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43
The Caucal hierarchy of infinite graphs in terms of logic and higherorder pushdown automata
 IN FSTTCS’03, VOLUME 2914 OF LNCS
, 2003
"... In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic secondorder (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to thi ..."
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Cited by 66 (9 self)
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In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic secondorder (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to this hierarchical approach, replacing the languagetheoretic operation of a rational mapping by an MSOtransduction and the unfolding by the treegraph operation. The second characterization is noniterative. We show that the family of graphs of the Caucal hierarchy coincides with the family of graphs obtained as the εclosure of configuration graphs of higherorder pushdown automata. While the different characterizations of the graph family show their robustness and thus also their importance, the characterization in terms of higherorder pushdown automata additionally yields that the graph hierarchy is indeed strict.
Finite Presentations of Infinite Structures: Automata and Interpretations
 Theory of Computing Systems
, 2002
"... We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations. ..."
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Cited by 54 (4 self)
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We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations.
Modular Data Structure Verification
 EECS DEPARTMENT, MASSACHUSETTS INSTITUTE OF TECHNOLOGY
, 2007
"... This dissertation describes an approach for automatically verifying data structures, focusing on techniques for automatically proving formulas that arise in such verification. I have implemented this approach with my colleagues in a verification system called Jahob. Jahob verifies properties of Java ..."
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Cited by 44 (21 self)
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This dissertation describes an approach for automatically verifying data structures, focusing on techniques for automatically proving formulas that arise in such verification. I have implemented this approach with my colleagues in a verification system called Jahob. Jahob verifies properties of Java programs with dynamically allocated data structures. Developers write Jahob specifications in classical higherorder logic (HOL); Jahob reduces the verification problem to deciding the validity of HOL formulas. I present a new method for proving HOL formulas by combining automated reasoning techniques. My method consists of 1) splitting formulas into individual HOL conjuncts, 2) soundly approximating each HOL conjunct with a formula in a more tractable fragment and 3) proving the resulting approximation using a decision procedure or a theorem prover. I present three concrete logics; for each logic I show how to use it to approximate HOL formulas, and how to decide the validity of formulas in this logic. First, I present an approximation of HOL based on a translation to firstorder logic, which enables the use of existing resolutionbased theorem provers. Second, I present an approximation of HOL based on field constraint analysis, a new technique that enables
LOGICS FOR UNRANKED TREES: AN OVERVIEW
 CONSIDERED FOR PUBLICATION IN LOGICAL METHODS IN COMPUTER SCIENCE
, 2006
"... Labeled unranked trees are used as a model of XML documents, and logical languages for them have been studied actively over the past several years. Such logics have different purposes: some are better suited for extracting data, some for expressing navigational properties, and some make it easy to ..."
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Cited by 40 (7 self)
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Labeled unranked trees are used as a model of XML documents, and logical languages for them have been studied actively over the past several years. Such logics have different purposes: some are better suited for extracting data, some for expressing navigational properties, and some make it easy to relate complex properties of trees to the existence of tree automata for those properties. Furthermore, logics differ significantly in their modelchecking properties, their automata models, and their behavior on ordered and unordered trees. In this paper we present a survey of logics for unranked trees.
Counting on CTL*: On the Expressive Power of Monadic Path Logic
, 2003
"... Monadic secondorder logic (MSOL) provides a general framework for expressing properties of reactive systems as modelled by trees. Monadic path logic (MPL) is obtained by restricting secondorder quantification to paths reflecting computation sequences. In this paper we show that the expressive powe ..."
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Cited by 21 (1 self)
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Monadic secondorder logic (MSOL) provides a general framework for expressing properties of reactive systems as modelled by trees. Monadic path logic (MPL) is obtained by restricting secondorder quantification to paths reflecting computation sequences. In this paper we show that the expressive power of MPL over trees coincides with the usual branching time logic CTL # embellished with a simple form of counting. As a corollary, we derive an earlier result that CTL # coincides with the bisimulationinvariant properties of MPL. In order to prove the main result, we first prove a new Composition Theorem for trees.
On the theory of structural subtyping
, 2003
"... We show that the firstorder theory of structural subtyping of nonrecursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ..."
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Cited by 18 (8 self)
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We show that the firstorder theory of structural subtyping of nonrecursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ≤ represents a subtype ordering. We introduce the notion of Σtermpower of C, which generalizes the structure arising in structural subtyping. The domain of the Σtermpower of C is the set of Σterms over the set of elements of C. We show that the decidability of the firstorder theory of C implies the decidability of the firstorder theory of the Σtermpower of C. This result implies the decidability of the firstorder theory of structural subtyping of nonrecursive types.
Deciding Monadic Theories of Hyperalgebraic Trees
"... We show that the monadic secondorder theory of any infinite tree generated by a higherorder grammar of level 2 subject to a certain syntactic restriction is decidable. By this we extend the result of Courcelle [7] that the MSO theory of a tree generated by a grammar of level 1 (algebraic) is decid ..."
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Cited by 18 (6 self)
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We show that the monadic secondorder theory of any infinite tree generated by a higherorder grammar of level 2 subject to a certain syntactic restriction is decidable. By this we extend the result of Courcelle [7] that the MSO theory of a tree generated by a grammar of level 1 (algebraic) is decidable. To this end, we develop a technique of representing infinite trees by infinite lambda terms, in such a way that the MSO theory of a tree can be interpreted in the MSO theory of a lambda term.
On the expression of graph properties in some fragments of monadic secondorder logic
 In Descriptive Complexity and Finite Models: Proceedings of a DIAMCS Workshop
, 1996
"... ABSTRACT: We review the expressibility of some basic graph properties in certain fragments of Monadic SecondOrder logic, like the set of MonadicNP formulas. We focus on cases where a property and its negation are both expressible in the same (or in closely related) fragments. We examine cases wher ..."
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Cited by 13 (1 self)
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ABSTRACT: We review the expressibility of some basic graph properties in certain fragments of Monadic SecondOrder logic, like the set of MonadicNP formulas. We focus on cases where a property and its negation are both expressible in the same (or in closely related) fragments. We examine cases where edge quantifications can be eliminated and cases where they cannot. We compare two logical expressions of planarity: one of them is constructive in the sense that it defines a planar embedding of the considered graph if it is planar and 3connected, and the other, logically simpler, uses the forbidden Kuratowski subgraphs.
Languages of Nested Trees
, 2006
"... We study languages of nested trees—structures obtained by augmenting trees with sets of nested jumpedges. These graphs can naturally model branching behaviors of pushdown programs, so that the problem of branchingtime software model checking may be phrased as a membership question for such langua ..."
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Cited by 11 (2 self)
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We study languages of nested trees—structures obtained by augmenting trees with sets of nested jumpedges. These graphs can naturally model branching behaviors of pushdown programs, so that the problem of branchingtime software model checking may be phrased as a membership question for such languages. We define finitestate automata accepting such languages—these automata can pass states along jumpedges as well as tree edges. We find that the modelchecking problem for these automata on pushdown systems is EXPTIMEcomplete, and that their alternating versions are expressively equivalent to NTµ, a recently proposed temporal logic for nested trees that can express a variety of branchingtime, “contextfree ” requirements. We also show that monadic second order logic (MSO) cannot exploit the structure: MSO on nested trees is too strong in the sense that it has an undecidable model checking problem, and seems too weak to capture NTµ.
Axiomatising Treeinterpretable Structures
 IN PROC. 19TH INT. SYMP. ON THEORETICAL ASPECTS OF COMPUTER SCIENCE, LNCS 2285, 2002
, 2001
"... We introduce the class of treeinterpretable structures which generalises the notion of a prefixrecognisable graph to arbitrary relational structures. We prove that every treeinterpretable structure is finitely axiomatisable in guarded secondorder logic with cardinality quantifiers. ..."
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Cited by 9 (1 self)
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We introduce the class of treeinterpretable structures which generalises the notion of a prefixrecognisable graph to arbitrary relational structures. We prove that every treeinterpretable structure is finitely axiomatisable in guarded secondorder logic with cardinality quantifiers.