Minimal and consistent evolution of knowledge bases

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

We investigate the query evaluation problem for fixed queries over fully dynamic databases where tuples can be inserted or deleted. The task is to design a dynamic data structure that can immediately report the new result of a fixed query after every database update. We consider unions of conjunctive queries (UCQs) and focus on the query evaluation tasks testing (decide whether an input tuple belongs to the query result), enumeration (enumerate, without repetition, all tuples in the query result), and counting (output the number of tuples in the query result). We identify three increasingly restrictive classes of UCQs which we call t-hierarchical, q-hierarchical, and exhaustively q-hierarchical UCQs. Our main results provide the following dichotomies: If the query's homomorphic core is t-hierarchical (q-hierarchical, exhaustively q-hierarchical), then the testing (enumeration, counting) problem can be solved with constant update time and constant testing time (delay, counting time). Otherwise, it cannot be solved with sublinear update time and sublinear testing time (delay, counting time), unless the OV-conjecture and/or the OMv-conjecture fails. We also study the complexity of query evaluation in the dynamic setting in the presence of integrity constraints, and we obtain similar dichotomy results for the special case of small domain constraints (i.e., constraints which state that all values in a particular column of a relation belong to a fixed domain of constant size).

We show that stable models of logic programs may be viewed as minimal models of programs that satisfy certain additional constraints. To do so, we transform the normal programs into disjunctive logic programs and sets of integrity constraints. We show that the stable models of the normal program coincide with the minimal models of the disjunctive program thatsatisfy the integrity constraints. As a consequence, the stable model semantics can be characterized using theextended generalized closed world assumption for disjunctive logic programs. Using this result, we develop a bottomup algorithm for function-free logic programs to find all stable models of a normal program by computing the perfect models of a disjunctive stratified logic program and checking them for consistency with the integrity constraints. The integrity constraints provide a rationale as to why some normal logic programs have no stable models.

Given a nonempty set \(X\) and \(Y\subseteq X\), we denote by \(T(X,Y)\) the subsemigroup of the full transformation semigroup \(\mathcal{T}_X\) on \(X\) consisting of all transformations whose range is contained in \(Y\). In this paper, we endow the semigroup \(T(X,Y)\) with the natural partial order and investigate when two elements are related, then find elements which are compatible with the order. Also, we characterize the minimal elements and the maximal elements.

Consistency enforcement aims at systematically modifying a database program such that the result is consistent with respect to a specified set of integrity constraints. This modification may be done at compile-time or at run-time. The commonly known run-time approach uses rule triggering systems (RTSs). It has been shown that these systems cannot solve the problem in general. As an alternative greatest consistent specializations (GCSs) have been studied. This approach requires the modified program specification to be a maximal consistent diminution of the original one with respect to some partial order. The chosen order is operational specialization. On this basis it is possible to derive a commutativity result and a compositionality result. The first one enables step-by-step enforcement for sets of constraints. The second one reduces the problem to providing the GCSs just for basic operations, whereas for complex programs the GCS can be easily determined. The approach turns out to be well-founded since the GCS for such complex programs is effectively computable if we require loops to be bounded. Despite its theoretical merits the GCS approach is still too coarse. This leads to the problem of modifying the chosen specialization order and to relax the requirement that the result should be unique. One idea is to exploit the fact that operational specialization is equivalent to the preservation of a set of transition invariants. In this case a reasonable order arises from a slight modification of this set, in which case we talk of a maximal consistent effect preserver (MCE). However, a strict theory of MCEs is still outstanding.

In the proposed data model for XML databases, an XML element is directly represented as a ground (variable-free) XML expression—a generalization of an XML element by incorporation of variables for representation of implicit information and enhancement of its expressive power—while a collection of XML documents as a set of ground expressions, each describing an XML element in the documents. Relationships among elements in the collection as well as integrity constraints are formalized as XML clauses. An XML database, consisting of: (i) a document collection (or an extensional database), (ii) a set of relationships (or an intensional database) and (iii) a set of integrity constraints, is therefore modeled as an XML declarative description comprising a set of ground XML expressions and XML clauses. Its semantics is a set of ground XML expressions, which are explicitly described by the extensional database or implicitly derived from the intensional database and satisfy all the specified set of constraints.KeywordsDescription LogicIntegrity ConstraintUnit ClauseImplicit InformationSemistructured DataThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The management of business processes has recently received a lot of attention. One of the most interesting problems is the description of a process model in a language that allows the checking of the compliance of a process execution (or trace) to the model. In this paper we propose a language for the representation of process models that is inspired to the SCIFF language and is an extension of clausal logic. A process model is represented in the language as a set of integrity constraints that allow conjunctive formulas as disjuncts in the head. We present an approach for inducing these models from data: we define a subsumption relation for the integrity constraints, we define a refinement operator and we adapt the algorithm ICL to the problem of learning such formulas. The system has been applied to the problem of inducing the model of a sealed bid auction and of the NetBill protocol. The data used for learning and testing were randomly generated from correct models of the processes.

Tree patterns form a natural basis to query tree-structured data such as XML and LDAP. To improve the efficiency of tree pattern matching, it is essential to quickly identify and eliminate redundant nodes in the pattern. In this paper, we study tree pattern minimization both in the absence and in the presence of integrity constraints (ICs) on the underlying tree-structured database. In the absence of ICs, we develop a polynomial-time query minimization algorithm called CIM, whose efficiency stems from two key properties: (i) a node cannot be redundant unless its children are; and (ii) the order of elimination of redundant nodes is immaterial. When ICs are considered for minimization, we develop a technique for query minimization based on three fundamental operations: augmentation (an adaptation of the well-known chase procedure), minimization (based on homomorphism techniques), and reduction. We show the surprising result that the algorithm, referred to as ACIM, obtained by first augmenting the tree pattern using ICs, and then applying CIM, always finds the unique minimal equivalent query. While ACIM is polynomial time, it can be expensive in practice because of its inherent non-locality. We then present a fast algorithm, CDM, that identifies and eliminates local redundancies due to ICs, based on propagating ”information labels” up the tree pattern. CDM can be applied prior to ACIM for improving the minimization efficiency. We complement our analytical results with an experimental study that shows the effectiveness of our tree pattern minimization techniques.

Repairing a database means bringing the database in accordance with a given set of integrity constraints by applying modifications that are as small as possible. In the seminal work of Arenas et al. on query answering in the presence of inconsistency, the possible modifications considered are deletions and insertions of tuples. Unlike earlier work, we also allow tuple updates as a repair primitive. Update-based repairing is advantageous, because it allows rectifying an error within a tuple without deleting the tuple, thereby preserving other consistent values in the tuple. At the center of the paper is the problem of query answering in the presence of inconsistency relative to this refined repair notion. Given a query, a trustable answer is obtained by intersecting the query answers on all repaired versions of the database. The problem arising is that, in general, a database can be repaired in infinitely many ways. A positive result is that for conjunctive queries and full dependencies, there exists a condensed representation of all repairs that permits computing trustable query answers.

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ –relation. The corresponding equivalence relation $$\simeq _N$$ is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

The absolute order on the symmetric group, constructible partially ordered sets and Cohen–Macaulay complexes

Several modules M over algebras with straightening law A have a structure which is similar to the structure of A itself: M has a system of generators endowed with a natural partial order, a standard basis over the ring B of coefficients, and the multiplication A × M → A satisfies a “straightening law”. We call them modules with straightening law, briefly MSLs. In section 1 we recall the notion of an algebra with straightening law together with those examples which will be important in the sequel. Section 2 contains the basic results on MSLs, whereas section 3 is devoted to examples: (i) powers of certain ideals and residue class rings with respect to them, (ii) “generic” modules defined by generic, alternating or symmetric matrices of indeterminates, (iii) certain modules related to differentials and derivations of determinantal rings. The essential homological invariant of a module is its depth. We discuss how to compute the depth of an MSL in section 4. The main tool are filtrations related to the MSL structure. The last section contains a natural strengthening of the MSL axioms which under certain circumstances leads to a straightening law on the symmetric algebra. The main examples of such modules are the “generic” modules defined by generic and alternating matrices. The notion of an MSL was introduced by the author in [Br.3] and discussed extensively during the workshop. The main differences of this survey to [Br.3] are the more detailed study of examples and the treatment of the depth of MSLs which is almost entirely missing in [Br.3]

Recently M. M. Kapranov [Kap] defined a poset KPAn−1, called the permuto-associahedron, which is a hybrid between the face poset of the permutohedron and the associahedron. Its faces are the partially parenthesized, ordered, partitions of the set {1, 2, …, n}, with a natural partial order. Kapranov showed that KPAn−1, is the face poset of a regular CW-ball, and explored its connection with a category-theoretic result of MacLane, Drinfeld's work on the Knizhnik-Zamolodchikov equations, and a certain moduli space of curves. He also asked the question of whether this CW-ball can be realized as a convex polytope. We show that indeed, the permuto-associahedron corresponds to the type An−1, in a family of convex polytopes KPW associated to the classical Coxeter groups, W = An−1, Bn, Dn. The embedding of these polytopes relies on the secondary polytope construction of the associahedron due to Gel'fand, Kapranov, and Zelevinsky. Our proofs yield integral coordinates, with all vertices on a sphere, and include a complete description of the facet-defining inequalities. Also we show that for each W, the dual polytope KPW* is a refinement (as a CW-complex) of the Coxeter complex associated to W, and a coarsening of the barycentric subdivision of the Coxeter complex. In the case W = An−1, this gives a combinatorial proof of Kapranov's original sphericity result.

For any amalgam ( S , T ; U ) (S,\,T;\,U) of inverse semigroups, it is shown that the natural partial order on S ∗ U T S{{\ast }_U}T , the (inverse semigroup) free product of S S and T T amalgamating U U , has a simple form on S ∪ T S \cup T . In particular, it follows that the semilattice of S ∗ U T S{{\ast }_U}T is a bundled semilattice of the corresponding semilattice amalgam ( E ( S ) , E ( T ) ; E ( U ) ) (E(S),\,E(T);\,E(U)) ; taken jointly with a result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered.