Transitive Closure Operator Research Articles

Reachability and the power of local ordering

The L ▪ NL question remains one of the major unresolved problems in complexity theory. Both L and NL have logical characterizations as the sets of totally ordered (⩽) structures expressible in first-order logic augmented with the appropriate Transitive Closure operator (Immerman, 1987): ( FO + DTC + ⩽) captures L and ( FO + TC + ⩽) captures NL. On the other hand, in the absence of ordering, ( FO + TC) is strictly more powerful than ( FO + DTC) (Grädel and McColm, 1992). An apparently quite different “structured” model of logspace machines is the Jumping Automaton on Graphs (JAG), (Cook and Rockoff, 1980). We show that the JAG model is intimately related to these logics on “one-way locally ordered” (1LO) structures. We argue that the usual JAG model is unreasonably weak and should be replaced, wherever possible, by the two-way JAG model, which we define. Furthermore, the language ( FO + DTC + 2 LO) over two-way locally ordered (2LO) graphs is more robust than even the two-way JAG model, and yet lower bounds remain accessible. We prove an upper bound on the power of TC over one-way locally ordered graphs, and three lower bounds on DTC.

Context-sensitive transitive closure operators

We introduce a new logical operator CSTC (standing for Context-Sensitive Transitive Closure) and show that incorporating this operator into first-order logic (with successor) enables as to capture the complexity class PSPACE. We also show that by varying how the operator is applied we can capture the complexity classes P, NP, the classes of the Polynomial Hierarchy PH, and PSPACE. As such, the operator CSTC can be regarded as a general purpose operator. We also give applications of these characterizations by showing that P and NP coincide with those problems accepted by two new classes of program schemes (involving arrays), by producing some new complete problems for various complexity classes where the reductions are from first principles and do not use known complete problems (that is, they are generic), and by giving a simple proof that first-order logic incorporated with the least fixed point operator captures P.

Algebraic optimization of recursive queries

Over the past few years, much attention has been paid to deductive databases. They offer a logic-based interface, and allow formulation of complex recursive queries. However, they do not offer appropriate update facilities, and do not support existing applications. To overcome these problems an SQL-like interface is required besides a logic-based interface. In the PRISMA project we have developed a tightly-coupled distributed database, on a multiprocessor machine, with two user interfaces: SQL and PRISMAlog. Query optimization is localized in one component: the relational query optimizer. Therefore, we have defined an eXtended Relational Algebra that allows recursive query formulation and can also be used for expressing executable schedules, and we have developed algebraic optimization strategies for recursive queries. In this paper we describe an optimization strategy that rewrites regular (in the context of formal grammars) mutually recursive queries into standard Relational Algebra and transitive closure operations. We also describe how to push selections into the resulting transitive closure operations. The reason we focus on algebraic optimization is that, in our opinion, the new generation of advanced database systems will be built starting from existing state-of-the-art relational technology, instead of building a completely new class of systems.

Knowledge representation using views in relational deductive data bases

A logic-based data base system combines data manipulation capabilities of conventional data base management systems (DBMS) and theorem-proving capabilities of logic systems. A deductive data base system is a homogeneous logic-based data base system in which only one programming system is used for both data base access operations and inferential functions. Logic systems such as Prolog are convenient for representing deductive axioms. However, because they lack the data base processing efficiency of conventional DBMSs, they are inadequate for very large data bases. In this paper, we propose to investigate the capabilities of traditional relational DBMSs, such as those based on the language SQL, as deductive data base systems. Since deductive axioms generate information in the form of virtual relations, they have a strong affinity to views in relational systems. Therefore, we can readily represent deductive axioms as views in relational data base systems. We demonstrate that Horn clauses as well as data base integrity constraints can be formulated as views without difficulty. Deductive axioms involving negation can also be formulated as views, provided the set difference operator is used for negated formulas. However, since the transitive closure operator is not implemented in conventional relational systems, such systems cannot process recursive deductive axioms. Relational DBMSs can be rendered almost fully deductive by incorporating into their design recursive axiom processing abilities and better view processing techniques.

Practical uses of operators in Sharp APL/HP

At APL86 in Manchester, the language features of SHARP APL/HP on a Hewlett Packard range of minicomputers were introduced in a paper written by this author, entitled 'APL PROCEDURES (USER DEFINED OPERATORS, FUNCTIONS AND TOKEN STRINGS)'. This is the first of several papers which have been planned to supplement this introductory work. The models of several new and existing operators are described in detail, all of which may be directly executed in SHARP APL/HP. The models take advantage of various features of SHARP APL/HP including extended assignment and procedure arrays [1] (often referred to as function arrays in their most common form). The paper will pursue a comparison of the less general implementation of operators in APL2, with special reference to Ed Eusebi's pioneering work in the field of 'Operators for Program Control and Recursion' [2 3], wherein an alternative definition for several of these operators will be proposed. The paper will then proceed to describe 'Operators applying to Enclosed Arrays' (for example a Pervasive Operator) and various 'Mathematical Operators' (instances of which are the Power and Transitive Closure operators), always highlighting these models with examples of their practical application.

Polynomial Space and Transitive Closure

A characterization of PSPACE in terms of the regular sets and certain algebraic closure operations is developed. It is shown that NP = PSPACE if and only if NP is closed under a form of the transitive closure operation.