Class Of Program Schemes Research Articles

Using Program Schemes to Capture Polynomial-Time Logically on Certain Classes of Structures

AbstractIn this paper, the study of the expressive power of certain classes of program schemes on finite structures is continued, in relation to more mainstream logics studied in finite model theory and to computational complexity. The author shows that there exists a program scheme – whose constructs are assignments and while-loops with quantifier-free tests and which has access to a stack – that can accept a P-complete problem, the deterministic path system problem, even in the absence of non-determinism, so long as problem instances are presented in a functional style. (The proof given here leans heavily on Cook's proof that the classes of formal languages accepted by deterministic and non-deterministic logspace auxiliary pushdown machines coincide.) However, whilst this result is of independent interest, in that it leads to a deterministic model of computation capturing P, whose non-deterministic variant also captures P, the program scheme can also be used to build a successor relation in certain classes of structures (namely: the class of strongly connected locally ordered digraphs, the class of connected planar embeddings, and the class of triangulations), with the consequence that on these classes of graphs, (a fragment of) path system logic (with no built-in relations) captures exactly the polynomial-time solvable problems.

Program schemes, arrays, Lindström quantifiers and zero-one laws

We characterize the class of problems accepted by a class of program schemes with arrays, NPSA, as the class of problems defined by the sentences of a logic formed by extending first-order logic with a particular uniform (or vectorized) sequence of Lindström quantifiers. A simple extension of a known result thus enables us to prove that our logic, and consequently our class of program schemes, has a zero-one law. However, we use another existing result to show that there are problems definable in a basic fragment of our logic, and so also accepted by basic program schemes, which are not definable in bounded-variable infinitary logic. As a consequence, the class of problems NPSA is not contained in the class of problems defined by the sentences of partial fixed-point logic even though in the presence of a built-in successor relation, both NPSA and partial fixed-point logic capture the complexity class PSPACE.

On the power of built-in relations in certain classes of program schemes

We completely classify the relative expressibilities of the program schemes of NPS augmented with the built-in relations: linear order; addition; multiplication; and BIT. We employ pebble games allied with some number theory.

A new programming technique for lazy functional languages

In this paper we present a new programming technique for lazy functional programming languages. The technique is embedded in a programming methodology which is based on divide and conquer: the division of problems into subproblems. Such a division will be represented by a call graph. A class of program schemes, which implement call graphs, is derived based on Johnsson's approach to attribute evaluation in attribute grammars. The key idea is to consider the passing of arguments to functions and the return of results by functions in a call graph as the propagation of inherited and synthesized attributes, respectively, in an attribute grammar. The new technique is illustrated by several small examples and a case study: the design and implementation of a compiler generator.

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.

An algorithm deciding functional equivalence in a new class of program schemes

An algorithm deciding functional equivalence in a new class of program schemes

�quivalente Transformationen f�r Flu�diagramme

In this paper we describe complete systems of transformations and algorithms for deciding 1) the operational and 2) the logic-termal equivalence problems in the whole class of program schemes, and 3) the functional equivalence problem in the class of so called “through-schemes”.