Michael G. Main

An O ( n log n ) algorithm for finding all repetitions in a string

Any nonempty string of the form xx is called a repetition. An O(n log n) algorithm is presented to find all repetitions in a string of lenght n. The algorithm is based on a linear algorithm to find all the new repetitions formed when two strings are concatenated. This linear algorithm is possible because new repetitions of equal length must occur in blocks with consecutive starting positions. The linear algorithm uses a variation of the Knuth-Morris-Pratt algorithm to find all partial occurrences of a pattern within a text string. It is also shown that no algorithm based on comparisons of symbols can improve O(n log n). Finally, some open problems and applications are suggested.

Detecting leftmost maximal periodicities

Periodicities are nonempty strings of the form pmq with m 12 and q a prefix of p. The integer m is called the power of the periodicity, “he length of p (written lpi) is the period length, and the periodicity length is the total length of pmq. This note presents a new algorithm to output a list of occurrences of periodicities within a string. The list does not contain all periodicities within the given string, but it is guaranteed to contain any periodicity which is leftmost and maximal. Informally, a substring of a string is a maximal periodicity provided that it is a periodicity such that neither the character which precedes nor the character which follows the substring can be added without destroying the periodicity. A substring of a string is leftmost provided that there is no previous occurrence of the substring within the given string. For a given string, the list of periodicities produced by the algorihm provides immediate answers to problems such as:

Linear Time Recognition of Squarefree Strings

A square is an immediately repeated nonempty string, e.g., aa, abab, abcabc. This paper presents a new O(n log n) algorithm to determine whether a string of length n has a substring which is a square. The algorithm is not as general as some previous algorithms for finding all squares [1,7,8,13], but it does have a simplicity which the others lack. Also, for a fixed alphabet of size k, the algorithm can be improved by a factor of log k (n), yielding an O(n) algorithm for determining whether a string contains a square.

FUNCTIONS DEFINED BY REACTION SYSTEMS

Reaction systems are a formal model of interactions between biochemical reactions. They consist of sets of reactions, where each reaction is classified by its set of reactants (needed for the reaction to take place), its set of inhibitors (each of which prevents the reaction from taking place), and its set of products (produced when the reaction takes place) – the set of reactants and inhibitors form the resources of the reaction. Each reaction system defines a (transition) function on its set of states. (States here are subsets of an a priori given set of biochemical entities.) In this paper we investigate properties of functions defined by reaction systems. In particular, we investigate how the power of defining functions depends on available resources, and we demonstrate that with small resources one can define functions exhibiting complex behavior.

COMBINATORICS OF LIFE AND DEATH FOR REACTION SYSTEMS

Reaction systems are a functional model of interactions between biochemical reactions. They define functions on finite sets (over a common finite domain). In this paper, we investigate combinatorial properties of functions defined by reaction systems. In particular, we provide analytical approximations of combinatorial properties of random reaction systems, with a focus on the probability of whether a system lives or dies. Based on these results, we can create parameterized random reaction systems that rarely die. We also empirically analyze the length of time before such a system enters cyclic behavior, and find that the time is related to the behavior of completely random functions on a smaller domain.

Restrictions on NLC graph grammars

Several models of ‘graph grammars’ have been studied with the objective of generating graphs from graphs using a finite set of derivation rules. In this way, possibly infinite sets of graphs (called graph languages) can be finitely defined. One aspect that must be addressed in any such model is the ‘embedding problem’, that is: When a production is applied, how does the new subgraph get reconnected to the original graph? Node-label controlled (NLC) grammars solve this problem in an elegant way that depends only on the labels of nodes in the new and original graphs. This paper examines certain restrictions on NLC grammars similar to the Chomsky or Greibach normal forms for context-free string grammars. For example, one restriction we consider requires each production to produce a terminal labeled node—similar to Greibach normal form. We also consider restrictions on the form which the embedding mechanism can take. Our result is that each of the restrictions we examine causes a reduction in generating power for the grammars. Finally, we discuss some directions for future research on NLC grammars.

Trace, failure and testing equivalences for communicating processes

A basic question in the theory of communicating processes is “When should two processes be considered equivalent?”. Attempts to answer this question have led to the concepts of observation equivalence, bisimulations, testing equivalence, failure equivalence, etc. The main point of this paper is to increase the understanding and motivation for two of these equivalences, namely failure and testing equivalences. The approach starts with the idea that the equivalence of processes should be reducible to the visible sequences of actions which a process performs in various contexts. This idea is implemented by a string-based semantic order for communicating processes where divergence is catastrophic. Under some assumptions about contexts, the resulting semantics is shown to be equivalent to theimproved failure semantics of Brookes and Roscoe(1) and also to themust testing-semantics of Hennessy and DeNicola.(2–4) This characterization gives independent support for the appropriateness of failures and testing.

Functional behavior of nondeterministic and concurrent programs

The functions behavior of a deterministic program segment is a function f:D→D, where D is some set of states for the computation. This notion of functional behavior can be extended to nondeterministic and concurrent programs using techniques from linear algebra. In particular, the functional behavior of a nondeterministic program segment is a linear transformation f:A→A, where A is a free semiring module. Other notions from linear algebra carry over into this setting. For example, weakest preconditions and predicate transformers correspond to well-studied concepts in linear algebra. Using multilinear algebra, programs with tuples of inputs and outputs can be handled. For nondeterministic concurrent programs, the functional behavior is a linear transformation f:A→A, where A is a free semiring algebra. In this case, f may also be an algebra morphism, which indicates that the program involves no interprocess communication. Finally, a model of syntax for programs is studied whose semantics is given using linear algebra. It is shown that in this model, free interpretations (essentially Herbrand universes) do not generally exist.

Free Constructions of Powerdomains

A powerdomain is a CPO together with extra algebraic structure for handling nondeterministic values. In the first powerdomains, the algebraic structure was a continuous binary operation or, which met certain axioms. Plotkin [9] and Smyth [14] showed how such a structure could be added to certain kinds of CPOs in a free or universal manner. This paper extends the work of Plotkin and Smyth by giving free constructions of powerdomains for a more adaptable algebraic structure: semiring modules. Prior to the constructions, three detailed examples are given, showing how the semiring module structure can capture information about nondeterministic behavior. By putting the available information in an algebraic framework, the algebraic properties can supplement the usual order-theoretic properties in program proofs.

Edge-label controlled graph grammars

Abstract We introduce a graph-grammar model based on edge-replacement, where both the rewriting and the embedding mechanisms are controlled by edge labels. The general power of this model is established—it turns out to have the complete power of recursive enumerability (in a sense to be made precise in the paper). In order to understand where this power originates, we identify three basic features of the embedding mechanism and examine how restrictions on these features affect the generative power. In particular, by imposing restrictions on all three features simultaneously, we obtain a graph-grammar model that was previously introduced by Kreowski and Habel.