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58
Some Complexity Results for Polynomial Ideals
, 1997
"... In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,.. ..."
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In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert’s Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.
Normalised Rewriting and Normalised Completion
, 1994
"... We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algor ..."
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Cited by 19 (2 self)
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We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...
Views and Queries: Determinacy and Rewriting
 In PODS
, 2005
"... We investigate the question of whether a query Q can be answered using a set V of views. We first define the problem in informationtheoretic terms: we say that V determines Q if V provides enough information to uniquely determine the answer to Q. Next, we look at the problem of rewriting Q in terms ..."
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Cited by 17 (1 self)
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We investigate the question of whether a query Q can be answered using a set V of views. We first define the problem in informationtheoretic terms: we say that V determines Q if V provides enough information to uniquely determine the answer to Q. Next, we look at the problem of rewriting Q in terms of V using a specific language. Given a view language V and query language Q, we say that a rewriting language R is complete for VtoQ rewritings if every Q ∈Qcan be rewritten in terms of V ∈ Vusing a query in R, whenever V determines Q. While query rewriting using views has been extensively investigated for some specific languages, the connection to the informationtheoretic notion of determinacy, and the question of completeness of a rewriting language, have received little attention. In this paper we investigate systematically the notion of determinacy and its connection to rewriting. The results concern decidability of determinacy for various view and query languages, as well as the power required of complete rewriting languages. We consider languages ranging from firstorder to conjunctive queries. 1.
The inference problem for template dependencies
, 1982
"... A template dependency is a formalized integrity constraint on a relational database, stating that whenever tuples exist in the database that agree on certain attributes, an additional tuple must also be present that agrees with the others in a specified way. It is shown that the inference problem fo ..."
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A template dependency is a formalized integrity constraint on a relational database, stating that whenever tuples exist in the database that agree on certain attributes, an additional tuple must also be present that agrees with the others in a specified way. It is shown that the inference problem for template dependencies is undecidable, that is, there can be no algorithm for determining whether a given dependency is a logical consequence of a given finite set of dependencies. The undecidability result holds whether or not databases are considered to be necessarily finite. INTROD UCTION The goal of dependency theory is to formalize constraints on the data comprising a relational database. In general, a dependency is a statement to the effect that when certain tuples are present in the database, so are certain others. Such statements can be used, for example, to capture the idea that attributes are functionally related or independent in some way. Many varieties of dependencies have been proposed in the literature; see the discussions in Fagin (1980) and Yannakakis and Papadimitriou (1980), for example. The proliferation of varieties is due in part to the desire to balance two opposing forces: on the one hand, dependencies should be of a form general enough to express interesting properties, but on the other hand, the form should not be so general that natural questions about dependencies become undecidable or computationally intractable. A significant question about any class of dependencies is its inference problem: Given a finite set D of dependencies and a single dependency D 0, to determine whether D O is true in every database in which each member of D is true. A solution to the inference problem carries with it the ability to determine whether two sets of
A New Lower Bound Construction for Commutative Thue Systems, with Applications
, 1997
"... For n 1; d 2, we describe a commutative Thue system that has ¸ 2n variables and O(n) rules, each rule of size d + O(1) and that counts to d 2 n in a certain technical sense. This gives a more "efficient" alternative to a wellknown construction of Mayr and Meyer. Using this construction, we sha ..."
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For n 1; d 2, we describe a commutative Thue system that has ¸ 2n variables and O(n) rules, each rule of size d + O(1) and that counts to d 2 n in a certain technical sense. This gives a more "efficient" alternative to a wellknown construction of Mayr and Meyer. Using this construction, we sharpen the known doubleexponential lower bounds for the maximum degrees D(n; d); I(n; d); S(n; d) associated (respectively) with Grobner bases, ideal membership problem and the syzygy basis problem: D(n; d) S(n; d) d 2 m ; I(n; d) d 2 m where m ¸ n=2, and n; d sufficiently large. For comparison, it was known that D(n; d) d 2 n and I(n; d) (2d) 2 n . Keywords: Lower bound, commutative Thue system, Grobner basis, ideal membership problem, syzygy basis problem This research is supported in part by NSF Grants DCR8401898, CCR8703458 and ONR Grant N0001485K0046. The work was carried out while visiting the Research Institute for Symbolic Computation (RISCLINZ), Johannes K...
Contrasting applications of logic in natural language syntactic description
 Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress
, 2005
"... Abstract. Formal syntax has hitherto worked mostly with theoretical frameworks that take grammars to be generative, in Emil Post’s sense: they provide recursive enumerations of sets. This work has its origins in Post’s formalization of proof theory. There is an alternative, with roots in the semanti ..."
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Abstract. Formal syntax has hitherto worked mostly with theoretical frameworks that take grammars to be generative, in Emil Post’s sense: they provide recursive enumerations of sets. This work has its origins in Post’s formalization of proof theory. There is an alternative, with roots in the semantic side of logic: modeltheoretic syntax (MTS). MTS takes grammars to be sets of statements of which (algebraically idealized) wellformed expressions are models. We clarify the difference between the two kinds of framework and review their separate histories, and then argue that the generative perspective has misled linguists concerning the properties of natural languages. We select two elementary facts about natural language phenomena for discussion: the gradient character of the property of being ungrammatical and the open nature of natural language lexicons. We claim that the MTS perspective on syntactic structure does much better on representing the facts in these two domains. We also examine the arguments linguists give for the infinitude of the class of all expressions in a natural language. These arguments turn out on examination to be either unsound or lacking in empirical content. We claim that infinitude is an unsupportable claim that is also unimportant. What is actually needed is a way of representing the structure of expressions in a natural language without assigning any importance to the notion of a unique set with definite cardinality that contains all and only the expressions in the language. MTS provides that.
Automatic Verification of DatabaseDriven Systems: A New Frontier
"... We describe a novel approach to verification of software systems centered around an underlying database. Instead of applying generalpurpose techniques with only partial guarantees of success, it identifies restricted but reasonably expressive classes of applications and properties for which sound a ..."
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We describe a novel approach to verification of software systems centered around an underlying database. Instead of applying generalpurpose techniques with only partial guarantees of success, it identifies restricted but reasonably expressive classes of applications and properties for which sound and complete verification can be performed in a fully automatic way. This leverages the emergence of highlevel specification tools for databasecentered applications that not only allow fast prototyping and improved programmer productivity but, as a side effect, provide convenient targets for automatic verification. We present theoretical and practical results on verification of databasedriven systems. The results are quite encouraging and suggest that, unlike arbitrary software systems, significant classes of databasedriven systems may be amenable to automatic verification. This relies on a novel marriage of database and model checking techniques, of relevance to both the database and the computer aided verification communities. 1.
Algorithmic problems in groups, semigroups and inverse semigroups
 Semigroups, Formal Languages and Groups
, 1995
"... ..."
The word problem for cancellation semigroups with zero
 Journal of Symbolic Logic
, 1984
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at. ..."
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at.
Databases and FiniteModel Theory
 IN DESCRIPTIVE COMPLEXITY AND FINITE MODELS
, 1997
"... Databases provide one of the main concrete scenarios for finitemodel theory within computer science. This paper presents an informal overview of database theory aimed at finitemodel theorists, emphasizing the specificity of the database area. It is argued that the area of databases is a rich sourc ..."
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Databases provide one of the main concrete scenarios for finitemodel theory within computer science. This paper presents an informal overview of database theory aimed at finitemodel theorists, emphasizing the specificity of the database area. It is argued that the area of databases is a rich source of questions and vitality for finitemodel theory.