model theory have a natural notion of an arity. The purpose of this article is to study the hierarchies which are formed by the fragments of such logics whose formulae are of bounded arity. Based on a construction of finite graphs with a certain property of homogeneity, we develop a method that allows us to prove that the arity hierarchies are strict for several logics, including fixedpoint logics, transitive cloure logic and its deterministic version, variants of the database language Datalog, and extensions of firstorder logic by implicit denitions. Furthermore, we show that all our results already hold on the class of finite graphs.
Guarded Quantification in Least Fixed Point Logic
, 2002
"... We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point ..."
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Cited by 2 (1 self)
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We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worstcase time complexity.
FixedPoint Logics and Solitaire Games
 THEORY COMPUT SYSTEMS
, 2004
"... The modelchecking games associated with fixedpoint logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time. We study SolitaireLFP, a fragment of least fixedpoint logic, whose evaluation games are nested soltaire games. ..."
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Cited by 2 (1 self)
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The modelchecking games associated with fixedpoint logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time. We study SolitaireLFP, a fragment of least fixedpoint logic, whose evaluation games are nested soltaire games. This means that on each strongly connected component of the game, only one player can make nontrivial moves. Winning sets of nested solitaire games can be computed efficiently. The model
A New Approach to Predicative Set Theory
"... We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an a ..."
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We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domainindependence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is typefree, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1