Linear Time Computable Problems and Logical Descriptions

Linear time computable problems and logical descriptions

It is a general problem to investigate the trade oo between the complexity of al-gorithmic problems, the structure of the input objects and the expressive power of problem description languages. The article concentrates on linear time algorithms and on rst order logic (FO) as problem description language. One of the main results is a proof that each FO-problem can be solved in linear time for a...

Linear Time Computable Problems and First-Order Descriptions

Computable choice functions for computable linear orderings

One direction of research in computability theory is the study of the effective content of combinatorial theorems. The last chapter of Rosenstein’s book [2] reports on this type of investigation for linear orderings. Questions raised in that chapter have been the subject of a number of papers, and a summary of results in the area can be found in Downey [1]. In this paper, we consider the proble...

Algebraic and logical descriptions of generalized trees

Quasi-trees generalize trees in that the unique “path” between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound of the rank-widths of its finite induced subgraphs. Join-trees are the corresponding directed trees. They are useful to define the modular decomposition of a...

Problems, Descriptions and Objects

The business of software development is solving problems. Following Polya, we can increase our ability to solve problems by focusing directly on problems themselves, on their parts and structures, and on the relationship between problem and solution method. This leads to an emphasis on describing the world outside the computer, and reasoning about it explicitly; to an approach to problem comple...