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Graph Logics with Rational Relations and the Generalized Intersection Problem
"... Abstract—We investigate some basic questions about the interaction of regular and rational relations on words. The primary motivation comes from the study of logics for querying graph topology, which have recently found numerous applications. Such logics use conditions on paths expressed by regular ..."
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Abstract—We investigate some basic questions about the interaction of regular and rational relations on words. The primary motivation comes from the study of logics for querying graph topology, which have recently found numerous applications. Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword (factor) or subsequence. Evaluating formulae in such extended graph logics boils down to checking nonemptiness of the intersection of rational relations with regular or recognizable relations (or, more generally, to the generalized intersection problem, asking whether some projections of a regular relation have a nonempty intersection with a given rational relation). We prove that for several basic and commonly used rational relations, the intersection problem with regular relations is either undecidable (e.g., for subword or suffix, and some generalizations), or decidable with nonmultiplyrecursive complexity (e.g., for subsequence and its generalizations). These results are used to rule out many classes of graph logics that freely combine regular and rational relations, as well as to provide the simplest problem related to verifying lossy channel systems that has nonmultiplyrecursive complexity. We then prove a dichotomy result for logics combining regular conditions on individual paths and rational relations on paths, by showing that the syntactic form of formulae classifies them into either efficiently checkable or undecidable cases. We also give examples of rational relations for which such logics are decidable even without syntactic restrictions. I.
Expressiveness of a spatial logic for trees
 In Proc. IEEE Symp. on Logic in Comp. Sci
, 2005
"... In this paper we investigate the quantifierfree fragment of the TQL logic proposed by Cardelli and Ghelli. The TQL logic, inspired from the ambient logic, is the core of a query language for semistructured data represented as unranked and unordered trees. The fragment we consider here, named STL, c ..."
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In this paper we investigate the quantifierfree fragment of the TQL logic proposed by Cardelli and Ghelli. The TQL logic, inspired from the ambient logic, is the core of a query language for semistructured data represented as unranked and unordered trees. The fragment we consider here, named STL, contains as main features spatial composition and location as well as a fixed point construct. We prove that satisfiability for STL is undecidable. We show also that STL is strictly more expressive than the Presburger monadic secondorder logic (PMSO) of Seidl, Schwentick and Muscholl when interpreted over unranked and unordered edgelabelled trees. We define a class of tree automata whose transitions are conditioned by arithmetical constraints; we show then how to compute from a closed STL formula a tree automaton accepting precisely the models of the formula. Finally, still using our tree automata framework, we exhibit some syntactic restrictions over STL formulae that allow us to capture precisely the logics MSO and PMSO. 1
Satisfiability of a spatial logic with tree variables
 In Proc. 21st Int. Workshop on Computer Science Logic (CSL
, 2007
"... Abstract. We investigate in this paper the spatial logic TQL for querying semistructured data, represented as unranked ordered trees over an infinite alphabet. This logic consists of usual Boolean connectives, spatial connectives (derived from the constructors of a tree algebra), tree variables and ..."
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Abstract. We investigate in this paper the spatial logic TQL for querying semistructured data, represented as unranked ordered trees over an infinite alphabet. This logic consists of usual Boolean connectives, spatial connectives (derived from the constructors of a tree algebra), tree variables and a fixpoint operator for recursion. Motivated by XMLoriented tasks, we investigate the guarded TQL fragment. We prove that for closed formulas this fragment is MSOcomplete. In presence of tree variables, this fragment is strictly more expressive than MSO as it allows for tree (dis)equality tests, i.e. testing whether two subtrees are isomorphic or not. We devise a new class of tree automata, called TAGED, which extends tree automata with global equality and disequality constraints. We show that the satisfiability problem for guarded TQL formulas reduces to emptiness of TAGED. Then, we focus on bounded TQL formulas: intuitively, a formula is bounded if for any tree, the number of its positions where a subtree is captured by a variable is bounded. We prove this fragment to correspond with a subclass of TAGED, called bounded TAGED, for which we prove emptiness to be decidable. This implies the decidability of the bounded guarded TQL fragment. Finally, we compare bounded TAGED to a fragment of MSO extended with subtree isomorphism tests. 1
On Spatial Conjunction as SecondOrder Logic
, 2004
"... Spatial conjunction is a powerful construct for reasoning about dynamically allocated data structures, as well as concurrent, distributed and mobile computation. While researchers have identified many uses of spatial conjunction, its precise expressive power compared to traditional logical constr ..."
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Spatial conjunction is a powerful construct for reasoning about dynamically allocated data structures, as well as concurrent, distributed and mobile computation. While researchers have identified many uses of spatial conjunction, its precise expressive power compared to traditional logical constructs was not previously known.
A Logic for Graphs with QoS
 In First International Workshop on Views On Designing Complex Architectures, ENTCS
, 2004
"... We introduce a simple graph logic that supports specification of Quality of Service (QoS) properties of applications. The idea is that we are not only interested in representing whether two sites are connected, but we want to express the QoS level of the connection. The evaluation of a formula in th ..."
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We introduce a simple graph logic that supports specification of Quality of Service (QoS) properties of applications. The idea is that we are not only interested in representing whether two sites are connected, but we want to express the QoS level of the connection. The evaluation of a formula in the graph logic is a value of a suitable algebraic structure, a csemiring, representing the QoS level of the formula and not just a boolean value expressing whether or not the formula holds. We present some examples and briefly discuss the expressiveness and complexity of our logic.
On Complexity Of ModelChecking For The TQL Logic
 IN 3RD IFIP INTERNATIONAL CONFERENCE ON THEORETICAL COMPUTER SCIENCE
, 2004
"... In this paper we study the complexity of the modelchecking problem for the tree logic introduced as the basis for the query language TQL [Cardelli and Ghelli, 2001]. We define two distinct fragments of this logic: TL containing only spatial connectives and TL containing spatial connectives and q ..."
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Cited by 6 (1 self)
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In this paper we study the complexity of the modelchecking problem for the tree logic introduced as the basis for the query language TQL [Cardelli and Ghelli, 2001]. We define two distinct fragments of this logic: TL containing only spatial connectives and TL containing spatial connectives and quantification. We show that the combined complexity of TL is PSPACEhard. We also study data complexity of modelchecking and show that it is linear for TL, hard for all levels of the polynomial hierarchy for TL and PSPACEhard for the full logic. Finally we devise a polynomial space modelchecking algorithm showing this way that the modelchecking problem for the TQL logic is PSPACEcomplete.
GRAPH LOGICS WITH RATIONAL RELATIONS
"... Abstract. We investigate some basic questions about the interaction of regular and rational relations on words. The primary motivation comes from the study of logics for querying graph topology, which have recently found numerous applications. Such logics use conditions on paths expressed by regular ..."
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Cited by 6 (2 self)
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Abstract. We investigate some basic questions about the interaction of regular and rational relations on words. The primary motivation comes from the study of logics for querying graph topology, which have recently found numerous applications. Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword or subsequence. Evaluating formulae in such extended graph logics boils down to checking nonemptiness of the intersection of rational relations with regular or recognizable relations (or, more generally, to the generalized intersection problem, asking whether some projections of a regular relation have a nonempty intersection with a given rational relation). We prove that for several basic and commonly used rational relations, the intersection problem with regular relations is either undecidable (e.g., for subword or suffix, and some generalizations), or decidable with nonprimitiverecursive complexity (e.g., for subsequence and its generalizations). These results are used to rule out many classes of graph logics that freely combine regular and rational relations, as well as to provide the simplest problem related to verifying lossy channel systems that has nonprimitiverecursive complexity. We then prove a dichotomy result for logics combining regular conditions on individual paths and rational relations on paths, by showing that the syntactic form of formulae classifies them into either efficiently checkable or undecidable cases. We also give examples of rational relations for which such logics are decidable even without syntactic restrictions.
A Spatial Equational Logic for the Applied πCalculus
 Distributed Computing
"... Abstract. Spatial logics have been proposed to reason locally and modularly on algebraic models of distributed systems. In this paper we define the spatial equational logic AπLwhose models are processes of the applied πcalculus. This extension of the πcalculus allows term manipulation and records ..."
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Cited by 5 (0 self)
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Abstract. Spatial logics have been proposed to reason locally and modularly on algebraic models of distributed systems. In this paper we define the spatial equational logic AπLwhose models are processes of the applied πcalculus. This extension of the πcalculus allows term manipulation and records communications as active substitutions in a frame, thus augmenting the underlying predefined equational theory. Our logic allows one to reason locally either on frames or on processes, thanks to static and dynamic spatial operators. We study the logical equivalences induced by various relevant fragments of AπL, and show in particular that the whole logic induces a coarser equivalence than structural congruence. We give characteristic formulae for some of these equivalences and for static equivalence. Going further into the exploration of AπL’s expressivity, we also show that it can eliminate standard term quantification. 1
Separating Graph Logic from MSO
"... Abstract. Graph logic (GL) is a spatial logic for querying graphs introduced by Cardelli et al. It has been observed that in terms of expressive power, this logic is a fragment of Monadic Second Order Logic (MSO), with quantification over sets of edges. We show that the containment is proper by exhi ..."
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Abstract. Graph logic (GL) is a spatial logic for querying graphs introduced by Cardelli et al. It has been observed that in terms of expressive power, this logic is a fragment of Monadic Second Order Logic (MSO), with quantification over sets of edges. We show that the containment is proper by exhibiting a property that is not GL definable but is definable in MSO, even in the absence of quantification over labels. Moreover, this holds when the graphs are restricted to be forests and thus strengthens in several ways a result of Marcinkowski. As a consequence we also obtain that Separation Logic, with a separating conjunction but without the magic wand, is strictly weaker than MSO over memory heaps, settling an open question of Brochenin et al. 1