Kevin Compton, Jean-Eric Pin ,Wolfgang Thomas (editors):

Automata Theory: Infinite Computations

Dagstuhl-Seminar-Report; 286.-10.1.92 (9202)

Kevin Compton, Jean-Eric Pin , Wolfgang Thomas (editors) :

Automata Theory: Infinite Computations

Dagstuhl-Seminar-Report; 28 6.-10.1.92 (9202)

ISSN 0940-1121

Copyright © 1992 by IBF I GmbH, Schloß Dagstuhl, W-6648 Wadern, GermanyTeI.: +49-6871 � 2458Fax: +49-6871 - 5942

Das Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI) ist eine gemein-nützige GmbH. Sie veranstaltet regelmäßig wissenschaftliche Seminare, welche nach Antragder Tagungsleiter und Begutachtung durch das wissenschaftliche Direktorium mit persönlicheingeladenen Gästen durchgeführt werden.

Verantwortlich für das Programm:Prof. Dr.-lng. Jose Encarnagao,Prof. Dr. Winfried Görke,Prof. Dr. Theo Härder,Dr. Michael Laska,Prof. Dr. Thomas Lengauer,Prof. Ph. D. Walter Tichy,Prof. Dr. Reinhard Wilhelm (wissenschaftlicher Direktor).

Gesellschafterz Universität des Saarlandes,Universität Kaiserslautern,Universität Karlsruhe,Gesellschaft für Informatik e.V.� Bonn

Träger: Die Bundesländer Saarland und Rheinland Pfalz.

Bezugsadresse: Geschäftsstelle Schloß DagstuhlInformatik, Bau 36Universität des SaarlandesW � 6600 Saarbrücken

GermanyTeI.: +49 -681 - 302 - 4396

Fax: +49 -681 - 302 - 4397

e-mail: office@dag.uni-sb.de

ISSN 0940-1121

Copyright © 1992 by IBFI GmbH, SchloB Dagstuhl, W-6648 Wadern, Germany Tel.: +49-6871 - 2458 Fax: +49-6871 - 5942

Das lntemationales Begegnungs- und Forschungszentrum fur lnformatik (IBFI) ist eine gemein­nutzige GmbH. Sie veranstaltet regelmaBig wissenschaftliche Seminare, welche nach Antrag der Tagungsleiter und Begutachtung durch das wissenschaftliche Direktorium mit personlich eingeladenen Gasten durchgefuhrt werden.

Verantwortlich fur das Programm: Prof. Dr.-lng. Jose Encarna9ao, Prof. Dr. Winfried Gorke, Prof. Dr. Theo Harder, Dr. Michael Laska, Prof. Dr. Thomas Lengauer, Prof. Ph. D. Walter Tichy, Prof. Dr. Reinhard Wilhelm (wissenschaftlicher Direktor).

Gesellschafter: Universitat des Saarlandes, Universitat Kaiserslautern, Universitat Karlsruhe, Gesellschaft fur lnformatik e.V., Bonn

Trager: Die Bundeslander Saarland und Rheinland Pfalz.

Bezugsadresse: Geschaftsstelle SchloB Dagstuhl lnformatik, Bau 36 Universitat des Saarlandes W - 6600 Saarbrucken Germany Tel.: +49 -681 - 302 - 4396 Fax: +49 -681 - 302 - 4397 e-mail: office@dag.uni-sb.de

Report on the Dagstuhl-Seminar

"Automata Theory: Infinite Computations"

(6.1.-10.1.1992)

Organizers: K. Compton (Ann Arbor), J .E. Pin (Paris), W. Thomas (Kiel)

The subject of the seminar, "in�nite computations in automata theory", has developedinto a very active �eld with a wide range of applications, including speci�cation of�nite state programs, ef�cient algorithms for program veri�cation, decidability oflogical theories, automata theoretic approaches to analysis and topology, and fractalgeometry.

The response to the invitations was very positive: 37 scientists from 12 countriesparticipated, which is about the maximum to be housed at Schloss Dagstuhl. Theseminar started or refreshed contacts especially between Eastem and Westem researchgroups; it was successful in joining the efforts to solve some of the fundamentalproblems in the theory, and helped to support promising new developments.

The program of talks is presented here in five sections:

Automata and in�nite sequencesAutomata on in�nite traces

Tree languages, tree automata, and in�nite gamesLogical aspectsCombinatorial aspects

In the first section, contributions on �nite automata accepting or generating in�nitesequences are collected. This includes new results on classical problems, like thecomplementation problem for (0-automata (N. Klarlund), the introduction ofappropriate syntactic congruences (B. Le Saec), and the extension of sernigroups byin�nite products (R.R. Redziejowski). The subject of multiplicities (measures for therange of different computations on given inputs) was addressed in the talks of D.Perrin as well as J. Karhumäki (who applied it to introduce a new way of computingreal functions by �nite automata). Other contributions were concemed with the classof locally testable (o�languages (Th. Wilke), with relations de�ned by two tape (o-automata (Ch. Frougny), and with a uni�ed framework for specifying and verifyingprograms using recursive automata on to-sequences (L. Staiger).

An interesting application of in�nite computations in automata theory is the analysisof concurrent (and usually nontemiinating) programs. A well developed theory for

Report on the Dagstuhl-Seminar

"Automata Theory: Infinite Computations"

(6.1.-10.1.1992)

Organizers: K. Compton (Ann Arbor), J.E. Pin (Paris), W. Thomas (Kiel)

The subject of the seminar, "infinite computations in automata theory", has developed into a very active field with a wide range of applications, including specification of finite state programs, efficient algorithms for program verification, decidability of logical theories, automata theoretic approaches to analysis and topology, and fractal geometry.

The response to the invitations was very positive: 37 scientists from 12 countries participated, which is about the maximum to be housed at Schloss Dagstuhl. The seminar started or refreshed contacts especially between Eastern and Western research groups; it was successful in joining the efforts to solve some of the fundamental problems in the theory, and helped to support promising new developments.

The program of talks is presented here in five sections:

Automata and infinite sequences Automata on infinite traces Tree languages, tree automata, and infinite games Logical aspects Combinatorial aspects

In the first section, contributions on finite automata accepting or generating infinite sequences are collected. This includes new results on classical problems, like the complementation problem for co-automata (N. Klarlund), the introduction of appropriate syntactic congruences (B. Le Saec), and the extension of semigroups by infinite products (R.R. Redziejowski ). The subject of multiplicities (measures for the range of different computations on given inputs) was addressed in the talks of D. Perrin as well as J. Karhumaki (who applied it to introduce a new way of computing real functions by finite automata). Other contributions were concerned with the class of locally testable co-languages (Th. Wilke), with relations defined by two tape co­automata (Ch. Frougny), and with a unified framework for specifying and verifying programs using recursive automata on co-sequences (L. Staiger).

An interesting application of infinite computations in automata theory is the analysis of concurrent (and usually nonterminating) programs. A well developed theory for

2

modelling concurrency is that of "Mazurkiewicz traces", in which certain partial orders(instead of words) are used as representations of computations. The recentdevelopment of a theory combining both features ("automata on in�nite traces") waspresented in three contributions (by A. Petit, P. Gastin, and V. Diekert).

A central subject of the seminar was the famous result of M.O. Rabin oncomplementation for �nite automata on in�nite trees. The special role of Rabin�stheorem rests on several facts. First, it makes possible an automata theoretic treatmentof logical systems (including proofs of decidability), in particular for monadic second-order theories and logics of programs. Secondly, its proof is tightly connected with theexistence of winning strategies in "in�nite games", a topic which recently attractsmuch attention in the study of nontenninating reactive systems. A third aspect is theintriguing dif�culty of the proof of Rabin�s theorem, which has motivated severalapproaches for simpli�cation. New results in this subject were presented in fourlectures during the seminar: by E.A. Emerson, A.W. Mostowski, P.E. Schupp, and S.Zeitman. In an additional evening session Paul Schupp gave a detailed exposition ofhis proof of Rabin�s'theorem. Two further contributions treated altemative and re�nedversions of Rabin tree automaton de�nability, namely de�nability of tree properties in�xed point calculi (D. Niwinski) and by restricted acceptance conditions for treeautomata (J. Skurczynski). '

Since about thirty years, an important application of �nite automata has been their usein the study of monadic second-order theories. These and related logical aspects weretreated in talks collected in the fourth section below. A.L. Semenov and An. A.

Muchnik gave a survey of Russian work on this subject, conceming extensions ofBiichi�s successor arithmetic and a strengthening of Rabin�s decidability result for themonadic theory of the in�nite binary tree. B. Courcelle and G. Sénizergues showedresults on monadic second-order logic over graphs (in particular, on de�nable graph

' transformations, and on graphs determined by automatic groups). Logical de�nabilityin connection with limit laws in model theory and with problems of circuit complexitywere presented by K. Compton and H. Straubing.

The final section summarizes lectures which are devoted to more combinatorial.

questions (in this case not necessarily applied to in�nite objects)". The subjects herewere decidability and complexity results on tiling pictures with given dominoes (D.Beauquier), the invertibility of automaton de�nable functions (C. Choffrut), and newschemes for language generation motivated by mechanisms of DNA splicing (T.Head).

The talks were supplemented by lively discussions, joint work in small groups, andthree long night sessions: On Tuesday evening, A. Muchnik presented his extensionof the Shelah-Stupp Theorem (showing that the tree unravelling of a structure has adecidable monadic theory if the given structure has). On Wednesday, H. Straubingexplained his reduction of general �rst-order de�nitions of regular word sets to first-

2

modelling concurrency is that of "Mazurkiewicz traces", in which certain partial orders (instead of words) are used as representations of computations. The recent development of a theory combining both features ("automata on infinite traces") was presented in three contributions (by A. Petit, P. Gastin, and V. Diekert).

A central subject of the seminar was the famous result of M.0. Rabin on complementation for finite automata on infinite trees. The special role of Rabin's theorem rests on several facts. First, it makes possible an automata theoretic treatment of logical systems (including proofs of decidability), in particular for monadic second­order theories and logics of programs. Secondly, its proof is tightly connected with the existence of winning strategies in "infinite games", a topic which recently attracts much attention in the study of nontenninating reactive systems. A third aspect is the intriguing difficulty of the proof of Rabin's theorem, which has motivated several approaches for simplification. New results in this subject were presented in four lectures during the seminar: by E.A. Emerson, A.W. Mostowski, P.E. Schupp, and S. Zeitman. In an additional evening session Paul Schupp gave a detailed exposition of his proof of Rabin' s · theorem. Two further contributions treated alternative and refined versions of Rabin tree automaton definability, namely definability of tree properties in fixed point calculi (D. Niwinski) and by restricted acceptance conditions for tree automata (J. Skurczynski).

Since about thirty years, an important application of finite automata has been their use in the study of monadic second-order theories. These and related logical aspects were treated in talks collected in the fourth section below. A.L. Semenov and An. A. Muchnik gave a survey of Russian work on this subject, concerning extensions of Biichi's successor arithmetic and a strengthening of Rabin's decidability result for the monadic theory of the infinite binary tree. B. Courcelle and G. Senizergues showed results on monadic second-order logic over graphs (in particular, on definable graph transformations, and on graphs detennined by automatic groups). Logical definability in connection with limit laws in model theory and with problems of circuit complexity were presented by K. Compton and H. Straubing.

The final section summarizes lectures which are devoted to more combinatorial questions (in this case not necessarily applied to infinite objects). The subjects here were decidability and complexity results on tiling pictures with given dominoes (D. Beauquier), the invertibility of automaton definable functions (C. Choffrut), and new schemes for language generation motivated by mechanisms of DNA splicing (T. Head).

The talks were supplemented by lively discussions, joint work in small groups, and three long night sessions: On Tuesday evening, A. Muchnik presented his extension of the Shelah-Stupp Theorem (showing that the tree unravelling of a structure has a decidable monadic theory if the given structure has). On Wednesday, H. Straubing explained his reduction of general first-order definitions of regular word sets to first-

3

order de�nitions with modulo counting predicates only. P. Schupp presented onThursday night his proof on simulation of alternating automata by nondeterrninistictree automata, thus giving a simpli�ed proof of Rabin�s theorem.

Summing up, the meeting made possible a very productive exchange of ideas andresults; it will stimulate the research of those who participated and already has led tothe solution of open problems. The friendly atmosphere in the house, the mostef�cient work of the secretary team and last not least the perfect kitchen wereappreciated very much. In the name of the participants, the organizers would like tothank the staff for their efforts in making the stay at Schloss Dagstuhl so pleasant andfruitful.

K. ComptonJ �E. Pin

W. Thomas

3

order definitions with modulo counting predicates only. P. Schupp presented on Thursday night his proof on simulation of alternating automata by nondeterministic tree automata, thus giving a simplified proof of Rabin 's theorem.

Summing up, the meeting made possible a very productive exchange of ideas and results; it will stimulate the research of those who participated and already has led to the solution of open problems. The friendly atmosphere in the house, the most efficient work of the secretary team and last not least the perfect kitchen were appreciated very much. In the name of the participants, the organizers would like to thank the staff for their efforts in making the stay at Schloss Dagstuhl so pleasant and fruitful.

K. Compton J.E. Pin W. Thomas

4

Abstracts of Talks

Page

I. Automata and Infinite Sequences 6

Ch. Frougny (Paris): Rational 0)-Relations, Application to the Representationof Real Numbers

J. Karhuméiki (Turku): Finite Automata Computing Real FunctionsN. Klarlund (Aarhus): Progress Measures for Complementation of to-AutomataB. Le Saec (Bordeaux): A Syntactic Approach to Deterministic to-AutomataD. Perrin (Paris): to-Automata with MultiplicitiesR.R. Redziejowski (Lidingö): Adding In�nite Product to a SemigroupL. Staiger (Siegen): Recursive Automata on In�nite Words and the Veri�cation

of Concurrent ProgramsTh. Wilke (Kiel): Locally Threshold Testable to-languages and FGHG5-Sets

II. Automata on In�nite Traces 10

A. Petit (Paris): Automata for In�nite TracesP. Gastin (Paris): Biichi Asynchronous Cellular AutomataV. Diekert (Stuttgart): Somc Open Problems on Deterministic Trace Automata

III. Tree Languages, Tree Automata, and In�nite Games 12

E.A. Emerson (Austin): Complexity of Logics of Programs and Automataon In�nite Objects

A.W. Mostowski (Gdansk): Games with Forbidden PositionsD. Niwinski (Warsaw): Problems in p-CalculusP.E. Schupp (Urbana): Simulating Altemating Automata by Nondeterministic

Automata

J. Skurczynski (Gdansk): Automata on In�nite Trees with Weak AcceptanceConditions

S. Zeitrnan (Detroit): Unforgettable Forgetful Detenninacy

IV. Logical Aspects ' 15

K. Compton (Ann Arbor and Roquencourt): A Monadic Second-Order Limit Lawfor Unary Function

4

Abstracts of Talks

I. Automata and Infinite Sequences

Ch. Frougny (Paris): Rational 0rRelations, Application to the Representation of Real Numbers

J. Karhumili (Turku): Finite Automata Computing Real Functions N. Klarlund (Aarhus): Progress Measures for Complementation of 0rAutomata B. Le Saec (Bordeaux): A Syntactic Approach to Deterministic 0rAutomata D. Perrin (Paris): 0rAutomata with Multiplicities R.R. Redziejowski (Lidingo): Adding Infinite Product to a Semigroup L. Staiger (Siegen): Recursive Automata on Infinite Words and the Verification

of Concurrent Programs Th. Wilke (Kiel): Locally Threshold Testable Or languages and F cr1G6-Sets

II. Automata on Infinite Traces

A. Petit (Paris): Automata for Infinite Traces P. Gastin (Paris): Biichi Asynchronous Cellular Automata V. Diekert (Stuttgart): Some Open Problems on Deterministic Trace Automata

III. Tree Languages, Tree Automata, and Infinite Games

E.A. Emerson (Austin): Complexity of Logics of Programs and Automata

on Infinite Objects A.W. Mostowski (Gdansk): Games with Forbidden Positions D. Niwinski (Warsaw): Problems in µ-Calculus P.E. Schupp (Urbana): Simulating Alternating Automata by Nondeterministic

Automata J. Skurczynski (Gdansk): Automata on Infinite Trees with Weak Acceptance

Conditions S. 2.eitman (Detroit): Unforgettable Forgetful Determinacy

IV. Logical Aspects

page 6

10

12

15

K. Compton (Ann Arbor and Roquencourt): A Monadic Second-Order Limit Law for Unary Function

P383

B. Courcelle (Bordeaux): Monadic Second-Order Definability Propertiesof In�nite Graphs

G. Sénizergues (Bordeaux): De�nability in Weak Second-Order Logicof Some In�nite Graphs

A.L. Semenov and An. A. Muchnik (Moscow): Automata on In�nite Objects,Monadic Theories, and Complexity

H. Straubing (Boston): Circuit Complexity, Finite Automata and GeneralizedFirst-Order Logic

V. Combinatorial Aspects 18

D. Beauquier (Paris): Games with DominoesC. Choffrut (Paris): Some Questions on Sequential Bijections Between

Free Monoids

T. Head (Binghamton): Splicing Schemes and DNA

5

B. Courcelle (Bordeaux): Monadic Second-Order Defmability Properties of Infinite Graphs

G. Senizergues (Bordeaux): Definability in Weak Second-Order Logic of Some Infinite Graphs

A.L. Semenov and An. A. Muchnik (Moscow): Automata on Infmite Objects, Monadic Theories, and Complexity

H. Straubing (Boston): Circuit Complexity, Finite Automata and Generalized First-Order Logic

V. Combinatorial Aspects

D. Beauquier (Paris): Games with Dominoes C. Choffrut (Paris): Some Questions on Sequential Bijections Between

Free Monoids T. Head (Binghamton): Splicing Schemes and DNA

page

18

I. AUTOMATA AND INFINITE SEQUENCES

Rational 0)-Relations. Application to the Representation of Real numbers

Ch. Frougny (Paris)

[joint work with J. Sakarovitch, Paris]

We consider rational (0-relations, i.e. relations of in�nite words which are recognizedby �nite 2-automata (with the Biichi acceptance condition). We show that a rationalco-relation recognized by a �nite 2-automaton with a bounded delay can besynchronized, that is, can be recognized by a letter-to-letter 2-automaton. We provethat a deterministic rational 0)-relation is a countable intersection of open sets. Thegeography of the set of rational 0)-relations is described as follows:

0)-DSynch2 C (o�DRat2n �

co-CwRat2 c o)�Synch2 c: co-Rat2

The inclusions 0)-DRat2 c: o)�Rat2 and (0-Synchz c: a)�Rat2 are undecidable; butco-DSynch2 c (0-Synchz is decidable. We have to-DSynch2 = a)�Synch2 n 0)-DRat2. Asan application, we consider the representation of real numbers in base B, where B isa real number >1. The normalization is the function which maps any B-representationof a number x onto the greedy one, called the B-expansion. We show that, if thenormalization is a rational function, then it is synchronized. This allows to prove thatthe normalization in base B is rational if and only if the set of in�nite words havingvalue O in base ß is recognizable by a �nite automaton.

Finite Automata Computing Real Functions

J. Karhumiiki (Turku)

[joint work with K. Culik II, Columbia, S.C., andD. Derencourt, M. Latteux, A. Terlutte, Lille]

We introduce a new application of �nite automata to compute real functions [0,1] -�> R.This research is motivated by computer graphics, a need to generate pictures, and isclosely related to the theory of rational fonnal power series. We use a

6

I. AUTOMATA AND INFINITE SEQUENCES

Rational co-Relations. Application to the Representation of Real numbers

Ch. Frougny (Paris)

[joint work with J. Sakarovitch, Paris]

We consider rational co-relations, i.e. relations of infinite words which are recognized

by finite 2-automata (with the Biichi acceptance condition). We show that a rational co-relation recognized by a finite 2-automaton with a bounded delay can be synchronized, that is, can be recognized by a letter-to-letter 2-automaton. We prove that a deterministic rational co-relation is a countable intersection of open sets. The geography of the set of rational co-relations is described as follows:

co-DSynch2 c co-DRati

" () co-CwRati c co-Synch2 c co-Ra¼

The inclusions co-DRati c co-Rati and co-Synch2 c co-Rati are undecidable; but

co-DSynch2 c co-Synch2 is decidable. We have co-DSynch2 = co-Synch2 f"'I co-DRa¼. As an application, we consider the representation of real numbers in base p, where p is a real number > 1. The normalization is the function which maps any P-representation of a number x onto the greedy one, called the P-expansion. We show that, if the normalization is a rational function, then it is synchronized. This allows to prove that the normalization in base P is rational if and only if the set of infinite words having value O in base p is recognizable by a finite automaton.

Finite Automata Computing Real Functions

J. Karhumaki (Turko)

[joint work with K. Culik II, Columbia, S.C., and D. Derencourt, M. Latteux, A. Terlutte, Lille]

We introduce a new application of finite automata to compute real functions [0,1] ➔ R. This research is motivated by computer graphics, a need to generate pictures, and is closely related to the theory of rational formal power series. We use a

7

nondeterrninistic automaton with multiplicities, R-2�.-automaton in terms of Eilenberg,to compute such a function as follows. A real x in [0,1] is identi�ed with an (n-wordin {O,1}�°, with its binary representation bin(x), and then the value of the function onx is the multiplicity of bin(x) determined by the considered automaton. In order toguarantee that such functions are always de�ned we restrict our considerations to so-called level automata which among other things do not allow any other loops thanloops from a state into itself.

We prove among other things that- such automata can compute all polynomials (in one or several unknowns), but

cannot compute any other smooth functions, i.e. functions having all thederivatives;

- to compute a polynomial of degree n at least n+1 states are needed;- there exists a four state automaton which computes a continuous function

having no derivative at any point;- given a level automaton there exists another one which computes the integral

of the function determined by the first automaton;- it is decidable whether an automaton de�nes a continuous or smooth function,

as well as whether two such automata define the same function.

Progress Measures for Complementation of co-Automata

N. Klarlund (Aarhus)

We discuss a new approach to the complementation problem for �nite�state automataon in�nite words. Instead of using usual combinatorial or algebraic properties oftransition relations, we show that a graph-theoretic approach based on progressmeasures is a potent tool for complementing (0-automata.

We apply progress measures to the classical problem of complementing Biichiautomata and obtain a simple, optimal method. Our technique also applies to Streettautomata for which we also obtain an optimal method.

7

nondetenninistic automaton with multiplicities, R-:E-automaton in terms of Eilenberg, to compute such a function as follows. A real x in [0,1] is identified with an co-word in {0,1 }co, with its binary representation bin(x), and then the value of the function on x is the multiplicity of bin(x) determined by the considered automaton. In order to guarantee that such functions are always defined we restrict our considerations to so­called level automata which among other things do not allow any other loops than loops from a state into itself.

We prove among other things that such automata can compute all polynomials (in one or several unknowns), but cannot compute any other smooth functions, i.e. functions having all the derivatives; to compute a polynomial of degree n at least n+l states are needed; there exists a four state automaton which computes a conti_nuous function having no derivative at any point; given a level automaton there exists another one which computes the integral of the function determined by the first automaton; it is decidable whether an automaton defines a continuous or smooth function, as well as whether two such automata define the same function.

Progress Measures for Complementation of co-Automata

N. Klarlund (Aarhus)

We discuss a new approach to the complementation problem for finite-state automata on infinite words. Instead of using usual combinatorial or algebraic properties of transition relations, we show that a graph-theoretic approach based on progress measures is a potent tool for complementing co-automata.

We apply progress measures to the classical problem of complementing Biichi automata and obtain a simple, optimal method. Our technique also applies to Streett automata for which we also obtain an optimal method.

A Syntactic Approach to Deterministic to-automata

B. Le Saec (Bordeaux)

[joint work with Do Long Van, I. Litovslci]

We propose uni�ed characterizations, by means of right congruence, to various classesof rational to-language. A family of o)-languages which admit a unique minimal tabletransition o)�automaton is introduced: The pre�x�recognizable to-languages. Thesyntactic congruence of those (o�languages is equal to the cycle congruence of theirminimal automaton. This cycle congruence is a new one we proposed for the on-automata.

Multiplicities in to-Automata

D. Perrin (Paris)

We propose to consider a Biichi automaton as an automaton with multiplicities in thearctic semiring it = <N U -oo, max, + > in such a way that the multiplicity of a wordw in A* is equal to the maximum of the number of �nal states on a path labeled wstarting where appropriate.

This gives first, by restriction to the Leduggdi semiring M! = <{-oo,O,l }, max, +>a matrix representation of the quotient of A" under the Biichi congruence. This maybe more interesting in some cases than the usual embedding into the Schiitzenbergerproduct of two copies of the ordinary transition monoid of the automaton.

Second, it is possible to associate with the automaton a new semigroup de�ned asfollows: we first compute the (in�nite) monoid M1 of matrices in the arctic semiring.We take the closure M2 of this monoid in the to-arctic semiring obtained by adding apoint at in�nity (O to N as a limit of increasing sequences. We fmally take the imageM3 of M3 in the reduced to-arctic semiring <{-oo,O,1,o)}, max,+>. We propose tocompute M3 using a method identical to the algorithm of H. Leung (see 1. Simon,MFCS 88). There remains to �nd a good interpretation of the coef�cients of theelements of M3.

8

A Syntactic Approach to Deterministic (!).automata

B. Le Saec (Bordeaux)

Uoint work with Do Long Van, I. Litovski]

We propose unified characterizations, by means of right congruence, to various classes of rational co-language. A family of co-languages which admit a unique minimal table transition co-automaton is introduced: The prefix-recognizable co-languages. The syntactic congruence of those co-languages is equal to the cycle congruence of their minimal automaton. This cycle congruence is a new one we proposed for the co­automata.

Multiplicities in ro-Automata

D. Perrin (Paris)

We propose to consider a Biichi automaton as an automaton with multiplicities in the arctic semiring ;t =<N u -00, max, + > in such a way that the multiplicity of a word w in A+ is equal to the maximum of the number of final states on a path labeled w starting where appropriate.

This gives firs4 by restriction to the reduced arctic semiring 1(..!( = < { -oo,O, 1 } , max, +> a matrix representation of the quotient of A+ under the Biichi congruence. This may be more interesting in some cases than the usual embedding into the Schiitzenberger product of two copies of the ordinary transition monoid of the automaton.

Second, it is possible to associate with the automaton a new semigroup defined as follows: we first compute the (infinite) monoid M1 of matrices in the arctic semiring. We take the closure M2 of this monoid in the ro-arctic semiring obtained by adding a point at infinity ro to N as a limit of increasing sequences. We finally take the image M3 of M2 in the reduced (J)-arctic semiring <{-00,0,1,ro}, max,+>. We propose to compute M3 using a method identical to the algorithm of H. Leung (see I. Simon, MFCS 88). There remains to find a good interpretation of the coefficients of the elements of M3•

Adding an Infinite Product to a Semigroup

R.R. Redziejowski (Lidingö)

A sernigroup (S,-) is extended with an o)�argument operation II: SN -9 Q where Q is acertain set, possibly disjoint with S. The operation l&#39;I is required to satisfy threeaxioms, expressing generalized associativity compatible with the semigroup operation.

It follows from the axioms and the Ramsey lemma that for a �nite S, II can have onlya �nite number of distinct values, all reducible to the form se�� where s,ee S and e2=e.

Homomorphism and a unique free algebra are de�ned for the class of semigroupsextended with the operation 1&#39;1. The purpose of the construction is to expressrecognizability of to-languages in a way analogous to that of �nite-word languages(directly in terms of a homomorphism into a �nite algebraic structure).

Recursive Automata on In�nite Words and

the Verification of Concurrent Programs

L. Staiger (Siegen)

In 1987 Vardi presented a framework based on recursive automata that uni�es a largenumber of trends in the area of speci�cation and veri�cation of concurrent programs.

Here the program and speci�cation are considered as recursive automata P and S,respectively, which accept the sets of possible execution systems. In this talk weshowed how several restrictions on the behaviour of the automata P and S, as theirbranching behaviour or the underlying alphabet, in�uence the complexity of theveri�cation problem, which was shown to be II;-complete for arbitrarynondeterministic speci�cations S.

As a general method we used the description of the accepted languages of in�nitewords via the arithmetical hierarchy of languages of �nite and in�nite words.

9

Adding an Infinite Product to a Semigroup

R.R. Redziejowski (Lidingo)

A semigroup (S,) is extended with an ro-argument operation TT: sN ➔ Q where Q is a certain set, possibly disjoint with S. The operation n is required to satisfy three axioms, expressing generalized associativity compatible with the semigroup operation.

It follows from the axioms and the Ramsey lemma that for a finite S, n can have only a finite number of distinct values, all reducible to the fonn seco where s,ee S and e2=e.

Homomorphism and a unique free algebra are defined for the class of semigroups extended with the operation n. The purpose of the construction is to express recognizability of ro-languages in a way analogous to that of finite-word languages (directly in tenns of a homomorphism into a finite algebraic structure).

Recursive Automata on Infinite Words and the Verification of Concurrent Programs

L. Staiger (Siegen)

In 1987 Vardi presented a framework based on recursive automata that unifies a large number of trends in the area of specification and verification of concurrent programs.

Here the program and specification are considered as recursive automata P and S, respectively, which accept the sets of possible execution systems. In this talk we showed how several restrictions on the behaviour of the automata P and S, as their branching behaviour or the underlying alphabet, influence the complexity of the verification problem, which was shown to be n;-complete for arbitrary nondetenninistic specifications S.

As a general method we used the description of the accepted languages of infinite words via the arithmetical hierarchy of languages of finite and infinite words.

l0

Locally Threshold Testable to-Languages and FGHG5 - Sets

Th. Wilke (Kiel)

It is proved that a o)-language is weakly locally threshold testable with bound L for thelength of counted factors if and only if it is locally threshold testable with the sameparameter and belongs to the Borel class Fa n G5. As a consequence we obtain thatit is decidable whether a formula in sequential calculus is equivalent to a first orderformula with successor.

II. AUTOMATA ON INFINITE TRACES

Automata for Infinite Traces

A. Petit (Paris)

[joint work with P. Gastin]

Trace languages provide a very natural semantics for concurrent processes. Inparticular, the behaviours of �nite state systems are described by recognizable tracelanguages, which form hence a basic family. For in�nite traces, this family has beenintroduced by means of recognizing morphisms and characterized by co-rationalexpressions. Here, we study �nite automata for recognizable trace languages. Diamondautomata (i.e. automata on in�nite traces) with classical acceptance conditions forin�nite word paths do not necessarily accept closed languages. Therefore, a majorissue addressed here is the de�nition of the notion of trace path and of the good classof (in�nite) trace automata. By construction, such an automaton accepts always arecognizable trace language. Conversely, our second aim is to prove that Biichi traceautomata characterize the family of recognizable languages of in�nite traces.

10

Locally Threshold ~estable co-Languages and F c11G6 - Sets

Th. Wilke (Kiel)

It is proved that a co-language is weakly locally threshold testable with bound L for the length of counted factors if and only if it is locally threshold testable with the same parameter and belongs to the Borel class Fan G6. As a consequence we obtain that it is decidable whether a formula in sequential calculus is equivalent to a first order formula with successor.

II. AUTOMATA ON INFINITE TRACES

Automata for Intlnite Traces

A. Petit (Paris)

[joint work with P. Gaslin]

Trace languages provide a very natural semantics for concurrent processes. In particular, the behaviours of finite state systems are described by recognizable trace languages, which form hence a basic family. For infinite traces, this family has been introduced by means of recognizing morphisms and characterized by co-rational expressions. Here, we study finite automata for recognizable trace languages. Diamond automata (i.e. automata on infinite traces) with classical acceptance conditions for infinite word paths do not necessarily accept closed languages. Therefore, a major issue addressed here is the definition of the notion of trace path and of the good class of (infinite) trace automata. By construction, such an automaton accepts always a recognizable trace language. Conversely, our second aim is to prove that Btichi trace automata characterize the family of recognizable languages of infinite traces.

11

Biichi Asynchronous Cellular Automata

P. Gastin (Paris)

[joint work with A. Petit]

Asynchronous automata, introduced by W. Zielonka, form a very important subclassof automata for �nite traces. In this talk we extend these automata to in�nite traces

using a suitable Biichi-like acceptance condition. Next we prove that the class of

language accepted by some Büchi asynchronous cellular automaton is exactly thefamily of recognizable languages of real traces. To this purpose, we give effectiveconstructions for the union, the intersection, the concatenation and the in�nite iteration

of asynchronous automata.

Some Open Problems on Deterministic Automata for Infinite Traces

V. Diekert (Stuttgart)

[joint work with P. Gastin and A. Petit]

The talk addresses to the problem whether every recognizable language of in�nitetraces can be accepted by some deterministic automaton. In fact, the hope is that therewill be an asynchronous automaton which is deterministic and has some local Muller-acceptance condition.

However, even the existence of a deterministic I-diamond automaton is open. Thecondition of being I-diamond is much weaker and demands only that q-ab = q-ba forall states q and independent letters a,beX ,(a,b)e I. As a candidate of trace languageswhich are accepted by I-diarnond�deterministic Biichi-automata we introduce tracelimits. As a first result we show that these languages are recognizable.

11

Biichi Asynchronous Cellular Automata

P. Gastin (Paris)

[joint work with A. Petit]

Asynchronous automata, introduced by W. Zielonka, form a very important subclass of automata for finite traces. In this talk we extend these automata to infinite traces

using a suitable Btichi-like acceptance condition. Next we prove that the class of language accepted by some Btichi asynchronous cellular automaton is exactly the family of recognizable languages of real traces. To this purpose, we give effective constructions for the union, the intersection, the concatenation and the infinite iteration of asynchronous automata.

Some Open Problems on Deterministic Automata for Infinite Traces

V. Dickert (Stuttgart)

[joint work with P. Gastin and A. Petit]

The talk addresses to the problem whether every recognizable language of infinite traces can be accepted by some deterministic automaton. In fact, the hope is that there will be an asynchronous automaton which is deterministic and has some local Muller­acceptance condition.

However, even the existence of a deterministic I-diamond automaton is open. The condition of being I-diamond is much weaker and demands only that q•ab = q•ba for all states q and independent letters a,be X ,( a,b )e I. As a candidate of trace languages which are accepted by I-diamond-deterministic Biichi-automata we introduce trace limits. As a first result we show that these languages are recognizable.

12

III. TREE LANGUAGES, TREE AUTOMATA, AND INFINITE GAMES

Complexity of Tree Automata and Logics of Programs

E.A. Emerson (Austin)

[joint work with C.S. Jutla, IBM Watson]

There is an intimate relationship between �nite state automata on in�nite trees and

temporal and modal logics of programs. The standard paradigm nowadays for testingsatis�ability of branching time temporal logics is reduction to the nonemptinessproblem for tree automata. We improve the complexity of testing nonemptiness of treeautomata and of testing satis�ability for a number of signi�cant temporal logics ofprograms.

We show that we can test nonemptiness of pairs (and complemented pairs) treeautomata in time polynomial in the number of states and exponential in the number ofpairs, while we also show the problem is NP-complete (coNP-complete, respectively)in general. This multiparameter analysis enables us to obtain exponentially improved,essentially tight bounds for a number of temporal logics of programs: CTL* isdeterministic double exponential time complete, while PDL-A and the Mu-Calculus aredeterministic exponential time complete.

We also show that the (full) Mu-calculus is expressively equivalent to pairs treeautomata. Since the Mu-Calculus is syntactically and semantically closed undercomplementation, we obtain the complementation lemma for tree automata as an easy

corollary.

We also believe that our techniques and results demonstrate the power of temporallogic and the Mu-Calculus for specifying and reasoning about automata and games.

Games with Forbidden Positions

A.W. Mostowski (Gdansk)

Games of in�nite trees using the table of an alternating automaton [Muller, Schupp,TCS 54 (l987)] are investigated. The �rst player chooses a transition, then the secondplayer chooses a direction (or, simultaneously with the �rst player, a transition from

12

m. TREE LANGUAGES, TREE AUTOMATA, AND INFINITE GAMES

Complexity of Tree Automata and Logics of Programs

E.A. Emerson (Austin)

Uoint work with C.S. Jutla, IBM Watson]

There is an intimate relationship between finite state automata on infinite trees and temporal and modal logics of programs. The standard paradigm nowadays for testing satisfiability of branching time temporal logics is reduction to the nonemptiness problem for tree automata. We improve the complexity of testing nonemptiness of tree automata and of testing satisfiability for a number of significant temporal logics of programs.

We show that we can test nonemptiness of pairs (and complemented pairs) tree automata in time polynomial in the number of states and exponential in the number of pairs, while we also show the problem is NP-complete (coNP-complete, respectively) in general. This multiparameter analysis enables us to obtain exponentially improved, essentially tight bounds for a number of temporal logics of programs: CIL * is deterministic double exponential time complete, while POL-A and the Mu-Calculus are deterministic exponential time complete.

We also show that the (full) Mu-calculus is expressively equivalent to pairs tree automata. Since the Mu-Calculus is syntactically and semantically closed under complementation, we obtain the complementation lemma for tree automata as an easy corollary.

We also believe that our techniques and results demonstrate the power of temporal logic and the Mu-Calculus for specifying and reasoning about automata and games.

Games with Forbidden Positions

A.W. Mostowski (Gdansk)

Games of infinite trees using the table of an alternating automaton [Muller, Schupp. TCS 54 (1987)) are investigated. The first player chooses a transition, then the second player chooses a direction (or, simultaneously with the first player, a transition from

13

a dual table). For the investigated games some nodes are forbidden to enter in somestates and the winning conditions are supposed to be strong Rabin chain conditions[Mostowski, TCS 83 (1991)].

The deterrninancy theorem for such games is proved and the winning strategies aredescribed. They are, loosely speaking, iterations of typical max-min strategies and ina node depend only on the last state reached and on the future which is given by thesubtree having a root in the node. Contrary to the strategies described by Gurevich andHarrington, Muchnik, and A. and V. Yakhnis they need no extra information from the

past of the game (even recognizable by an automaton as in [Yakhnis, Yakhnis, Ann.Pure Appl. Logik 48 (1990)). The paper is self contained and uses neitherdetermination theorems for other games nor complementation lemma.

Problems in u-Calculus

D. Niwinski (Warsaw)

Many phenomena of in�nite computations can be adequately characterized by �xedpoint de�nitions, usually involving both least and greatest �xed point operators. Weinvestigate de�nability of sets of in�nite trees by �xed points, and compare it to othermodes of de�nability, namely by automata and logics. Nice coincidences andcorrespondences can be seen, involving the concept of a "�xed point hierarchy" ofaltemation of least and greatest �xed point operators. However, a problem of in�nityof this hierarchy, in the presence of intersection as a basic operation, remains stillopen and amounts to a problem if a hierarchy of Rabin indices for altematingautomata is in�nite. Another important problem conceming u-calculus is that ofcharacterizing the valid formulas in terms of some complete �nitary proof system. Itseems that application of infinite games (played on tableaux) may give some evidencehere.

Simulating Alternating Automata by-Nondeterministic Automata

P.E. Schupp (Urbana)

We give a new and simple proof that an altemating automaton M on in�nite trees canbe simulated by a nondeterrninistic automaton N. This gives a new proof of Rabin�sand, simultaneously, of McNaughton�s Theorem in the strong version due to Safra,and generalizes Safra�s result. Let IMI denote the number of states of M. If M uses the

de�ned acceptence by one complemented pair (this condition includes Biichi and co-

13

a dual table). For the investigated games some nodes are forbidden to enter in some states and the winning conditions are supposed to be strong Rabin chain conditions [Mostowski, TCS 83 (1991)].

The detenninancy theorem for such games is proved and the winning strategies are described. They are, loosely speaking, iterations of typical max-min strategies and in a node depend only on the last state reached and on the future which is given by the subtree having a root in the node. Contrary to the strategies described by Gurevich and Harrington, Muchnik, and A. and V. Y akhnis they need no extra information from the past of the game (even recognizable by an automaton as in [Yakhnis, Yakhnis, Ann. Pure Appl. Logik 48 (1990)). The paper is self contained and uses neither determination theorems for other games nor complementation lemma.

Problems in µ-Calculus

D. Niwinski (Warsaw)

Many phenomena of infinite computations can be adequately characterized by fixed point definitions, usually involving both least and greatest fixed point operators. We investigate definability of sets of infinite trees by fixed points, and compare it to other modes of definability, namely by automata and logics. Nice coincidences and correspondences can be seen, involving the concept of a "fixed point hierarchy" of alternation of least and greatest fixed point operators. However, a problem of infinity of this hierarchy, in the presence of intersection as a basic operation, remains still open and amounts to a problem if a hierarchy of Rabin indices for alternating automata is infinite. Another important problem concerning µ-calculus is that of characterizing the valid formulas in terms of some complete finitary proof system. It seems that application of infinite games (played on tableaux) may give some evidence here.

Simulating Alternating Automata by.Nondeterministic Automata

P.E. Schupp (Urbana)

We give a new and simple proof that an alternating automaton Mon infinite trees can be simulated by a nondeterministic automaton N. This gives a new proof of Rabin's and, simultaneously, of McNaughton's Theorem in the strong version due to Safra, and generalizes Safra' s result. Let IMI denote the number of states of M. If M uses the defined acceptence by one complemented pair (this condition includes Biichi and co-

14

Büchi) then INI is 2O(&#39;M"&#39;°3&#39;M&#39;). If M uses Muller acceptance, then INI is twoexponentials in IMI.

In proving the result we consider, as is now usual, the acceptance of M as beingde�ned by the existence of a winning strategy in an in�nite game. In contrast with thearguments of Gurevich and Harrington and Muchnik this proof is 953 by induction orrank functions.

Automata on Trees with Weak Acceptance Conditions

J. Skurczynski (Gdansk)

We construct a weak SkS formula describing the set of in�nite trees accepted by anautomaton with weak Muller conditions. Making use of the theorem on in�nity of ahierarchy of weak SkS formulas, we show that the weak Muller index can be

arbitrarily great.

Unforgettable Forgetful Determinacy

S. Zeitman (Detroit) \

In�nite games between two players have recently found application in computerscience as models of nonterminating computations in which, for example, an operatingsystem must interact with a potentially hostile environment. Natural speci�cations onsuch computations give rise to a class of games to which the Forgetful DeterminacyTheorem of Yuri Gurevich and Leo Harrington apply. This theorem �rst appeared aspart of a simpler proof of the most dif�cult part of Michael Rabin�s proof that themonadic second-order theory of the in�nite binary tree is decidable. An independentproof of Rabin�s result was given by Andre Muchnik. The original proof of theForgetful Determinacy Theorem was sketchy and became a source of frustration formany. J. Donald Monk presented an expanded and complete version of the proof aspart of a seminar on Rabin�s result and related topics. Recently, Alexander andVladimir Yakhnis gave a proof that strenthened the theorem by providing moreexplicit strategies for the players. Still, there seems to be interest in a more compactproof to this much-used result, and this talk provides one. It should be stressed that the

proof presented is conceptually the same as the one by Yakhnis and Yakhnis, althoughit is done in the framework of the original result, and it illuminates some notions

involved by de�ning them in a related game which we call the "forgetful game".

14

Bilchi) then INI is 20(IMl·loglMI)_ If M uses Muller acceptance, then INI is two

exponentials in IMI.

In proving the result we consider, as is now usual, the acceptance of M as being defined by the existence of a winning strategy in an infinite game. In contrast with the arguments of Gurevich and Harrington and Muchnik this proof is not by induction or rank functions.

Automata on Trees with Weak Acceptance Conditions

J. Skurczynski (Gdansk)

We construct a weak SkS formula describing the set of infinite trees accepted by an automaton with weak Muller conditions. Making use of the theorem on infinity of a hierarchy of weak SkS formulas, we show that the weak Muller index can be arbitrarily great.

Unforgettable Forgetful Determinacy

S. Zeitman (Detroit)

I

Infinite games between two players have recently found application in computer science as models of nonterminating computations in which, for example, an operating system must interact with a potentially hostile environment. Natural specifications on such computations give rise to a class of games to which the Forgetful Determinacy Theorem of Yuri Gurevich and Leo Harrington apply. This theorem first appeared as part of a simpler proof of the most difficult part of Michael Rabin' s proof that the monadic second-order theory of the infinite binary tree is decidable. An independent proof of Rabin' s result was given by Andre Muchnik. The original proof of the Forgetful Determinacy Theorem was sketchy and became a source of frustration for many. J. Donald Monk presented an expanded and complete version of the proof as part of a seminar on Rabin's result and related topics. Recently, Alexander and Vladimir Y akhnis gave a proof that strenthened the theorem by providing more explicit strategies for the players. Still, there seems to be interest in a more compact proof to this much-used result, and this talk provides one. It should be stressed that the proof presented is conceptually the same as the one by Y akhnis and Y akhnis, although it is done in the framework of the original result, and it illuminates some notions involved by defining them in a related game which we call the "forgetful game".

15

IV. LOGICAL ASPECTS

A Monadic Second-Order Limit Law for Unary Functions

K. Compton (Roquencourt and Ann Arbor)

Let C be a class of structures, C� be the structures in C with universe {O,...,n-1}, and�for a sentence (p, let pn(<p) be the fraction structures in Cu satisfying (p. We say that Chas a limit law for a logic L if u((p) = lin|n_,°,pn((p) is de�ned for all sentences (p in L.We show that-the class of unary functions has a limit law for monadic second-order

logic.

The proof uses generating series techniques and a type of theorem known in classicalanalysis as a Tauberian theorem. The idea is that generating series enumerating classesof structures can be developed using ideas from the theory of N-rational series and theasymptotics of the coef�cients of the series may than be obtained using a Tauberiantheorem.

Finally, we note that many limit laws for finite random structures have analogues forinfinite random structures. We pose the problem of �nding an appropriate setting toinvestigate limit laws for in�nite unary functions.

Monadic Second-Order De�nability Properties of In�nite Graphs

B. Courcelle (Bordeaux)

Equational graphs can be characterized in several equivalent ways: (1) as initialsolutions of systems of graph equations, (2) as graphs having regular tree-decompositions, (3) as graphs which are both de�nable in monadic-second order logicand have �nite tree-width. Graphs de�nable (by monadic second-order formulas) in

equational graphs are equational.

The talk has shown the important role of monadic second-order graph transductions inthese results.

15

IV. LOGICAL ASPECTS

A Monadic Second-Order Limit Law for Unary Functions

K. Compton (Roquencourt and Ann Arbor)

Let C be a class of structures, ½i be the structures in C with universe {0, ... ,n-1 }, and, for a sentence cp, let Jln(cp) be the fraction structures in C

0 satisfying cp. We say that C

has a limit law for a logic L ifµ( cp) = lilJ\i_..,J10 ( cp) is defined for all sentences cp in L. We show that the class of unary functions has a limit law for monadic second-order logic.

The proof uses generating series techniques and a type of theorem known in classical analysis as a Tauberian theorem. The idea is that generating series enumerating classes of structures can be developed using ideas from the theory of N-rational series and the asymptotics of the coefficients of the series may than be obtained using a Tauberian theorem.

Finally, we note that many limit laws for finite random structures have analogues for infinite random structures. We pose the problem of finding an appropriate setting to investigate limit laws for infinite unary functions.

Monadic Second-Order Defmability Properties of Infinite Graphs

B. Courcelle (Bordeaux)

Equational graphs can be characterized in several equivalent ways: (1) as initial solutions of systems of graph equations, (2) as graphs having regular tree­decompositions, (3) as graphs which are both definable in monadic-second order logic and have finite tree-width. Graphs definable (by monadic second-order formulas) in equational graphs are equational.

The talk has shown the important role of monadic second-order graph transductions in these results.

16

De�nability in Weak Monadic Second-Order Logic of Some In�nite Graphs

G. Sénizergues (Bordeaux)

We prove the following result, which is a partial solution to a conjecture of B.Courcelle [stated in ICALP 89]:

Let G be an equational 1-graph such that(1) G has a spanning tree of �nite degree(2) for every pair of vertices (v,v�) there are only �nitely many edges

then G is de�nable in weak monadic second-order logic.

The main ideas of the proof are:

v a notion of automatic graph (which is a natural extension of that of

automatic group, de�ned by [Cannon-Epstein-Holt-Paterson-Thurston,Research report of Warwick�s universityl)

v such graphs are de�nable by �rst-order "colored" formula

- each graph G ful�lling (1), (2) is the result of a "half-tum-relabelling"of some automatic graph.

Automata on In�nite Objects, Monadic Theories, and Complexity

A. Muchnik, A.L. Semenov (Moscow)

A survey of results obtained by the Mathematical Logic and Computer Science Groupat the Institute of New Technologies (Moscow and St. Petersburg) is presented. Theresults belong to Konstantin Gorbunov, Andrei Muchnik, Alexei Semenow, Anatol�Slisenko, Sergey Soprunov. Some of the results were proved in parallel with Westemresearch.

Major topics are the following. Conditions of decidability for the monadic theory ofa structure «n; .<.� P>� where P is an unary predicate (or a tuple of unary predicates).Almost periodic to-words P and the corresponding monadic theories. Double in�nitewords. The Muchnik proof for Rabin Theorem. A generalization of Shelah-StuppTheorem by adding the unary predicate of equality of two last symbols in a sequence.The uniformization problem in different monadic theories. The complexity of

16

Definability in Weak Monadic Second-Order Logic of Some Infinite Graphs

G. Senizergues (Bordeaux)

We prove the following result, which is a partial solution to a conjecture of B. Courcelle [stated in ICALP 89]:

Let G be an equational I-graph such that (1) G has a spanning tree of finite degree (2) for every pair of vertices (v,v') there are only finitely many edges

then G is definable in weak monadic second-order logic.

The main ideas of the proof are:

• a notion of automatic graph (which is a natural extension of that of automatic group, defined by [Cannon-Epstein-Holt-Paterson-Thurston, Research report of Warwick's university])

• such graphs are definable by first-order "colored" formula

• each graph G fulfilling (1), (2) is the result of a "half-turn-relabelling" of some automatic graph.

Automata on Infinite Objects, Monadic Theories, and Complexity

A. Muchnik, A.L. Semenov (Moscow)

A survey of results obtained by the Mathematical Logic and Computer Science Group at the Institute of New Technologies (Moscow and St. Petersburg) is presented. The results belong to Konstantin Gorbunov, Andrei Muchnik, Alexei Semenow, Anatol' Slisenko, Sergey Soprunov. Some of the results were proved in parallel with Western research.

Major topics are the following. Conditions of decidability for the monadic theory of a structure <ro; ~. P>, where P is an unary predicate (or a tuple of unary predicates). Almost periodic c.o-words P and the corresponding monadic theories. Double infinite words. The Muchnik proof for Rabin Theorem. A generalization of Shelah-Stupp Theorem by adding the unary predicate of equality of two last symbols in a sequence. The uniformization problem in different monadic theories. The complexity of

17

complementation and deterrninization operations. Arbitrary graphs, their combinatorialproperties, new bounds, connections with complexity for �nite graphs and decidabilityfor infinite. An example of predicate on tree for which the weak monadic theory isdecidable and the monadic theory is undecidable. Non-existence of a maximaldecidable weak monadic theory.

Some of the results were discussed in Semenov�s paper at MFCS�84 (Springer LectureNotes in Computer Science) and some were proved in Semenov�s, Math. USSRIzvestija, 1984.

Circuit Complexity, Finite Automata and Generalized First-Order Logic

H. Straubing (Boston)

Let FO[<] denote the class of languages in A* de�ned by first-order sentences inwhich all the atomic formulas are of the form x<y and Qax (meaning that the symbolin position x is a). It is well-known that FO[<] is exactly the class of star-free, or

aperiodic, regular languages. (McNaughton)

We can generalize this in two ways: introduce new modular guanti�ers: 3E x (p(x)means the number of positions such that <p(x) is congruent to r modulo q.Alternatively, introduce new numerical p_rQicates: these are atomic formulas, such asx<y, that do not depend on which input symbol appears in the given positions.Extending the above notation, letter 9\£denotes the class of all numerical predicates:

Conjecture

(FO+{3�s&#39;: OSs<q})[9\(] n Regular languages =(FO+{32: 0.<_s<q})[<,{x s O(mod r): r>O}]

This conjecture is closely connected to problems conceming the complexity ofconstant-depth circuits. It is known to be true when q = 1 or q is a prime power. It isequivalent to the conjecture that the circuit complexity class ACC is strictly containedin NC�. The conjecture has been proved in a special case: Let M denote the class of93g numerical predicates. Then

Theorem (FO+{3g: OSs<q})[<,M] n Regular languages =

(FO+{3§: OSs<q})[<,{x s O(mod r): r>O}].

17

complementation and determinization operations. Arbitrary graphs, their combinatorial properties, new bounds, connections with complexity for finite graphs and decidability for infinite. An example of predicate on tree for which the weak monadic theory is decidable and the monadic theory is undecidable. Non-existence of a maximal decidable weak monadic theory.

Some of the results were discussed in Semenov's paper at MFCS'84 (Springer Lecture Notes in Computer Science) and some were proved in Semenov's, Math. USSR Izvestija, 1984.

Circuit Complexity, Finite Automata and Generalized First-Order Logic

H. Straubing (Boston)

Let FO[<] denote the class of languages in A• defined by first-order sentences in which all the atomic formulas are of the form x<y and Q1x (meaning that the symbol in position x is a). It is well-known that FO[<] is exactly the class of star-free, or aperiodic, regular languages. (McNaughton)

We can generalize this in two ways: introduce new modular quantifiers: ':f! x cp(x) means the number of positions such that cp(x) is congruent to r modulo q. Alternatively, introduce new numerical predicates: these are atomic formulas, such as x<y, that do not depend on which input symbol appears in the given positions. Extending the above notation, letter ?ldenotes the class of all numerical predicates:

Coniecture

(FO+{~: lli;s<q})[~ n Regular languages= (FO+{~: lli;s<q})[<,{x = O(mod r): r>O}]

This conjecture is closely connected to problems concerning the complexity of constant-depth circuits. It is known to be true when q = 1 or q is a prime power. It is equivalent to the conjecture that the circuit complexity class ACC is strictly contained in NC1• The conjecture has been proved in a special case: Let !M denote the class of ~ numerical predicates. Then

Theorem

(FO+{~: lli;s<q})[<,$\(1 n Regular languages =

(FO+{~: ~s<q})[<,{x = O(mod r): r>O}].

18

V. COMBINATORIAL ASPECTS

Games with Dominoes

D. Beauquier (Paris)

There exists a linear algorithm to decide whether a �nite picture (union of unitsquares) is tilable by dominoes. Secondly there is a sufficient condition for a picturewithout holes to decide whether it has no unique tiling by horizontal bars of somelength m 2 2, and vertical bars of length n 2 2 (for dominoes the condition holdsalso for pictures with holes).

Thirdly, if a finite picture without holes admits a tiling T with bars hm, v�, it admitsa unique tiling iff there is no subtiling of T covering the rectangle R(m,n) (fordominoes, the condition is available for pictures with holes).

Some Questions on Sequential Bijections Between Free Monoids

C. Choffrut (Paris)

Given two free �nitely generated monoids A* and B*, a sequential function is amapping realized by a deterministic transducer, i.e., by a finite deterministic automatonwith input alphabet A and equipped with an output alphabet B. If A = {a,b,c} andB = {x,y}, then the following sequential transducer realizes a function f in which, forinstance, f(cc) = yxyyy and f(abab) = xyxxy holds.

18

V. COMBINATORIAL ASPECTS

Games with Dominoes

D. Beauquier (Paris)

There exists a linear algorithm to decide whether a finite picture (union of unit squares) is tilable by dominoes. Secondly there is a sufficient condition for a picture without holes to decide whether it has no unique tiling by horizontal bars of some length m ~ 2, and vertical bars of length n ~ 2 (for dominoes the condition holds also for pictures with holes).

Thirdly, if a finite picture without holes admits a tiling T with bars 1\n, v0 , it admits a unique tiling iff there is no subtiling of T covering the rectangle R(m,n) (for dominoes, the condition is available for pictures with holes).

Some Questions on Sequential Bijections Between Free Monoids

C. Choffrut (Paris)

Given two free finitely generated monoids A• and B*, a sequential function is a mapping realized by a deterministic transducer, i.e., by a finite deterministic automaton with input alphabet A and equipped with an output alphabet B. If A = { a,b,c} and B = { x,y}, then the following sequential transducer realizes a function f in which, for instance, f(cc) = yxyyy and f(abab) = xyxxy holds.

b/y

a/x

bty >-------t.,_----{ 2

c/yxy

a/x b/xy

a/x b/xy

c/yy

19

The question we raise originates from Eilenberg�s textbook where an example is givenof a sequential function de�ning a bijection from A* onto B* ("Automata, Languagesand Machines", Vol. A, Academic Press, 1974, p. 305; the above deterministictransducer also realizes a sequential bijection). In Eilenberg�s example, the conditionCard(A) = 3 and Card(B) = 2 holds.

More generally, we prove that a necessary and sufficient condition for a sequentialbijection to exist is: Card(A)2Card(B). Our construction makes use of complete suf�xcodes of the cardinality of A, but we ignore whether or not this is necessary (in thecurrent example we have the suffix code {x,xy,yy}). A further step would be toconstruct all sequential bijections. Yet, though we have some clues of how to get newexamples from old ones, we ignore what the general method could be. It is notdifficult to see that the problem reduces to obtaining the characteristic series of B* asthe solution of certain types of linear systems of equations where the unknowns arerational series in the indeterminates B with coefficients in the natural numbers N. For

instance the current example gives rise to the following system where L0 = B*.Intuitively Li, i = O,...,3, represents all the outputs obtained when starting to read aninput in state i:

L0: 1 +xI.0+yL1+yxyL3L1 = 1 + yL2 + (x+yyy)L3L2 = l + (x+xy+y)L3L3 = 1 + (x+xy+yy)L3

The idea would be to transform this system while still preserving B* as a solution.

� Splicing Schemes and DNA

T. Head (Binghamton)

Splicing systems were introduced by the speaker (Bull. Math. Biology (1987) 737-759)as language generating devices that imitate the action of sets or restriction enzymesand a ligase on double stranded DNA molecules. The existing literature on theseformal systems is summarized. This previous work has been concerned only withstrings as models of linear molecules. However, DNA molecules occur naturally inboth linear and circular form. It is now proposed that splicing theory deal with bothlinear and circular strings and their interactions. New problems are proposed which thespeaker hopes members of the audience or their students will solve. An extendedversion of this talk is available on request from the speaker.

19

The question we raise originates from Eilenberg's textbook where an example is given of a sequential function defining a bijection from A* onto B * (" Automata, Languages and Machines", Vol. A, Academic Press, 1974, p. 305; the above deterministic transducer alsQ realizes a sequential bijection). In Eilenberg's example, the condition Card(A) = 3 and Card(B) = 2 holds.

More generally, we prove that a necessary and sufficient condition for a sequential bijection to exist is: Card(A)~ard(B). Our construction makes use of complete suffix codes of the cardinality of A, but we ignore whether or not this is necessary (in the current example we have the suffix code { x,xy ,yy} ). A further step would be to construct all sequential bijections. Yet, though we have some clues of how to get new examples from old ones, we ignore what the general method could be. It is not difficult to see that the problem reduces to obtaining the characteristic series of B * as the solution of certain types of linear systems of equations where the unknowns are rational series in the indetenninates B with coefficients in the natural numbers N. For instance the current example gives rise to the following system where Lo = B*. Intuitively Li, i = 0, ... ,3, represents all the outputs obtained when starting to read an input in state i:

Lo = 1 + xl..o + yL1 + yxyL3 L1 = 1 + Y½ + (x+yyy)L3 ½ = 1 + (x+xy+y)L3 L3 = 1 + (x+xy+yy)L3

The idea would be to transform this system while still preserving B* as a solution.

· Splicing Schemes and DNA

T. Head (Binghamton)

Splicing systems were introduced by the speaker (Bull. Math. Biology (1987) 737-759) as language generating devices that imitate the action of sets or restriction enzymes and a ligase on double stranded DNA molecules. The existing literature on these formal systems is summarized. This previous work has been concerned only with strings as models of linear molecules. However, DNA molecules occur naturally in both linear and circular form. It is now proposed that splicing theory deal with both linear and circular strings and their interactions. New problems are proposed which the speaker hopes members of the audience or their students will solve. An extended version of this talk is available on request from the speaker.

Dagstuhl-Seminar 9202 Participants

Daniele Beauquier GermanyLITP diekert@informatik.uni-stuttgart.deUniversité Paris VII teI.: +49-711 78 16 32975252 Paris Cedex 05France Werner Ebingerbeauquier@Iitp.ibp.fr Universität StuttgartteI.: +33 46 60 55 15 Institut für Informatik

Breitwiesenstraße 20-22

Véronique Bruyére W-7000 Stuttgart 80Université de Mons-Hainaut GermanFaculté des Sciences ebinger informatik-uni-stuttgart.de15 Avenue Maistriau teI.: +49-711 78 16 4017000 MonsBelgium E. Allen Emersonsbruyere@rm1.vmh.ac.be University of Texas at AustinteI.: +32-65-37 3444 Department of Computer Sciences

Taylor Hall 2.124Christian Choffrut Austin TX 78712-1188LITP USAUniversité Paris VII emerson@cs.utexas.edu2 Place Jussieu teI.: +1 512 471 953775252 Paris Cedex 05France Christiane Frougnycc@litp.ibp.fr LITP

Institut Blaise PascalKevin Compton 4 Place JussieuINRIA 75252 Paris Cedex 05Domaine de Voluceau FranceFiocquencourt frougny@litp.ibp.frBP 105 teI.: +33 1 44 27 70 9578153 Le Chesnay CedexFrance Paul Gastincompton@margaux.inria.fr LITP - IBPteI.: +33 1 39 63 56 88 Université Paris VI

4 Place JussieuBmno Courcelle 75252 Paris Cedex 05Université Bordeaux - I FranceLaboratoire Bordelais de gastin@litp.ibp.frRecherche en Informatique teI.: +33 1 44 27 53 91351 Cours de la Liberation33405 Talence Cedex Leucio GuerraFrance LITPcourceIl@geocub.greco-prog.fr Université Paris VIIteI.: +33 56 84 60 86 2 Place Jussieu

75252 Paris Cedex 05Max Dauchet FranceUniversité de Lille 1LIFL Thomas J. HeadBatiment M 3 SUNY at Binghamton59655 Villeneuve d&#39;Ascq Cedex Dept. of Mathematical SciencesFrance Binghamton NY 13902dauchet@IifI.IifI.fr USAteI.: +33-20 43 45 88 tjhead@bingvaxu.cc.binghamton.edu

teI.: +1 607 777 21 47Volker DiekertUniversität Stuttgart Hendrik Jan HoogeboomInstitut für Informatik University of LeidenBreitwiesenstraße 20-22 Department of Computer ScienceW-7000 Stuttgart 80 P.O. Box 9512

Dagstuhl-Seminar 9202

Daniele Beauquter LITP Universite Paris VII 75252 Paris Cedex 05 France beauquier@litp.ibp.fr tel. : +33 46 60 55 15

Veronique Bruyere Universite de Mons-Hainaut Faculte des Sciences 15 Avenue Maistriau 7000 Mons Belgium sbruyere@rm1 .vmh.ac.be tel. : +32-65-37 3444

Christian Choffrut LITP Universite Paris VII 2 Place Jussieu 75252 Paris Cedex 05 France cc@litp.ibp.fr

Kevin Compton INRIA Domaine de Voluceau Rocquencourt BP 105 78153 Le Chesnay Cedex France compton@margaux.inria.fr tel. : +33 1 39 63 56 88

Bruno Courcelle Universite Bordeaux - I Laboratoire Bordelais de Recherche en lnformatique 351 Cours de la Liberation 33405 Talence Cedex France courcell@geocub.greco-prog.fr tel. : +33 56 84 60 86

Max Oauchet Universite de Lilla 1 LIFL Batiment M 3 59655 Villeneuve d'Ascq Cedex France dauchet@lifl. lifl . fr tel.: +33-20 43 45 88

Volker Diekert Universitat Stuttgart lnstitut fur lnformatik BreitwiesenstraBe 20-22 W-7000 Stuttgart 80

Participants

Germany diekert@informatik.uni-stuttgart.de tel. : +49-711 78 16 329

Werner Ebinger Universitat Stuttgart lnstitut fur lnformatik BreitwiesenstraBe 20-22 W-7000 Stuttgart 80 Germany ebinger@informatik-uni-stuttgart.de tel.: +49-711 78 16 401

E. Allen Emerson University of Texas at Austin Department of Computer Sciences Taylor Hall 2.124 Austin TX 78712-1188 USA emerson@cs.utexas.edu tel. : +1512471 9537

Christiane Frougny LITP lnstitut Blaise Pascal 4 Place Jussieu 75252 Paris Cedex 05 France frougny@litp.ibp.fr tel. : +33 1 44 27 70 95

Paul Gastin LITP - IBP Universite Paris VI 4 Place Jussieu 75252 Paris Cedex 05 France gastin@litp.ibp.fr tel. : +33 1 44 27 53 91

Leucio Guerra LITP Universite Paris VII 2 Place Jussieu 75252 Paris Cedex 05 France

Thomas J. Head SUNY at Binghamton Dept. of Mathematical Sciences Binghamton NY 13902 USA tjhead@bingvaxu.cc.binghamton.edu tel. : + 1 607 777 21 47

Hendrik Jan Hoogeboom University of Leiden Department of Computer Science P.O. Box 9512

NL-2300 RA Leiden Damian NiwinskiThe Netherlands Universit of Warsawhjh@ruIwinw.LeidenUniv.nI Inst. of athematicsteI.: +31 71 27 7062/7065 (Secr.) U1. Banacha 2

00-913 Warsaw 59Juhani Karhumäki PolandUniversity of Turku NIWINSKI@PLEARN.PLDept. of Mathematics teI.: +48 22 22 10 50 (home)20500 Turku 50Finland Dominique Perrinkarhumak@utu.cs.fi LITPteI.: +358-21-6335613 Université Paris VII

2 Place JussieuNils Klarlund 75252 Paris Cedex 05Universit of Aarhus FranceDept. of omputer Science dp@litp.ibp.frNy Munkegade 116 tel.: +33 1 44 27 28 458000 Aarhus CDenmark Antoine Petitklarlund@daimi.aau.dk Université Paris Sud

Laboratoire de Recherche en InformatiqueMichel Latteux Bat 490Université de Lille 1 91405 Orsay CedexLIFL FranceBatiment M 3 petit@ln&#39;.lri.fr59655 Villeneuve d�Ascq Cedex teI.: +33 1 69 41 66 09Francelatteux@lifI.IitI.fr Jean-Eric PinteI.: +33 20 43 42 67 LITP

Université Paris VIBertrand Le Saec 4 Place JussieuUniversité de Bordeaux I 75252 Paris Cedex 05Laboratoire Bcrdelais de France .Recherche en Informatique pin@litp.ibp.fr351 Cours de Ia Liberation33405 Talence Cedex Roman R. RedziejowskiFrance IBM Nordic Laboratorytel.: +33 56 84 65 29 P.O. Box 962

18109 LidingöIgor Lltovski SwedenUniversité de Bordeaux l teI.: +46 8 636 6263Departement d�lnformatique351 Cours de la Liberation Antonio Restivo33405 Talence Cedex Universita degli Studi di PalermoFrance Dipt. di Matematica ed Applicazionilito@geocub.greco-prog.fr Via Archirafi 34

90123 PalermoA.W. Mostowski ItalyUniwersytet Gdanski restivo@ipamat.cres.itInstitut Matematyki tel.: +39 91 60 40 307Wita Stwosza 57 e80-952 Gdansk Paul E. SchuppPoland University of IllinoisteI.: +48-58-522212 Dept. of Mathematics.

273 Alt eld HallAndrej A. Muchnik 1409 est Green StreetInstitute of New Technologies Urbana IL 61801Kirovogradskaya ul. 11 USAMoscow 1 13587 schupp@symcom.math.uiuc.eduRussia teI.: +1 217 333 1610/ 3350postmaster@gIobIab.msk.sute|.: +7 O95 311 4427

NL-2300 RA Leiden The Netherlands hjh@rulwinw. LeidenUniv. nl tel. : +31 71 27 7062 / 7065 (Seer.)

Juhani Karhumlkl University of Turku Dept. of Mathematics 20500 Turku 50 Finland karhumak@utu.es. fi tel.: +358-21-6335613

Nils Klarlund University of Aarhus Dept. of Computer Science Ny Munkegade 116 8000 Aarhus C Denmark klarlund@daimi.aau.dk

Michel Latteux Universite de Lille 1 LIFL Batiment M 3 59655 Villeneuve d'Ascq Cedex France latteux@lifl .lifl. fr tel.: +33 20 43 42 67

Bertrand Le Saec Universite de Bordeaux I Laboratoire Bordelais de Recherche en lnformatique 351 Cours de la Liberation 33405 Talence Cedex France tel.: +33 56 84 65 29

Igor Lltovskl Universite de Bordeaux I Departement d'lnformatique 351 Cours de la Liberation 33405 Talence Cedex France lito@geocub.greco-prog.fr

A. W. Mostowskl Uniwersytet Gdanski lnstitut Matematyki Wita Stwosza 57 80-952 Gdansk Poland tel.: +48-58-522212

Andrej A. Muchnlk Institute of New Technologies Kirovogradskaya ul. 11 Moscow 113587 Russia postmaster@globlab.msk.su tel.: +7 095 311 4427

Damian Nlwlnskl University of Warsaw Inst. of Mathematics U 1. Banach a 2 00-913 Warsaw 59 Poland NIWINSKl@PLEARN.PL tel. : +48 22 22 1 O 50 (home)

Dominique Perrin LITP Universite Paris VII 2 Place Jussieu 75252 Paris Cedex 05 France dp@litp.ibp.fr tel. : +33 1 44 27 28 45

Antoine Petit Universite Paris Sud Laboratoire de Recherche en I nformatique Bat 490 91405 Orsay Cedex France petit@lri.lri.fr tel.: +33 1 69 41 66 09

Jean-Eric Pin LITP Universite Paris VI 4 Place Jussieu 75252 Paris Cedex 05 France pin@litp.ibp.fr

Roman R. Redzlejowskl IBM Nordic Laboratory P.O. Box 962 1a109 Lidingo Sweden tel.: +46 8 636 6263

Antonio Restivo Universita degli Studi di Palermo Dipt. di Matematica ed Applicazioni Via Archirafi 34 90123 Palermo Italy restivo@ipamat.cres.it tel. : +39 91 60 40 307

PaulE.Schupp University of Illinois Dept. of Mathematics . 273 Altgeld Hall 1409 West Green Street Urbana IL 61801 USA schupp@symcom.math.uiuc.edu tel. : +1 217 333 1610 / 3350

Alexei L. SemenovInstitute of New TechnologiesKirovogradskaya ul. 11Moscow 1 13587Russia

postmaster@gIob|ab.msk.suteI.: +7 O95 315 08 O8

Geraud SenizerguesUniversity of Warwick (till July 1992)Mathematics InstituteCoventry CV4 7ALGreat Britain

ges@maths.warwick.ac.uk

Jerzy SkurczynskiUniwersytet Gdanskilnstitut MatematykiWita Stwosza 5780-952 GdanskPolandteI.: +48 58 41 49 14

Ludwig StalgerUniversität - GH - Siegen

Programmiersprachen Postfach 10 12 405900 SiegenGermanyteI.: +49-2238 51 580 (home)

Howard StraubingBoston CollegeDept. of Computer ScienceChestnut Hill MA 02167USAstraubin@beuxs1 .bo.eduteI.: +1 617 552 39 77

Wolfgang ThomasUniversität KielInstitut für InformatikOlshausenstraße 40W-2300 Kiel 1

Germanywt@informatik.uni-kieI.dbp.deteI.: +49-431 880 44 80

Thomas WllkeUniversität KieIInstitut für InformatikOlshausenstraße 40W-2300 Kiel 1

Germany

tw@informatik.uni-kiel.dbp.de teI.: +49-431 880 46 62Suzanne ZeitmanWayne State UniversityDepartment of Computer ScienceDetroit Michigan 48202USAsze@cs.wayne.edu

Alexei L. Semenov Institute of New Technologies Kirovogradskaya ul. 11 Moscow 113587 Russia postmaster@globlab.msk.su tel. : +7 095 315 08 08

Geraud Senlzergues University of Warwick (till July 1992) Mathematics Institute Coventry CV4 ?AL Great Britain ges@maths.warwick.ac.uk

Jerzy Skurczynski Uniwersytet Gdanski lnstitut Matematyki Wita Stwosza 57 80-952 Gdansk Poland tel. : +48 58 41 49 14

Ludwig Staiger Universitat - GH - Siegen P rogrammiersprachen Postfach 1 O 12 40 5900 Siegen Germany tel.: +49-2238 51 580 (home)

Howard Straublng Boston College Dept. of Computer Science Chestnut Hill MA 02167 USA straubin@beuxs1 .bc.edu tel. : +1617552 39 77

Wolfgang Thomas Universitat Kiel lnstitut fur lnformatik OlshausenstraBe 40 W-2300 Kiel 1 Germany wt@informatik.uni-kiel.dbp.de tel.: +49-431 880 44 80

Thomas WIike Universitat Kiel lnstitut fur lnformatik OlshausenstraBe 40 W-2300 Kiel 1 Germany tw@informatik.uni-kiel.dbp.de tel.: +49-431 880 46 62

Suzanne Zeltman Wayne State University Department of Computer Science Detroit Michigan 48202 USA sze@cs. wayne .edu

Zuletzt erschienene und geplante Titel:

J. Berstel , J.E. Pin, W. Thomas (editors):Automata Theory and Applications in Logic and Complexity, Dagstuhl-Seminar-Report; 5, 14.-18.1.1991 (9103)

B. Becker, Ch. Meinel (editors):Entwerfen, Prüfen, Testen, DagstuhI-Seminar-Report; 6, 18.-22.2.1991 (91 O8)

J. P. Finance, S. Jähnichen, J. Loeckx, M. Wirsing (editors):Logical Theory for Program Construction, Dagstuhl-Seminar-Report; 7, 25.2.-1.3.1991 (9109)

E. W. Mayr, F. Meyer aut der Heide (editors):Parallel and Distributed Algorithms, Dagstuhl-Seminar-Report; 8, 4.-8.3.1991 (9110)

M. Broy, P. Deussen, E.-R. Olderog, W.P. de Roever (editors):Concurrent Systems: Semantics, Specification, and Synthesis, Dagstuhl-Seminar-Report; 9, 11.-15.3.1991 (9111)

K. Apt, K. Indermark, M. Rodriguez-Artalejo (editors):Integration of Functional and Logic Programming, Dagstuhl-Seminar-Report; 10, 18.-22.3.1991(9112)

E. Novak, J. Traub, H. Wozniakowski (editors):Algorithms and Complexity for Continuous Problems, Dagstuhl-Seminar-Report; 11, 15-19.4.1991 (9116)

B. Nebel, C. Peltason, K. v. Luck (editors):Terminological Logics� DagstuhI-Seminar-Report; 12, 6.5.-18.5.1991 (9119)

R. Giegerich, S. Graham (editors):Code Generation - Concepts, Tools, Techniques, Dagstuhl-Seminar-Report; 13, 20.-24.5.1991(9121)

M. Karpinski, M. Luby, U. Vazirani (editors):Randomized Algorithms, Dagstuhl-Seminar-Report; 14, 10.-14.6.1991 (9124)

J. Ch. Freytag, D. Maier, G. Vossen (editors):Query Processing in Object-Oriented, Complex-Object and Nested Relation Databases, Dag-stuhl-Seminar-Report; 15, 17.-21.6.1991 (9125)

M. Droste, Y. Gurevich (editors): .Semantics of Programming Languages and Model Theory, Dagstuhl-Seminar-Report; 16, 24.-28.6.1991 (9126)

G. Farin, H. Hagen, H. Noltemeier (editors):Geometric Modelling, Dagstuhl-Seminar-Report; 17, 1.-5.7.1991 (9127)

A. Karshmer, J. Nehmer (editors):Operating Systems of the 1990s, Dagstuhl-Seminar-Report; 18, 8.-12.7.1991 (9128)

H. Hagen, H. Müller, G.M. Nielson (editors):Scientific Visualization, DagstuhI-Seminar-Report; 19, 26.8.-30.8.91 (9135)

T. Lengauer, R. Möhring, B. Preas (editors):Theory and Practice of Physical Design of VLSI Systems, Dagstuhl-Seminar-Report; 20, 2.9.-6.9.91 (9136)

F. Bancilhon, P. Lockemann, D. Tsichritzis (editors):Directions of Future Database Research, Dagstuhl-Seminar-Report; 21, 9.9.-12.9.91 (9137)

H. Alt , B. Chazelle, E. Welzl (editors):Computational Geometry, Dagstuhl-Seminar-Report; 22, 07.10.-1 1.10.91 (9141)

F.J. Brandenburg , J. Berstel, D. Wotschke (editors):Trends and Applications in Formal Language Theory, Dagstuhl-Seminar-Report; 23, 14.10.-18.10.91 (9142)

Zuletzt erschienene und geplante Tltel: J. Berstel , J.E. Pin, W. Thomas (editors):

Automata Theory and Applications in Logic and Complexity, Dagstuhl-Seminar-Report; 5, 14.-18.1.1991 (9103)

B. Becker, Ch. Meinel (editors): Entwerfen, PrOfen, Testen, Dagstuhl-Seminar-Report; 6, 18.-22.2.1991 (9108)

J. P. Finance, S. Jahnichen, J . Loeckx, M. Wirsing (editors): Logical Theory for Program Construction, Dagstuhl-Seminar-Report; 7, 25.2.-1.3.1991 (9109)

E.W. Mayr, F. Meyer auf der Heide (editors): Parallel and Distributed Algorithms, Dagstuhl-Seminar-Report; 8, 4.-8.3.1991 (9110)

M. Broy, P. Deussen, E.-R. Olderog, W.P. de Roever (editors) : Concurrent Systems: Semantics, Specification, and Synthesis, Dagstuhl-Seminar-Report; 9, 11.-15.3 .1991 (91 11)

K. Apt, K. lndermark, M. Rodriguez-Artalejo (editors) : Integration of Functional and Logic Programming, Dagstuhl-Seminar-Report; 10, 18.-22.3.1991 (9112)

E. Novak, J. Traub, H. Wozniakowski (editors): Algorithms and Complexity for Continuous Problems, Dagstuhl-Seminar-Report; 11 , 15-19.4.1991 (9116)

B. Nebel, C. Peltason, K. v. Luck (editors): Terminological Logics, Dagstuhl-Seminar-Report ; 12, 6.5.-18.5.1991 (9119)

R. Giegerich, S. Graham (editors): Code Generation - Concepts, Tools, Techniques, Dagstuhl-Seminar-Report; 13, 20.-24.5.1991 (9121 )

M. Karpinski, M. Luby, U. Vazirani (editors) : Randomized Algorithms, Dagstuhl-Seminar-Report; 14, 10.-14.6.1991 (9124)

J. Ch. Freytag, D. Maier, G. Vossen (editors): Query Processing in Object-Oriented, Complex-Object and Nested Relation Databases, Dag­stuhl-Seminar-Report; 15, 17.-21.6.1991 (9125)

M. Droste, Y. Gurevich (editors) : Semantics of Programming Languages and Model Theory, Dagstuhl-Seminar-Report; 16, 24.-28.6.1991 (9126)

G. Farin, H. Hagen, H. Noltemeier (editors): Geometric Modelling, Dagstuhl-Seminar-Report; 17, 1.-5 .7.1991 (9127)

A. Karshmer, J . Nehmer (editors) : Operating Systems of the 1990s, Dagstuhl-Seminar-Report; 18, 8.-12.7.1991 (9128)

H. Hagen, H. MOiier, G.M. Nielson (editors): Scientific Visualization, Dagstuhl-Seminar-Report; 19, 26.8.-30.8.91 (9135)

T. Lengauer, A. Mohring, B. Preas (editors): Theory and Practice of Physical Design of VLSI Systems, Dagstuhl-Seminar-Report; 20, 2.9.-6.9 .91 (9136)

F. Bancilhon, P. Lockemann, D. Tsichritzis (editors): Directions of Future Database Research, Dagstuhl-Seminar-Report; 21 , 9.9.-12.9 .91 (9137)

H. Alt , B. Chazelle, E. Welzl (editors): Computational Geometry, Dagstuhl-Seminar-Report; 22, 07. 10.-11 .10.91 (9141)

F.J. Brandenburg , J. Berstel, D. Wotschke (editors): Trends and Applications in Formal Language Theory, Dagstuhl-Seminar-Report; 23, 14.10.-18.10.91 (9142)

H. Comon , H. Ganzinger, C. Kirchner, H. Kirchner, J.-L. Lassez , G. Smolka (editors):Theorem Proving and Logic Programming with Constraints, Dagstuhl-Seminar-Report; 24,21.10.-25.10.91 (9143)

H. Noltemeier, T. Ottmann, D. Wood (editors):Data Structures, Dagstuhl-Seminar-Report; 25, 4.11.-8.1 1.91 (9145)

A. Dress, M. Karpinski, M. Singer(editors):Efficient Interpolation Algorithms, Dagstuhl-Seminar-Report; 26, 2.-6.12.91 (9149)

B. Buchberger, J. Davenport, F. Schwarz (editors):Algorithms of Computeralgebra, Dagstuhl-Seminar-Report; 27, 16..-20.12.91 (9151)

K. Compton, J.E. Pin , W. Thomas (editors):Automata Theory: Infinite Computations, Dagstuhl-Seminar-Report; 28, 6.-10.1.92 (9202)

H. Langmaack, E. Neuhold, M. Paul (editors):Software Construction - Foundation and Application, Dagstuhl-Seminar-Fleport; 29, 13..-17.1.92(9203)

K. Ambos-Spies, S. Homer, U. Schöning (editors):Structure and Complexity Theory, Dagstuhl-Seminar-Report; 30, 3.�7.02.92 (9206)

B. Booß, W. Coy, J.-M. Pflüger (editors):Limits of Modelling with Programmed Machines, Dagstuhl-Seminar-Report; 31, 10.-14.2.92(9207)

K. Compton, J.E. Pin , W. Thomas (editors):Automata Theory: Infinite Computations, Dagstuhl-Seminar-Report; 28, 6.-10.1.92 (9202)

H. Langmaack, E. Neuhold, M. Paul (editors):Software Construction - Foundation and Application, Dagstuhl-Seminar-Report; 29, 13..-17.1.92(9203)

K. Ambos-Spies, S. Homer, U. Schöning (editors):Structure and Complexity Theory, Dagstuhl-Seminar-Report; 30, 3.-7.02.92 (9206)

B. Booß, W. Coy, J.-M. Pflüger (editors):Limits of Modelling with Programmed Machines, Dagstuhl-Seminar-Report; 31, 10.-14.2.92(9207)

N. Habermann, W.F. Tichy (editors):Future Directions in Software Engineering, Dagstuhl-Seminar-Rep°ort; 32; 17.02.-21.02.92 (9208)

Ft. Cole, E.W. Mayr, F. Meyer auf der Heide (editors):Parallel and Distributed Algorithms; Dagstuhl-Seminar-Report; 33; 02.03.-06.03.92 (9210)

P. Klint, T. Reps (Madison, Wisconsin), G. Snelting (editors):Programming Environments; Dagstuhl-Seminar-Fleport; 34; 09.03.-13.03.92 (9211)

H.-D. Ehrich, J.A. Goguen, A. Sernadas (editors): .Foundations of Information Systems Specification and Design; Dagstuhl-Seminar-Report; 35;16.03.-19.03.9 (9212)

W. Damm, Ch. Hankin, J. Hughes (editors):Functional Languages: -Compiler Technology and Parallelism; Dagstuhl-Seminar-Report; 36; 23.03.-27.03.92 (9213)

Th. Beth, W. Diffie, G.J. Simmons (editors): _System Security; Dagstuhl-Seminar-Report; 37; 30.03.-03.04.92 (9214)

C.A. Ellis, M. Jarke (editors):Distributed Cooperation in Integrated Information Systems; Dagstuhl-Seminar-Fleport; 38; 06.04.-08.04.92 (9215)

H. Comon, H. Ganzinger, C. Kirchner, H. Kirchner, J.-L. Lassez, G. Smolka (editors) : Theorem Proving and Logic Programming with Constraints, Dagstuhl-Seminar-Report; 24, 21 .10.-25.10.91 (9143)

H. Noltemeier, T. Ottmann, D. Wood (editors): Data Structures, Dagstuhl-Seminar-Report; 25, 4.11 .-8.11 .91 (9145)

A. Dress, M. Karpinski, M. Singer(editors): Efficient Interpolation Algorithms, Dagstuhl-Seminar-Report; 26, 2.-6.12.91 (9149)

B. Buchberger, J. Davenport, F. Schwarz (editors): Algorithms of Computeralgebra, Dagstuhl-Seminar-Report; 27, 16 .. -20. 12. 91 (9151)

K. Compton, J.E. Pin , w. Thomas (editors): Automata Theory: Infinite Computations, Dagstuhl-Seminar-Report; 28, 6. -10.1.92 (9202)

H. Langmaack, E. Neuhold, M. Paul (editors): Software Construction - Foundation and Application, Dagstuhl-Seminar-Report; 29, 13 .. -17. 1.92 (9203)

K. Ambos-Spies, S. Horner, U. SchOning (editors): Structure and Complexity Theory, Dagstuhl-Seminar-Report; 30, 3.-7.02.92 (9206)

B. BooB, W. Coy, J.-M. Pfluger (editors): Limits of Modelling with Programmed Machines, Dagstuhl-Seminar-Report; 31 , 10.-14.2.92 (9207)

K. Compton, J.E. Pin , W. Thomas (editors): Automata Theory: Infinite Computations, Dagstuhl-Seminar-Report; 28, 6. -10.1 .92 (9202)

H. Langmaack, E. Neuhold, M. Paul (editors): Software Construction - Foundation and Application, Dagstuhl-Seminar-Report; 29, 13 .. -17.1.92 (9203)

K. Ambos-Spies, S. Homer, U. SchOning (editors): Structure and Complexity Theory, Dagstuhl-Seminar-Report; 30, 3.-7.02.92 (9206)

B. BooB, W. Coy, J .-M. Pfluger (editors): Limits of Modelling with Programmed Machines, Dagstuhl-Seminar-Report; 31 , 10.-14.2.92 (9207)

N. Habermann, W.F. Tichy (editors): Future Directions in Software Engineering, Dagstuhl-Seminar-Report; 32; 17.02.-21.02.92 (9208)

A. Cole, E.W. Mayr, F. Meyer auf der Heide (editors): Parallel and Distributed Algorithms; Dagstuhl-Seminar-Report; 33; 02.03.-06.03.92 (9210)

P. Klint, T. Reps (Madison, Wisconsin), G. Snelting (editors): Programming Environments; Dagstuhl-Seminar-Report; 34; 09.03.-13.03.92 (9211)

H.-D. Ehrich, J.A. Goguen, A. Sernadas (editors): Foundations of Information Systems Specification and Design; Dagstuhl-Seminar-Report; 35; 16.03.-19.03.9 (9212)

W. Damm, Ch. Hankin, J. Hughes (editors): Functional Languages: Compiler Technology and Parallelism; Dagstuhl-Seminar-Report; 36; 23.03.-27.03.92 (9213)

Th. Beth, W. Diffie, G.J. Simmons (editors): System Security; Dagstuhl-Seminar-Report; 37; 30.03.-03.04.92 (9214)

C.A. Ellis, M. Jarke (editors): Distributed Cooperation in Integrated Information Systems; Dagstuhl-Seminar-Report; 38; 06.04.-08.04.92 (9215)