Zoltán Ésik

Free shuffle algebras in language varieties

We 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.

Group Axioms for Iteration

Iteration theories provide a sound and complete axiomatization of the equational properties of the iteration (or fixed point) operation in many models of theoretical computer science including ordered and metric structures, trees and synchronization trees. All known equational axiomatizations of iteration theories consist of a small set of equational axioms for Conway theories and a complicated equation scheme, the commutative identity. Here we associate an identity with each finite semigroup. We prove that the set consisting of the Conway identities and the group identities associated with the finite (simple) groups is complete. Moreover, we prove that the Conway identities and a subcollection of the semigroup identities associated with a subclass of the finite semigroups is complete iff each finite (simple) group divides one of the semigroups in the subclass. We also formulate a conjecture and study its consequences. The results are a generalization of Krobs axiomatization of the equational theory of the regular sets.

Completeness of Park induction

Abstract The (in)equational properties of iteration, i.e., least (pre-)fixed point solutions over cpos, are captured by the axioms of iteration theories. All known axiomatizations of iteration theories consist of the Conway identities and a complicated equation scheme, the commutative identity. The results of this paper show that the commutative identity is implied by the Conway identities and a weak form of the Park induction principle . Hence, we obtain a simple first order axiomatization of the (in)equational theory of iteration. It follows that a few simple identities and a weak form of the Scott induction principle , formulated to involve only inequations, are also complete. We also show that the Conway identities and the Park induction principle are not complete for the universal Horn theory of iteration.

Estimation of state complexity of combined operations

It appears that the state complexity of each combined operation has its own special features. Thus, it is important and practical to obtain good estimates for some commonly used general cases. In this paper, we consider the state complexity of combined Boolean operations and give an exact bound for all of them in the case when the alphabet is not fixed. Moreover, we show that for any fixed alphabet, this bound can be reached in infinitely many cases. We also consider the state complexity of multiple catenations. The state complexities are obtained in the cases of the catenations of three and four languages. An estimate for the catenation of an arbitrary number of languages is given, which is very close to the state complexities in the three and four languages cases.

Recent advances in formal languages and applications

Basic Notation and Terminology.- Janusz Brzozowski.- Maxime Crochemore, Thierry Lecroq.- Jozef Gruska.- Tom Head, Dennis Pixton.- Lucian Ilie.- Jarkko Kari.- Satoshi Kobayashi.- Hans-Jorg Kreowski, Renate Klempien-Hinrichs, Sabine Kuske.- Mitsunori Ogihara.- Friedrich Otto.- Holger Petersen.- Shuly Wintner.- Hsu-Chun Yen.

Equational Properties of Iteration in Algebraically Complete Categories

The main result is the following completeness theorem: If the fixed point operation over a category is defined by initiality, then the equations satisfied by the fixed point operation are exactly those of iteration theories. Thus, in such categories, the equational axioms of iteration theories provide a sound and complete axiomatization of the equational properties of the fixed point operation.

Iteration theories of synchronization trees

Synchronization trees are shown to form an iteration theory in a natural way. The class of grove iteration theories is introduced. The regular synchronization trees are shown to be the free theories in the subclass of synchronization theories. Moreover, the bisimulation equivalence classes of regular synchronization trees are shown to be the free synchronization theories satisfying an ‘infinite’ idempotency law.

Axiomatizing schemes and their behaviors

The syntax and the semantics of flowchart algorithms may be conveniently separated. The syntax of a flowchart algorithm is the underlying flowchart scheme. The semantics is given by an interpretation of each symbol appear ing on the nodes of the scheme. One aspect of this theory is the determination of when two schemes have the same “behavior” under all interpretations. When “behavior” is interpreted in the strongest sense as “performing the same stepwise computations,” the collection of equivalence classes of schemes was shown by Elgot [E-MC] to have the structure of an “iterative theory.” The strong behavior of a scheme is precisely captured by the (labeled, locally ordered, locally finite) tree obtained by “unfolding” the scheme in the familiar way. (When the “behavior” of a scheme F is considered to be the partial input-output function computed by F in each interpretation, certain quotients of the trees are the appropriate interpretations.) The collection of labeled schemes and trees is equipped with the same structure, which is preserved by the map which takes a scheme F to its strong behavior [F]. This structure is simple to describe: starting from the atomic and “base” schemes, using just three operations (composition, tupling, and iteration), any scheme may be constructed. The operation of iteration (or “looping”) on schemes and trees was not defined in all cases in the setup of [E-MC]; for example, consider the scheme 1 -+ 1 consisting of only an exit (and the tree which forms its strong behavior). Thus the schemes and trees form a partial algebra. It was to remedy this situation on the semantic algebras of trees and other iterative theories that the notion of an “iteration” theory was introduced (in [BEW]). Iteration theories are (total) algebras with the same operations as iterative theories. The collection of trees which are strong behaviors of schemes may be given the structure of an iteration theory by completing the iteration operation in an almost arbitrary way. The collection of iteration theories is the class of all mode ls of the set of equations valid in all iteration theories of trees. (See [BEW] for details.)

Inductive *-semirings

One of the most well-known induction principles in computer science is the fixed point induction rule, or least pre-fixed point rule. Inductive *-semirings are partially ordered semirings equipped with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms. Inductive *-semirings are extensions of continuous semirings and the Kleene algebras of Conway and Kozen.We develop, in a systematic way, the rudiments of the theory of inductive *-semirings in relation to automata, languages and power series. In particular, we prove that if S is an inductive *-semiring, then so is the semiring of matrices Sn×n, for any integer n ≥ 0, and that if S is an inductive *-semiring, then so is any semiring of power series S«A*». As shown by Kozen, the dual of an inductive *-semiring may not be inductive. In contrast, we show that the dual of an iteration semiring is an iteration semiring. Kuich proved a general Kleene theorem for continuous semirings, and Bloom and Esik proved a Kleene theorem for all Conway semirings. Since any inductive *-semiring is a Conway semiring and an iteration semiring, as we show, there results a Kleene theorem applicable to all inductive *-semirings. We also describe the structure of the initial inductive *-semiring and conjecture that any free inductive *-semiring may be given as a semiring of rational power series with coefficients in the initial inductive *-semiring. We relate this conjecture to recent axiomatization results on the equational theory of the regular sets.

Matrix and matricial iteration theories, Part II

This paper extends Part 1 of the paper with the same title. Here, matricial iteration theories Matr(S; V) are characterized by identities involving theory operations, a star operation S → S and an omega operation S → V. The initial matricial iteration theory is described explicitly. One answer is given to the following question: If T0 is a submatricial theory of the matricial theory T which is an iteration theory, when can the star and omega operations on T0 be extended to T so that T becomes an iteration theory? Applications to program correctness logic and to finding equational axioms for the regular sets are indicated.