An Extension Theorem with an Application to Formal Tree Series

A grove theory is a Lawvere algebraic theory T for which each hom-set T(n,p) is a commutative monoid; composition on the right distributes over all finite sums: (\sum f_i) . h = \sum f_i . h. A matrix theory is a grove theory in which composition on the left and right distributes over finite sums. A matrix theory M is isomorphic to a theory of all matrices over the semiring S = M(1,1). Examples of grove theories are theories of (bisimulation equivalence classes of) synchronization trees, and theories of formal tree series over a semiring S . Our main theorem states that if T is a grove theory which has a matrix subtheory M which is an iteration theory, then, under certain conditions, the fixed point operation on M can be extended in exactly one way to a fixed point operation on T such that T is an iteration theory. A second theorem is a Kleene-type result. Assume that T is an iteration grove theory and M is a sub iteration grove theory of T which is a matrix theory. For a given collection Sigma of scalar morphisms in T we describe the smallest sub iteration grove theory of T containing all the morphisms in M union Sigma

Simulations of Weighted Tree Automata

Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, Noetherian semirings, semiring of natural numbers) that all equivalent wta M and N can be joined by a finite chain of simulations. More precisely, in all mentioned cases there exists a single wta that simulates both M and N. Those results immediately yield decidability of equivalence provided that the semiring is finitely (and effectively) presented.Comment: 17 pages, 2 figure

A connection between concurrency and language theory

We show that three fixed point structures equipped with (sequential) composition, a sum operation, and a fixed point operation share the same valid equations. These are the theories of (context-free) languages, (regular) tree languages, and simulation equivalence classes of (regular) synchronization trees (or processes). The results reveal a close relationship between classical language theory and process algebra

Fixed Points on Abstract Structures without the Equality Test

In this paper we present a study of definability properties of fixed points of effective operators on abstract structures without the equality test. In particular we prove that Gandy theorem holds for abstract structures. This provides a useful tool for dealing with recursive definitions using Sigma-formulas. One of the applications of Gandy theorem in the case of the reals without the equality test is that it allows us to define universal Sigma-predicates. It leads to a topological characterisation of Sigma-relations on |R

Comparative Methods for Gene Structure Prediction in Homologous Sequences

The increasing number of sequenced genomes motivates the use of evolutionary patterns to detect genes. We present a series of comparative methods for gene finding in homologous prokaryotic or eukaryotic sequences. Based on a model of legal genes and a similarity measure between genes, we find the pair of legal genes of maximum similarity. We develop methods based on genes models and alignment based similarity measures of increasing complexity, which take into account many details of real gene structures, e.g. the similarity of the proteins encoded by the exons. When using a similarity measure based on an exiting alignment, the methods run in linear time. When integrating the alignment and prediction process which allows for more fine grained similarity measures, the methods run in quadratic time. We evaluate the methods in a series of experiments on synthetic and real sequence data, which show that all methods are competitive but that taking the similarity of the encoded proteins into account really boost the performance

An Extension Theorem with an Application to Formal Tree Series

A grove theory is a Lawvere algebraic theory T for which each hom-set T(n,p) is a commutative monoid; composition on the right distributes over all finite sums: (\sum f_i) . h = \sum f_i . h. A matrix theory is a grove theory in which composition on the left and right distributes over finite sums. A matrix theory M is isomorphic to a theory of all matrices over the semiring S = M(1,1). Examples of grove theories are theories of (bisimulation equivalence classes of) synchronization trees, and theories of formal tree series over a semiring S . Our main theorem states that if T is a grove theory which has a matrix subtheory M which is an iteration theory, then, under certain conditions, the fixed point operation on M can be extended in exactly one way to a fixed point operation on T such that T is an iteration theory. A second theorem is a Kleene-type result. Assume that T is an iteration grove theory and M is a sub iteration grove theory of T which is a matrix theory. For a given collection Sigma of scalar morphisms in T we describe the smallest sub iteration grove theory of T containing all the morphisms in M union Sigma