It is shown that the syntactic algebra of the characteristic series of a rational language L is semisimple in the following two cases: L is a free submonoid generated by a bifix code, or L is a cyclic language.

This chapter has two appendices, one on semisimple algebras (without proofs) and another on simple semigroups, with concise proofs. We use the symbols A1 and A2 to refer to them.

Bifix codes

Let E be a set of endomorphisms of a finite dimensional vector space V. Recall that E is called irreducible if there is no subspace of V other than 0 and V itself which is invariant under all endomorphisms in E. Similarly, we say that E is completely reducible if V is a direct sum V = V1 ⊕…⊕ Vk of subspaces such that for each i, the set of induced endomorphisms e∣Vi of Vi, for e ϵ E, is irreducible.

A set of matrices in Kn×n (K being a field) is irreducible (resp. completely reducible) if it is so, viewed as a set of endomorphisms acting at the right on K1×n, or equivalently at the left on Kn×1 (for this equivalence, see Exercises 1.1 and 1.2).

A linear representation (λ, μ, γ)ofaseries S ϵ K⟪A⟫ is irreducible (resp. completely reducible) if the set of matrices {μa ∣ a ϵ A}(or equivalently the sets μA* or μ(K⟨A⟩)) is so.

This chapter gives a presentation of results concerning the minimization of linear representations of recognizable series. A central concept of this study is the notion of syntactic algebra, which is introduced in Section 2.1. Rational series are characterized by the fact that their syntactic algebras are finite dimensional (Theorem 1.2). The syntactic right ideal leads to the notion of rank and of Hankel matrix; the quotient by this ideal is the analogue for series of the minimal automaton for languages.

Section 2.2 is devoted to the detailed study of minimal linear representations. The relations between representations and syntactic algebra are given. Two minimal representations are always similar (Theorem 2.4), and an explicit form of the minimal representation is given (Corollary 2.3).

The minimization algorithm is presented in Section 2.3. We start with a study of prefix sets. The main tool is a description of bases of right ideals of the ring of noncommutative polynomials (Theorem 3.2).

Several important consequences are given. Among them are Cohn's result on the freeness of right ideals, the Schreier formula for right ideals and linear recurrence relations for the coefficients of a rational series. A detailed description of the minimization algorithm completes the chapter.

Syntactic ideals

We start by assuming that K is a commutative ring. The algebra of polynomials K ⟨A⟩ is a free K-module having as a basis the free monoid A*. Consequently, the set K ⟪A⟫ of formal series can be identified with the dual of K⟨A⟩.

This chapter describes the relations between rational series and languages.

We start with Kleene's theorem, presented as a consequence of Schützenberger's theorem. Then we describe the cases where the support of a rational series is a rational language. The most important result states that if a series has finite image, then its support is a rational language (Theorem 2.10).

The family of languages which are supports of rational series have closure properties given in Section 3.4. The iteration theorem for rational series is proved in Section 3.5. The last section is concerned with an extremal property of supports which forces their rationality; to prove it, we use a remarkable characterization of rational languages due to Ehrenfeucht, Parikh and Rozenberg.

Kleene's theorem

Definition A congruence in a monoid is an equivalence relation which is compatible with the operation in the monoid. A language L is recognizable if there exists a congruence with finite index in A* that saturates L (that is L is a union of equivalence classes).

It is equivalent to say that L is recognizable if there exists a finite monoid M, a morphism of monoids ϕ: A* → M and a subset P of M such that L = ϕ−1(P).

The product of two languages L1 and L2 is the language L1L2 ={xy∣x ϵ L1, y ϵ L2}. If L is a language, the submonoid generated by L is ∪n≥0Ln.

Formal power series have long been used in all branches of mathematics. They are invaluable in algebra, analysis, combinatorics and in theoretical computer science.

Historically, the work of M.-P. Schützenberger in the algebraic theory of finite automata and the corresponding languages has led him to introduce noncommutative formal power series. This appears in particular in his work with Chomsky on formal grammars. This last point of view is at the origin of this book.

The first part of the book, composed of Chapters 1–4, is especially devoted to this aspect: Formal power series may be viewed as formal languages with coefficients, and finite automata (and more generally weighted automata) may be considered as linear representations of the free monoid. In this sense, via formal power series, algebraic theory of automata becomes a part of representation theory.

The first two chapters contain general results and discuss in particular the equality between rational and recognizable series (Theorem of Kleene–Schützenberger) and the construction of the minimal linear representation. The exposition illustrates the synthesis of linear algebra and syntactic methods inherited from automata theory.

The next two chapters are concerned with the comparison of some typical properties of rational (regular) languages, when they are transposed to rational series. First, Chapter 3 describes the relationship with the family of regular languages studied in theoretical computer science. Next, the chapter contains iteration properties for rational series, also known as pumping lemmas, which are much more involved than those for regular languages.

This chapter contains the definitions of the basic concepts, namely rational and recognizable series in several noncommuting variables.

We start with the definition of a semiring, followed by the notation for the usual objects in free monoids and formal series. The topology on formal series is briefly introduced.

Section 1.4 contains the definition of rational series, together with some elementary properties and the fact that certain morphisms preserve the rationality of series.

Recognizable series are introduced in Section 1.5. An algebraic characterization is given. We also prove (Theorem 5.1) that the Hadamard product preserves recognizability.

Weighted automata are presented in Section 1.6.

The fundamental theorem of Schützenberger (equivalence between rational and recognizable series, Theorem 7.1) is the concern of the last section. This theorem is the starting point for the developments given in the subsequent chapters.

Semirings

Recall that a semigroup is a set equipped with an associative binary operation, and a monoid is a semigroup having a neutral element for its law.

A semiring is, roughly speaking, a ring without subtraction. More precisely, it is a set K equipped with two operations + and ·(sum and product) such that the following properties hold:

1. (i) (K, +) is a commutative monoid with neutral element denoted by 0.

2. (ii) (K, ·) is a monoid with neutral element denoted by 1.

3. (iii) The product is distributive with respect to the sum.

4. (iv) For all a in K,0a = a0 = 0.

If K is a subsemiring of a semiring L, each K-rational series is clearly L-rational. The main problem considered in this chapter is the converse: how to determine which of the L-rational series are rational over K. This leads to the study of semirings of a special type, and also shows the existence of remarkable families of rational series.

In the first section, we examine principal rings from this aspect. Fatou's Lemma is proved and the rings satisfying this lemma are characterized (Chabert's Theorem 1.5).

In the second section, Fatou extensions are introduced. We show in particular that ℚ+ is a Fatou extension of ℕ (Theorem 2.2 due to Fliess).

In the third section, we apply Shirshov's theorem on rings with polynomial identities to prove criteria for rationality of series and languages. This is then applied, in the last section, to Fatou ring extensions.

Rational series over a principal ring

Let K be a commutative principal ring and let F be its quotient field. Let S ϵ K⟪A⟫ be a formal series over A with coefficients in K. If S is a rational series over F, is it also rational over K? This question admits a positive answer, and there is even a stronger result, namely that S has a minimal linear representation with coefficients in K.

Theorem 1.1 (Fliess 1974a) Let S ϵ K⟪A⟫ be a series which is rational of rank n over F.