Free shuffle algebras in language varieties

AbstractWe give simple concrete descriptions of the free algebras in the varieties generated by the “shuffle semirings” LΣ := (P(Σ∗),+,., ⊗, 0,1), or the semirings RΣ := (R(Σ∗),+,., ⊗,∗,0,1), where P(Σ∗) is the collection of all subsets of the free monoid Σ∗, and R(Σ∗) is the collection of all regular subsets. The operation x ⊗ y is the shuffle product

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

Axiomatizing Omega and Omega-op Powers of Words

In 1978, Courcelle asked for a complete set of axioms and rules for the equational theory of (discrete regular) words equipped with the operations of product, omega power and omega-op power. In this paper we find a simple set of equations and prove they are complete. Moreover, we show that the equational theory is decidable in polynomial time

Completing categorical algebras : Extended abstract

Let Σ be a ranked set. A categorical Σ-algebra, cΣa for C, for short, is a small category C equipped with a functor σC : C n each σ ∈ Σn , n ≥ 0. A continuous categorical Σ-algebra is a cΣa which C; has an initial object and all colimits of ω-chains, i.e., functors N each functor σC preserves colimits of ω-chains. (N is the linearly ordered set of the nonnegative integers considered as a category as usual.) We prove that for any cΣa C there is an ω-continuous cΣa C ω , unique up to equivalence, which forms a “free continuous completion” of C. We generalize the notion of inequation (and equation) and show the inequations or equations that hold in C also hold in C ω . We then find examples of this completion when – C is a cΣa of finite Σ-trees – C is an ordered Σ algebra – C is a cΣa of finite A-sychronization trees – C is a cΣa of finite words on A.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

Some Remarks on Regular Words

In the late 1970's, Courcelle introduced the class of ``arrangements'', or labeled linear ordered sets, here called just ``words''. He singled out those words which are solutions of finite systems of fixed point equations involving finite words, which we call the ``regular words''. The current paper contains some new descriptions of this class of words related to properties of regular sets of binary strings, and uses finite automata to decide various natural questions concerning these words. In particular we show that a countable word is regular iff it can be defined on an ordinary regular language (which can be chosen to be a prefix code) ordered by the lexicographical order such that the labeling function satisfies a regularity condition. Those regular words whose underlying order is ``discrete'' or ``scattered'' are characterized in several ways