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84
Homomorphism Preservation Theorems
, 2008
"... The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in fin ..."
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The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the ̷Lo´sTarski theorem and Lyndon’s positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existentialpositive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a firstorder formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existentialpositive formula of equal quantifierrank.
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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Cited by 22 (6 self)
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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
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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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.
Vertexminors, Monadic Secondorder Logic, and a Conjecture by Seese
, 2006
"... We prove that one can express the vertexminor relation on finite undirected graphs by formulas of monadic secondorder logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that set ..."
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Cited by 17 (7 self)
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We prove that one can express the vertexminor relation on finite undirected graphs by formulas of monadic secondorder logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic secondorder logic have bounded cliquewidth. We also obtain a polynomialtime algorithm to check that the rankwidth of a graph is at most k for any fixed k. The proofs use isotropic systems.
On Combining Theories with Shared Set Operations
"... Abstract. We explore the problem of automated reasoning about the nondisjoint combination of theories that share set variables and operations. We prove a combination theorem and apply it to show the decidability of the satisfiability problem for a class of formulas obtained by applying propositional ..."
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Abstract. We explore the problem of automated reasoning about the nondisjoint combination of theories that share set variables and operations. We prove a combination theorem and apply it to show the decidability of the satisfiability problem for a class of formulas obtained by applying propositional operations to quantified formulas belonging to several expressive decidable logics. 1
Model theory makes formulas large
 In Proceedings of the 34th International Colloquium on Automata, Languages and Programming
, 2007
"... Gaifman’s locality theorem states that every firstorder sentence is equivalent to a local sentence. We show that there is no elementary bound on the length of the local sentence in terms of the original. Gaifman’s theorem is an essential ingredient in several algorithmic meta theorems for first ord ..."
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Cited by 16 (4 self)
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Gaifman’s locality theorem states that every firstorder sentence is equivalent to a local sentence. We show that there is no elementary bound on the length of the local sentence in terms of the original. Gaifman’s theorem is an essential ingredient in several algorithmic meta theorems for first order logic. Our result has direct implications for the running time of the algorithms. The classical Ło´sTarski theorem states that every firstorder sentence preserved under extensions is equivalent to an existential sentence. We show that there is no elementary bound on the length of the existential sentence in terms of the original. Recently, variants of the Ło´sTarski theorem have been proved for certain classes of finite structures, among them the class of finite trees and more generally classes of structures of bounded tree width. Our lower bound also applies to these variants. The firstorder theory of trees is decidable. We prove that there is no elementary decision algorithm. Notably, our lower bounds do not apply to restrictions of the results to structures of bounded degree. For such structures, we obtain elementary upper bounds in all cases. However, even there we can prove at least doubly exponential lower bounds. 1
Decision Procedures for Multisets with Cardinality Constraints
"... Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of ..."
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Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of linked data structure with duplicate elements leads to multisets. Interactive theorem provers such as Isabelle specify theories of multisets and prove a number of theorems about them to enable their use in interactive verification. However, the decidability and complexity of constraints on multisets is much less understood than for constraints on sets. The first contribution of this paper is a polynomialspace algorithm for deciding expressive quantifierfree constraints on multisets with cardinality operators. Our decision procedure reduces in polynomial time constraints on multisets to constraints in an extension of quantifierfree Presburger arithmetic with certain “unbounded sum ” expressions. We prove bounds on solutions of resulting constraints and describe a polynomialspace decision procedure for these constraints. The second contribution of this paper is a proof that adding quantifiers to a constraint language containing subset and cardinality operators yields undecidable constraints. The result follows by reduction from Hilbert’s 10th problem. 1
The firstorder theory of sets with cardinality constraints is decidable
, 2004
"... Data structures often use an integer variable to keep track of the number of elements they store. An invariant of such data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. Using a program analysis framework that supports abstractio ..."
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Data structures often use an integer variable to keep track of the number of elements they store. An invariant of such data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. Using a program analysis framework that supports abstraction of data structures as sets, such constraints can be expressed using the language of sets with cardinality constraints. The same language can be used to express preconditions that guarantee the correct use of the data structure interfaces, and to express invariants useful for the analysis of the termination behavior of programs that manipulate objects stored in data structures. In this paper we show the decidability of valid formulas in one such language. Specifically, we examine the firstorder theory that combines 1) Boolean algebras of sets of uninterpreted elements and 2) Presburger arithmetic operations. Our language allows relating the cardinalities of sets to the values of integer variables. We use quantifier elimination to show the decidability of the resulting firstorder theory. We thereby disprove a recent conjecture that this theory is undecidable. We describe a basic quantifierelimination algorithm and its more sophisticated versions. From the analysis of our algorithms we obtain an elementary upper bound on the complexity of the resulting combination. Furthermore, our algorithm yields the decidability of a combination of sets of uninterpreted elements with any decidable extension of Presburger arithmetic. For example, we obtain decidability of monadic secondorder logic of nsuccessors extended with sets of uninterpreted elements and their cardinalities, a result which is in contrast to the undecidability of extensions of monadicsecond order logic over strings with equicardinality operator on sets of strings. POPL’05 submission #181. 1
Word equations over graph products
 In Proceedings of the 23rd Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2003), Mumbai (India), number 2914 in Lecture Notes in Computer Science
, 2003
"... For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those fac ..."
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For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.