MOTIVATION: Tandem repeats (TRs) are associated with human disease, play a role in evolution and are important in regulatory processes. Despite their importance, locating and characterizing these patterns within anonymous DNA sequences remains a challenge. In part, the difficulty is due to imperfect conservation of patterns and complex pattern structures. We study recognition algorithms for two complex pattern structures: variable length tandem repeats (VLTRs) and multi-period tandem repeats (MPTRs). RESULTS: We extend previous algorithmic research to a class of regular tandem repeats (RegTRs). We formally define RegTRs, as well as two important subclasses: VLTRs and MPTRs. We present algorithms for identification of TRs in these classes. Furthermore, our algorithms identify degenerate VLTRs and MPTRs: repeats containing substitutions, insertions and deletions. To illustrate our work, we present results of our analysis for two difficult regions in cattle and human data which reflect practical occurrences of these subclasses in GenBank sequence data. In addition, we show the applicability of our algorithmic techniques for identifying Alu sequences, gene clusters and other distant regions of similarity. We illustrate this with an example from yeast chromosome I.

In this paper a new property of sets, near-testability, is introduced. A set S is near-testable$(S \in NT)$ if the membership relation for all immediate neighbors is polynomially computable; i.e., if the function $t(x) = \chi _S (x) + \chi _S (x - 1)(\bmod 2)$ is polynomially computable. The near-testable sets form a subclass of the class $ \oplus P$ (parity polynomial time), introduced by Papadimitriou and Zachos, and Goldschlager and Parberry. $ \oplus P$ has a complete set $ \oplus SAT$ that has recently been shown by Valiant and Vazirani to be hard for $NP$ under randomized polynomial-time reductions. It is proved that there is a uniform polynomial one-one reduction that takes every set in $ \oplus P$ to a near-testable set, and it is shown that the image of $ \oplus SAT$ under this reduction (which we call $NTSAT$) is polynomially isomorphic to $ \oplus SAT$. As corollaries it is shown that $NTSAT$ is complete for both $NT$ and for $ \oplus SAT$, that $NTSAT$ is hard for $NP$ under randomized polynomial-time reductions, and that the existence of one-way functions implies the existence of sets that are near-testable but not polynomially decidable. It is then asked whether near-testability is preserved under p-isomorphisms. This leads to a generalization, $NT^ * $, of $NT$ similar to those introduced by Meyer and Paterson and by Ko for self-reducible sets. With this more general definition, $NT^ * $ is shown to be closed under polynomial-time isomorphisms while remaining a subclass of $ \oplus P$. It is conjectured that it is a proper subclass. In fact it is shown that, relative to a random oracle, the containments $P \subseteq NT \subseteq NT^ * \subseteq \oplus P$ are proper with probability one. It is also shown that, relative to a random oracle, with probability one $NT$ and $NT^ * $ are incomparable with both $NP$ and with $coNP$. Finally, the effects that the distribution and density of elements have on the complexity of near-testable sets are considered.

This paper proves three results relating to the isomorphism question for NP-complete sets. Result 1: We construct an oracle A such that SATA is ≤mP- complete for NPA and all ≤mP-complete sets for NPA are pA- isomorphic to SATA. Result 2: We construct a time function T(n) such that DTIME(T(n)) contains btt-complete sets, which are many-one equivalent, but are not p-isomorphic. The proof of this result has two corollaries: 1) There is an oracle, D, such that NPD contains non-p-isomorophic ≤m(D),P-complete sets. 2) There is a ≤mP-degree that contains non-p-isomorphic sets. Result 3: We show that no simple modification of the diagonalization argument used by Ko, Long and Du can be used to produce sets that are both EXPtime-complete w.r.t, polynomial many-one reducibility and not p-isomorphic.

The mover’s problem is the following: can an object in 3-dimensional space be moved from one given position to another while avoiding obstacles? It is known that the general version of this problem involving objects with movable joints is PSPACE hard, even for a simple tree-like structure moving in a 3-dimensional region. In this paper, we investigate a 2-dimensional mover’s problem in which the object is a robot arm with an arbitrary number of joints. In particular, we give a polynomial time algorithm for moving an arm confined within a circle from one given configuration to another. We also give a polynomial time algorithm for moving the arm from its initial position to a position in which the end of the arm reaches a given point within the circle. Finally, we show that 148 circles suffice to cover the boundary of the reachable region of a joint in an arm enclosed in a circle and that the boundary can be computed in polynomial time.

This paper is motivated by questions concerning the planning of motion in robotics. In particular, it is concerned with the motion of planar linkages from the complexity point of view. There are two main results. First, a planar linkage can be constrained to stay inside a bounded region whose boundary consists of straight lines by the addition of a polynomial number of new links. Second, the question of whether a planar linkage in some initial configuration can be moved so that a designated joint reaches a given point in the plane is PSPACE-hard.

The classical mover's problem is the following: can a rigid object in 3-dimensional space be moved from one given position to another while avoiding obstacles? It is known that a more general version of this problem involving objects with movable joints is PSPACE-complete, even for a simple tree-like structure. In this paper, we investigate a 2-dimensional mover's problem in which the object being moved is a robot arm with an arbitrary number of joints. We reduce the mover's problem for arms constrained to move within bounded regions whose boundaries are made up of straight lines to the mover's problem for a more complex linkage that is not constrained. We prove that the latter problem is PSPACE-hard even in 2-dimensional space and then turn to special cases of the mover's problem for arms. In particular, we give a polynomial time algorithm for moving an arm confined within a circle from one given configuration to another. We also give a polynomial time algorithm for moving the arm from its initial position to a position in which the end of the arm reaches a given point within the circle.

In this paper we study two alternative approaches for investigating whether NP complete sets have fast algorithms. One is to ask whether there are long initial segments on which such sets are easily decidable by relatively short programs. The other approach is to ask whether there are weak fragments of arithmetic for which it is consistent to believe that P = NP. We show, perhaps surprisingly, that the two questions are equivalent: It is consistent to believe that P = NP in certain models of weak arithmetic theories iff it is true (in the standard model of computation) that there are infinitely many initial segments on which satisfiability is polynomially decidable by programs that are much shorter than the length of the initial segment.

Although there has been considerable additional work discussing limitations of formal proof techniques for Computer Science ([YO-73&77], [HAR-76], [HAR&HO-77], [HAJ-77&79], [GO-79]), these papers show only very general consequences of incompleteness: the stated results hold for all sufficiently powerful formal systems for Computer Science. Only the work of O'Donnell and of Lipton directly addresses the question of just how powerful formal axioms for Computer Science should be, and these two authors make rather radically different suggestions.