2. Automata, semigroups and rational sets

Jean-Eric Pin

Abstract

This is a survey paper on the connections between formal logic and the theory of automata. The logic we have in mind is the sequential calculus of Buchi, a system which allows to formalize properties of words. In this logic, there is a predicate for each letter and the unique extra non logical predicate is the relation symbol, which is interpreted as the usual order on the integers.

Several famous classes have been classied within this logic.

We shall briey review the main results concerning second order, which covers classes like PH, NP, P, etc. and then study in more detail the results concerning the monadic second order and the rst order logic.

1. Introduction

The aim of this paper is to survey the connections between formal logic and the theory of automata. The logic we have in mind is the \sequential calculus" of Buchi, a system which allows to formalize properties of words. A typical formula of this logic looks like

9x⁹y (x < y)^{^}(Rax)^{^}(Rby);

and can intuitively be interpreted on a worduby \there exist two integersx < ysuch that, inu, the letter in positionxis anaand the letter in positiony is ab".

Thus, ifA=^fa;b^gis the alphabet, the set of nite words satisfying our formula is the set of all words containing an occurrence ofafollowed (but not necessarily immediately) by an occurrence ofb, and can be described by the rational expressionAaAbA. Similarly, the set of innite (resp. biinnite) words satisfying the formula isAaAbA^!(resp. A^!~aAbA^!).

This example illustrates the logical point of view to dene a set of words, but there are other approaches to do so, including automata, rational expressions and semigroups. As we shall see in this paper, these various points of view complement each other and are, to some extent, equivalent. This leads to a remarkable theory and to numerous problems, some of which are still open.

In this paper, we focus on the logical point of view. We shall classify the most fundamental logical problems that arise in this framework into three categories:

(1) Descriptive power. Given a set S of sentences (such as rst order sentences, 1

formul, etc.,) characterize the sets of words that can be dened by a formula ofS. (2) Decision problems. Given a set S of sentences and a rational set of words X, is it decidable whetherX can be dened by a sentence ofS?

(3) Elementary equivalence. Given a set S of sentences, two words are said to be S- equivalent if they satisfy exactly the same sentences ofS. The problem is to describe these

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equivalence relations. The following question, proposed by Parikh, falls into this category:

\if two biinnite wordsuandv have the same factors, do they satisfy the same rst order formul of the theory of successor?"

But before we tackle such logical problems, we need to survey the three other ap- proaches: automata, semigroups and rational expressions.

2. Automata, semigroups and rational sets

In this section, we recall the basic denitions of the theory of nite automata needed in this article. Most of them are quite standard, but the reader might not be familiar with some of them, in particular those relative to biinnite words and to semigroups.

2.1. Words

LetA be a nite set called an alphabet, whose elements are letters. A nite word is a nite sequence of letters, that is, a functionufrom a nite set of the form^f0;1;2;:::;n^g intoA. If one putsu(i) =aifor 0in, the worduis usually denoted bya0a1 an, and the integer^ju^j=n+ 1 is the length ofu. The unique word of length 0 is the empty word, denoted by 1. An innite word (on the right) is a functionufrom^NintoA, usually denoted bya0a1a2 , whereu(i) =aifor alli^{2 N}. An innite word on the left is a functionufrom the set of non positive integers intoA, usually denoted by a 2a 1a0, where a i 2 A for all i^{2 N}. Finally, a biinnite word is a functionu from ^Zinto A, usually denoted by

a 2a 1a0a1a2 . Two biinnite wordsu:A ^{! Z} andv : A^{! Z} are shift equivalent if there exists an integer nsuch that, for alli^{2 Z}, v(i) =u(i+n). A bilateral word is an equivalence class for the shift equivalence. It is usually denoted by a 2a 1a0a1a2 . A word is either a nite word, an innite word on the right, an innite word on the left or a bilateral word.

Intuitively, the concatenation or product of two wordsuandvis the worduvobtained by writing u followed byv. More precisely, if u is nite and v is nite or innite on the right, thenuvis the word dened by

(uv)(i) =u(i) ifi <^ju^j v(i ^ju^j) if i ^ju^j

ifuis innite on the left andv is nite, thenuv is the word dened by (uv)(i) =u(i) ifi0

v(i 1) ifi >0

Finally, ifuis innite on the left andv is innite on the right, thenuvis the shift class of the biinnite worduv dened by

(uv)(i) =u(i) ifi0 v(i 1) ifi >0

We denote respectively byA, A⁺, A^N, A ^N andA^Z the set of all nite words, nite non- empty words, innite words (on the right), innite words on the left and bilateral words.

A word xis a factor of a wordw if there exist two words u and v (possibly empty) such thatw=uxv. A factor xof a biinnite wordu= a 2a 1a0a1a2 occurs \innitely often on the right" (respectively left) ofuif for everyn, x is a factor of the innite word anan+1an+2 (resp. an+2an+1an). A biinnite word u is recurrent if every factor ofu occurs innitely often on the right and on the left. Since these notions are invariant under shift, they can be extended to bilateral words.

For every nite wordu=a0a1 an, we set ~u=anan 1 a0. Similarly, for every innite word on the rightu, we denote by ~uthe innite word on the left dened by ~u n=un.

2.2. Rational sets

The rational operations are the three operations union, product and star, dened on the set of subsets ofAas follows

L1[L2=^fu^ju²L1 oru²L2g

(1) Union :

L1L2=^fu1u2ju12L1andu22L2g

(2) Product :

L=^fu1 unjn0 andu1;:::;un2L^g (3) Star :

It is also convenient to introduce the operator

L⁺=LL=^fu1 unjn >0 andu1;:::;un2L^g

The set of rational subsets ofAis the smallest set of subsets ofAcontaining the nite sets and closed under nite union, product and star. For instance, (a^[ab)ab^[(bab) denotes a rational set. The rational subsets ofA⁺are the rational subsets ofAthat do not contain the empty word.

It is possible to generalize the concept of rational sets to innite words as follows. First, the product can be extended toAA^N, by setting, forX Aand Y A^N,

XY =^fxy^jx²X andy²Y^g:

Next, we dene an innite iteration! by setting, for every subsetX ofA⁺ X^!=^fx0x1 j for alli0;xi2X^g

Equivalently,X^! is the set of innite words obtained by concatenating an innite sequence of words ofX. In particular, ifu=a0a1 an, we set

u^!=a0a1 ana0a1 ana0a1 ana0a1

By denition, a subset ofA^N is^N-rationalif and only if it can be written as a nite union of subsets of the formXY^! whereX andY are non-empty rational subsets ofA⁺.

For biinnite words, we rst extend the product toA ^NA^N, by setting, forX A ^N andY A^N,

XY =^fxy^jx²X andy²Y^g: Next, innite iteration on the left is dened by

X^!~ =^fu²A ^Nju~²( ~X)^!g

where ~X =^fx~^jx²X^g. By denition, a subset ofA^Z is^Z-rationalif and only if it can be written as a nite union of subsets of the formX^!~Y Z^! whereX,Y andZ are non-empty rational subsets ofA⁺.

Example 2.1.

The set of innite words on the alphabet^fa;b^ghaving only a nite number ofb's is given by the expression^fa;b^ga^!. The set of biinnite words on the alphabet^fa;b^g having only a nite number ofb's is given by the expressiona^!~fa;b^ga^!.

1 2 a

b a

b

2.3. Finite automata and recognizable sets

A nite (non deterministic) automaton is a triple ^A= (Q;A;E) where Qis a nite set (the set of states),Ais an alphabet, and E is a subset ofQAQ, called the set of transitions. Two transitions (p;a;q) and (p⁰;a⁰;q⁰) are consecutive ifq=p⁰. A path in ^Ais a nite sequence of consecutive transitions

e0= (q0;a0;q1); e1= (q1;a1;q2); ::: ; en = (qn;an;qn+1) also denoted

q0 a!0 q1 a1 ! q2 qn a!n qn+1

The stateq0 is the origin of the path, the stateqn+1is its end, and the wordx=a0a1 an

is its label.

An^N-path in^Ais a sequence pof consecutive transitions indexed by^N, e0= (q0;a0;q1); e1= (q1;a1;q2); :::

also denoted

q0 a!0 q1 a!1 q2

The stateq0 is the origin of the innite path and the innite word a0a1 is its label. A stateq occurs innitely often inpifqn =q for innitely manyn. Similarly, a^Z-path in ^A is a sequencepof consecutive transitions indexed by^Z. A stateqoccurs innitely often on the right(respectively on the left) in p ifqn =q for innitely many positive (respectively negative)n.

Example 2.2.

Let^A= (Q;A;E) where

Q=^f1;2^g;A=^fa;b^g andE=^f(1;a;1);(2;b;1);(1;a;2);(2;b;2)^g be the automaton represented below.

Figure 2.1.

Then (1;a;2)(2;b;2)(2;b;1)(1;a;2)(2;b;2)(2;b;1)(1;a;2)(2;b;2)(2;b;1) is an innite path of^A.

A nite Buchi automaton is a quintuple^A= (Q;A;E;I;F) where (1) (Q;A;E) is a nite automaton,

(2) I andF are subsets ofQ, called the set of initial and nal states, respectively.

A nite path in ^A is successful if its origin is in I and its end is in F. An ^N-path p is successfulif its origin is in Iand if some state ofF occurs innitely often inp. A^Z-pathp is successful if some state ofIoccurs innitely often on the left inpand if some state ofF occurs innitely often on the right inp.

The set of nite (respectively innite, bilateral) words recognized by ^A is the set, denotedL⁺(^A) (respectivelyL^N(^A),L^Z(^A)), of the labels of all successful nite (respectively

N-, ^Z-) paths of^A. A set of nite (respectively innite, bilateral) words X is recognizable if there exists a nite Buchi automaton^Asuch thatX =L⁺(^A) (respectivelyX =L^N(^A), X=L^Z(^A)).

1 2 a

b a

b

Example 2.3.

Let^Abe the Buchi automaton obtained from example 2.1 by takingI=^f1^g and F = ^f2^g. Initial states are represented by an incoming arrow and nal states by an arrow going out.

Figure 2.2.

A Buchi automaton.

Then L⁺(^A) = a^fa;b^g is the set of all nite words whose rst letter is an a, L^N(^A) = a(ab)^! is the set of innite words whose rst letter is an a and containing an innite number ofb's and L^Z(^A) = (ab)^!~(ab)^! is the set of innite words containing innitely manya's on the left and innitely manyb's on the right.

The relationship between rational and recognizable sets of nite words is given by the famous theorem of Kleene.

Theorem 2.1.

A subset of A is rational if and only if it is recognizable.

The counterpart of Kleene's theorem for innite words is due to Buchi [12] and for bilateral words to Nivat and Perrin [43].

Theorem 2.2.

A subset ofA^N(resp. A^Z) is^N-rational (resp. ^Z-rational) if and only if it is recognizable.

2.4. Semigroups

A semigroup is a set equipped with an internal associative operation which is usually written in a multiplicative form. A monoid is a semigroup with identity (usually denoted by 1). IfSis a semigroup,S¹denotes the monoid equal toSifShas an identity and toS^[f1^g otherwise. In the latter case, the multiplication on S is extended by setting s1 = 1s= s for every s ² S¹. An element e of a semigroup S is idempotent ife² = e. A zero is an element 0 such that, for every s ² S, s0 = 0s = 0. Given two semigroups, S and T, a (semigroup) morphism':S ^!T is an application ofS into T such that for allx;y² S, '(xy) ='(x)'(y). A semigroupS is a quotient of a semigroupT if there exists a surjective morphism from T onto S. A semigroup S divides a semigroup T if S is a quotient of a subsemigroup ofT. Division is a quasi-order on nite semigroups (up to an isomorphism).

A semiring is a setK equipped with two operations, called respectively addition and multiplication, denoted (s;t)^!s+tand (s;t)^!st, and an element, denoted 0, such that:

(1) (K;+;0) is a commutative monoid, (2) (K;:) is a semigroup,

(3) for all s;t1;t22K,s(t1+t2) =st1+st2 and (t1+t2)s=t1s+t2s, (4) for all s²K, 0s=s0 = 0.

Thus the only dierence with a ring is that inverses with respect to addition may not exist.

We denote by^Bthe boolean semiring dened by the following operations

+ 0 1

0 0 1

1 1 1

0 1

0 0 0

1 0 1

Given a semiringK, the set Kⁿⁿ of nnmatrices overK is naturally equipped with a structure of semiring. In particular,Kⁿⁿ is a monoid under multiplication dened by

(rs)i;j= ^X

1kn

ri;ksk;j

Let be a countable alphabet. Given two wordsu, vof ⁺ (resp. ), a semigroup (resp.

monoid)S satises the equationu=v if, for every semigroup (monoid) morphism'from ⁺ () into S, '(u) = '(v). For instance, a semigroup is commutative if and only if it satises the equationxy=yx. Let (un=vn)n2N be a sequence of equations. A semigroup (resp. monoid)S ultimately satisesthe sequence of equations (un =vn)n2N if there exists an integernS such that, for allnnS, S satises the equationun =vn. For instance, one can show that a nite monoid is a group if and only if it ultimately satises the equations (u^n!= 1)n>0.

Green's relationson a semigroup S are dened as follows. If sandt are elements of S, we note

s^Lt if there existx;y²S¹such thats=xtandt=ys, s^Rt if there existx;y²S¹such thats=txandt=sy, s^J t if there existx;y;u;v²S¹ such thats=xty andt=usv. s^Ht ifs^Rtands^Lt.

For nite semigroups, these four equivalence relations can be represented as follows. The elements of a given ^R-class (resp. ^L-class) are represented in a row (resp. column). The intersection of an^R-class and a^L-class is an^H-class. Each^J-class is a union of^R-classes (and also of ^L-classes). It is not obvious to see that this representation is consistent : it relies in particular on the non-trivial fact that, in nite semigroups, the relations ^R and

L commute. An idempotent is represented by a star. One can show that each ^H-class containing an idempotenteis a subsemigroup of S, which is in fact a group with identity e. Furthermore, all ^R-classes (resp. ^L-classes) of a given ^J-class have the same number of elements.

Each row is an^R-class

8<

:

Each column is an^L-class

z }| {

a1;a2 a3;a4 a5;a6

a7;a8 a9;a10 a11;a12

A

^J

-class.

A semigroupS is ^L-trivial (resp. ^R-trivial,^J-trivial,^H-trivial) if two element of S which are^L-equivalent (resp. ^R-equivalent,^J-equivalent,^H-equivalent) are equal.

1 2 a

b a

a,b

2.5. Semigroups and recognizable sets

In this section, we turn to a more algebraic denition of the recognizable sets, using semigroups in place of automata. Although this denition is more abstract than the de- nition using automata, it is more suitable to handle the ne structure of recognizable sets.

Indeed, as discovered by Eilenberg [21], semigroups provide a powerful and systematic tool to classify recognizable sets. We will see in particular that most of the classes of recognizable sets associated with standard classes of formul (rst order, existential, etc.) admit some simple algebraic characterizations.

The abstract denition of recognizable sets is based on the following observation.

Let^A= (Q;A;E;I;F) be a nite automaton. To each word u²A, there corresponds a boolean square matrix of size Card(Q), denoted (u), and dened by

(u)p;q=ⁿ1 if there exists a path fromptoqwith label u 0 otherwise

Example 2.4.

Let^A= (Q;A;E;I;F) be the automaton represented below

Figure 2.3.

A non deterministic automaton.

ThenQ=^f1;2^g, A=^fa;b^g andE =^f(1;a;1);(1;a;2);(2;a;2);(2;b;1);(2;b;2)^g, I=^f1^g, F=^f2^g, whence

(a) =1 1 0 1

(b) =0 0 1 1

(aa) =(a) (ab) =1 1

1 1

(ba) =(bb) =(b)

It is not dicult to see that is a semigroup morphism from A⁺ into the multiplicative semigroup of square boolean matrices of size Card(Q). Furthermore, a worduis recognized by ^A if and only if there exists a successful path from 1 to 2 with label u, that is, if (u)1;2= 1. Therefore, a word is recognized by^Aif and only if(u)1;2= 1. The semigroup (A⁺) =^{f 0 0}_{1 1}; ^{1 1}_{0 1}; ^{1 1}_{1 1}^gis called the transition semigroup of^A.

Let us now give the formal denitions. Let':A^{+ !}Sbe a semigroup morphism. A subsetX ofA⁺is recognized by'if there exists a subsetP ofSsuch thatX =' ¹(P). As shown by the previous example, a set recognized by a nite automaton is recognized by the transition semigroup of this automaton. Conversely, given a nite semigroup recognizing a subsetXofA⁺, one can build a nite automaton recognizingX. Therefore, the two notions of recognizable sets (by nite automata and by nite semigroups) are equivalent:

Theorem 2.3.

A subset of A⁺ is recognizable if and only if it is recognized by a nite semigroup.

LetX be a recognizable set ofA⁺. Amongst the nite semigroups that recognizeX, there is a minimal one (with respect to division). This nite semigroup is called the syntactic semigroupofX. It can be dened directly as the quotient of A⁺ under the congruenceX

dened byuX v if and only if, for every x;y²A, xuy²X ^,xvy²X. It is also equal to the transition semigroup of the minimal automaton of^A. See [54] for more details.

2.6.

^!

-semigroups

It is possible to extend the previous results to innite words by replacing semigroups by!-semigroups, which are, basically, algebras in which innite products are dened. Al- though these algebras do not have a nitary signature, standard results on algebras still hold in this case. In particular,A¹ appears to be the free algebra on the setAand recognizable sets can be dened, as before, as the sets recognized by nite algebras. However, a problem arises since nite algebras have an innitary signature and thus are not really nite! This problem can be solved by a Ramsey type argument showing that the structure of these nite algebras can be totally determined by only two operations of nite signature. This denes a new type of algebra of nite signature, the Wilke algebras, that suce to deal with innite products dened on nite sets. ⁽⁾

We now come to the precise denitions. An!-semigroupis an algebraS = (Sf;S!) equipped with the following operations:

| A binary operation dened onSf and denoted multiplicatively,

| A mapping Sf S! ! S!, called mixed product, that associates to each couple (s;t)²SfS! an element ofS! denotedst,

| A mapping:Sf^N!S!, called innite product

These three operations should satisfy the following properties : (1) Sf, equipped with the binary operation, is a semigroup, (2) for everys;t²Sf and for everyu²S!,s(tu) = (st)u,

(3) for every increasing sequence (kn)n>0and for all (sn)n2N2Sf^N,

(s0s1 sk1 1;sk1sk1+1 sk2 1; :::) =(s0;s1;s2;:::) (4) for everys²Sf and for every (sn)n2N2Sf^N

s(s0;s1;s2;:::) =(s;s0;s1;s2;:::)

Conditions (1) and (2) can be thought of as an extension of associativity. Conditions (3) et (4) show that one can replace the notation (s0;s1;s2;:::) without ambiguity by the notation s0s1s2 . We shall use this simplied notation in the sequel. Intuitively, an

!-semigroup is a sort of semigroup in which innite products are dened.

Example 2.5.

We denote by A¹ the !-semigroup (A⁺;A^N) equipped with the usual concatenation product.

() Actually, the chronology is a little bit dierent. Ramsey type arguments have been used for a long time in semigroup theory [33], the Wilke algebras were introduced by Wilke in [83] under the name ofbinoids to clarify the approach of Arnold [4], Pecuchet [45] and Perrin [46] and the idea of using innite products on semigroups came last [51].

Given two!-semigroups S= (Sf;S!) andT = (Tf;T!), a morphism of!-semigroup is a couple'= ('f;'!) consisting of a semigroup morphism'f :Sf !Tf and of a mapping '! :S! !T! preserving the mixed product and the innite product: for every sequence (sn)n2N of elements ofSf,

'!(s0s1s2 ) ='f(s0)'f(s1)'f(s2) and for everys²Sf,t²S!,

'f(s)'!(t) ='!(st)

In the sequel, we shall omit the subscripts, and use the simplied notation'instead of'f

and'!.

Algebraic concepts like congruence,!-subsemigroup, quotient and division are easily adapted to !-semigroups. The semigroup A⁺ is called the free semigroup on the set A because it satises the following property (which denes free objects in the general setting of category theory): every map fromAinto a semigroupS can be extended in a unique way into a semigroup morphism fromA⁺ intoS. Similarly, it is not dicult to see that the free

!-semigroup on (A;^;) is the!-semigroupA¹.

A key result is that whenS is nite, the innite product is totally determined by the elements of the forms^!=sss , according to the following result

Theorem 2.4.

(Wilke) LetSf be a nite semigroup and let S! be a nite set. Suppose that there exists a mixed product Sf S! ! S! and a map from Sf into S!, denoted s^!s^!, satisfying, for every s;t²Sf, the equations

s(ts)^!= (st)^!

(sⁿ)^!=s^! for everyn >0

Then the couple S = (Sf;S!) can be equipped, in a unique way, with a structure of !- semigroup such that for everys²S, the productsss is equal tos^!.

This is a non trivial result, based on a consequence of Ramsey's theorem which is worth mentioning:

Theorem 2.5.

Let':A^{+ !}S be a morphism fromA⁺ into a nite semigroup S. For every innite word u² A^N, there exist a couple (s;e) of elements ofS such that se = s, e²=e, and a factorizationu=u0u1 ofuas a product of words ofA⁺such that'(u0) =s and'(un) =efor everyn >0.

A morphism of!-semigroups ' : A^{1 !} S recognizes a subset X of A^N if, there exists a subsetP ofS!such thatX=' ¹(P). By extension, a!-semigroupS recognizesX if there exists a morphism of!-semigroup':A^1!S that recognizes X. As for nite words, the following result holds:

Theorem 2.6.

A subset of A^N is recognizable if and only if it is recognized by a nite

!-semigroup.

We now give the construction to pass from a nite (Buchi) automaton to a nite

!-semigroup. This construction is much more involved than the corresponding construction for nite words.

Given a nite Buchi automaton^A= (Q;A;E;I;F) recognizing a subset ofX ofA^N, we would like to obtain a nite!-semigroup recognizingX. Our construction makes use of the semiringk=^{f 1};0;1^gin which addition is the maximum for the ordering ¹<0<1 and multiplication is given in the following table

1 0 1

1 1 1 1

0 ¹ 0 1

1 ¹ 1 1

To each lettera²Ais associated a matrix (a) with entries ink dened by (a)p;q =

8<

:

1 if (p;a;q)²=E

0 if (p;a;q)²E andp =²F andq =²F 1 if (p;a;q)²E and (p²F or q²F)

We have already used a similar technique to encode automata, but now we discriminate paths that go through a nal state. We would like to extendinto a morphism of!-semigroup. It is easy to extend into a semigroup morphism fromA⁺ into the multiplicative semigroup ofQQ-matrices overk. Ifuis a nite word, one gets

(u)p;q =

8>

>>

<

>>

>:

1 if there exists no path of labelufrom ptoq, 1 if there exists a path fromptoq with labelu

going through a nal state,

0 if there exists a path fromptoq with labelu but no such path goes through a nal state

However, trouble arises when one tries to equipk^QQ with a structure of!-semigroup. The solution consists in coding innite paths not by square matrices, but by column matrices, in such a way that each coecient(u)p codes the existence of an innite path of labelu starting atp.

LetS= (Sf;S!) where Sf =k^QQ is the set of square matrices of size CardQwith entries inkandS!=k^Q is the set of column matrices with entries in^{f 1};1^g.

In order to dene the operation!on square matrices, we need a convenient denition.

Ifsis a matrix ofSf, we call innites-path starting atpa sequencep=p0;p1;:::of elements ofQsuch that, for 0in 1,spi;pi+1 6= ¹.

Thes-path is successful ifspi;pi+1= 1 for an innite number of coecients. Thens^! is the element ofS!dened, for everyp²Q, by

s^!p =ⁿ1 if there exists a successfuls-path of originp,

1 otherwise

Note that the coecients of this matrix can be eectively computed. Indeed, computings^!p

amounts to check the existence of circuits containing a given edge in a nite graph. Then one can verify thatS, equipped with these operations, is a!-semigroup. Furthermore, we have the following result

Proposition 2.7.

The morphism of!-semigroup fromA¹intoSinduced byrecognizes the setL^N(^A).

The!-semigroup(A¹) is called the!-semigroup associated with^A.

Example 2.6.

Let X = (a^fb;c^{g[ f}b^g)^!. This set is recognized by the Buchi automaton represented below:

1 2 a, b

b, c a

b, c

Figure 2.4.

The!-semigroup associated with this automaton contains 9 elements a= 1 1

1 1

b=1 ¹ 1 0

c= 1¹ 0¹

ba=1 1 1 1

ca= 1¹ 1¹

a^!= 1

1

b^!=1 1

c^!= ¹₁ (ca)^!= 1¹

It is dened by the following relations:

a²=a ab=a ac=a b²=b bc=c cb=c c²=c c^!= 0 (ba)^!=b^! aa^!=a^! ab^!=a^! a(ca)^!=a^! ba^!=b^! bb^!=b^! b(ca)^!= (ca)^! ca^!= (ca)^! cb^!= (ca)^! c(ca)^!= (ca)^! As in the case of nite words, there exists a minimal nite !-semigroup (with respect to division) recognizing a given recognizable setX. This!-semigroup is called the syntactic!- semigroupofX. It can also be dened directly as the quotient ofA¹under the congruence of !-semigroup X dened on A⁺ by uX v if and only if, for every x;y ² A and for everyz²A⁺,

xuyz^!2X ⁽⁾xvyz^!2X

x(uy)^!2X ⁽⁾x(vy)^!2X ⁽²:1) and onA^NbyuXv if and only if, for everyx²A,

xu²X ⁽⁾xv²X (2:2)

One can also compute the syntactic!-semigroup of a recognizable set given a nite Buchi automaton^ArecognizingX. One rst computes the nite!-semigroupS associated with

Aand the imageP ofX in S. Then the syntactic!-semigroup is the quotient ofS by the congruence P dened on Sf by u P v if and only if, for every r;s ²S¹_f and for every t²Sf

rust^!2P ⁽⁾rvst^!2P r(us)^!2P ⁽⁾r(vs)^!2P and onS! byuP v if and only if, for everyr²Sf¹,

ru²P ⁽⁾rv²P

This provides an algorithm to compute the syntactic!-semigroup of a recognizable set.

For bilateral words, there is a corresponding notion of-semigroup, obtained by con- sidering innite products indexed by intervals of ^Z instead of !-products. Formally, a -semigroup is a multisorted algebra S = (Sf;S!;S!~;S) where Sf, S!, S!~, S are sets intuitively representing the nite products, the innite products on the right, the innite product on the left and the biinnite products, equipped with a productSfSf !Sf and three mixed products Sf S! ! S!, S!~ S! ! S, and S!~ Sf ! S!~ satisfying the following axioms:

(1) For all s;t;u²Sf, (st)u=s(tu), (2) For all s;t²Sf,u²S!, (st)u=s(tu), (3) For all s²S!~,t;u²Sf, (st)u=s(tu), (4) For all s²S!~,t²Sf, u²S!, (st)u=s(tu).

Theorem 2.4 can be adapted to-semigroups as follows

Theorem 2.8.

LetSf be a nite semigroup and letS!,S!~ andS be nite sets. Suppose that there exist three mixed productsSfS!!S!,S!~Sf!S!~ andS!~S!!S and two maps Sf ! S!, denoted s ^!s^!, and Sf !S!~ denoted s ^!s^!~ satisfying, for every s;t²Sf, the equations

s(ts)^!= (st)^! (ts)^!~= (st)^!~s

(sⁿ)^!=s^! (sⁿ)^!~=s^!~ for everyn >0

Then the setS = (Sf;S!;S!~;S) can be equipped, in a unique way, with a structure of -semigroup such that for every s²S, the productsss is equal tos^! and the product

sssis equal tos^!~.

Then one can dene the syntactic -semigroup of a recognizable set. As in the case of innite words, one can eectively compute this nite object, for any given nite Buchi automaton.

3. The sequential calculus

We now come to the major topic of this article, the denition of sets of words by logical formul.

3.1. Denitions

For each lettera²A, letRa denote a unary predicate. In the sequential calculus, a worduis represented as a structure of the form

Dom(u);(Ra)a2A;S;<; where

Dom(u) =

8<

:

f0;:::;^ju^j 1^g ifuis a non empty nite word,

N ifuis an innite word,

Z ifuis a biinnite word, and where

Ra =^fi²Dom(u)^ju(i) =a^g:

Thus, if u = abbaab, then Dom(u) = ^f0;1;:::;5^g, Ra = ^f0;3;4^g and Rb = ^f1;2;5^g. If u= (aba)^!, then

Ra =^fn^{2 N j}n0 mod 3 orn2 mod 3^g and Rb =^fn^{2 N j}n1 mod 3^g:

We shall also use two other non-logical symbols, the binary relation symbols<andS, which are interpreted as the usual order and as the successor relation on Dom(u).

Now terms, atomic formul, rst order formul and second order formul are formed in the usual way. In rst order logic, all variables are element variables, while in second order logic, relation variables are allowed. Monadic second order logic is the restriction of second order logic in which only element variables and set variables are allowed. Weak monadic second order logic is a variant of monadic second order logic in which set variables are restricted to range over nite subsets of the domain.

We shall use the notations F1(<), MF2(<), WMF2(<), F2(<), for the set of rst order, monadic second order, weak monadic second order and second order formul with signature^f<;(Ra)a2Ag. Similarly,F1(S),MF2(S),WMF2(S),F2(S), denote the same sets of formul with signature^fS;(Ra)a2Ag. If we need to specify the domain, we shall use the notationsF1(^N;S), F1(^Z;S), etc. In fact, the distinction between the signatures S and<

is irrelevant for second order theories according to the following lemma.

Lemma 3.1.

The relation S can be expressed in F1(<), and the relation < can be expressed inWMF2(S) and in MF2(S).

Proof.

We give the proof for the monadic second order theory in ^Z and leave the other cases as exercises. First,S(i;j) can be dened by the formula

(i < j)^{^ 8}k(i < k)^!((j=k)^_(j < k))

which says thatj =i+ 1 ifi is smaller thanj and if there is no element betweeniand j. Conversely,i < j can be expressed inMF2(S) as follows:

9X ⁸x⁸y (x²X)^{^}S(x;y)^!(y²X)^{^}(j²X)^{^}(i =²X)

!

which intuitively means there is an interval of the form [k;+¹[ containingj but noti. The relation<can also be expressed inWMF2(S) (left as an exercise), but not inF1(S).

To each sentence', one associates the sets of words that satisfy ': L⁺(') =^fu²A^+jusatises'^g L^N(') =^fu²A^Njusatises'^g L^Z(') =^fu²A^Zjusatises'^g

This last denition requires a justication: normally, one should dene the set of innite words satisfying a given formula. But since this set is always shift invariant, it naturally denes a set of bilateral words.

Example 3.1.

Let'=⁹i Rai. Then

L⁺(') =AaA; L^N(') =AaA^!; L^Z(') =A^!~aA^!:

Example 3.2.

Let'=⁹min ⁸i mini^{^} Ramin. Then, according to intuition, min is interpreted as the minimum of the domain and thus

L⁺(') =aA; L^N(') =aA^!; L^Z(') =^;:

Example 3.3.

Let

'=⁹min ⁸i(mini) ^{^ 9}max ⁸i(imax)

^ 9X min²X^{^}max²= X^{^ 8}i⁸j (S(i;j)^!(i²X ^$j =²X)) Again, min and max have their natural interpretation and the set X can be informally described as a set containing the minimum of the domain, not containing the maximum and such thati²X if and only if i+ 1²= X. ThusX is interpreted as the empty set over the domains^Nand ^Z(because these domains don't have any maximum). On a nite domain, X represents a set of even numbers of the form ^f0;2;4;6:::^g. Since X does not contain the maximum (that is, ^ju^j 1), this number is odd, and thus ^ju^j is even. It follows that L^N(') =^;,L^Z(') =^; andL⁺(') is the set of nite words of even length.

Two sentences'and are said to be equivalent (resp. ^N-equivalent,^Z-equivalent) if L⁺(') =L⁺( ) (resp. L^N(') =L^N( ),L^Z(') =L^Z( )).

3.2. Full second order

As we have already pointed out, there is no dierence between the second order theories of<and ofS. The full second order theory denes the polynomial hierarchy. This hierarchy is often dened in terms of Turing machines with oracles, but a direct denition is also possible. LetA be an alphabet. A polynomial relation on A is a relation which is accepted in polynomial time by a deterministic Turing machine on the alphabetA. For every k >0, we dene the set ^Pk(A) as follows. A subset Lof A belongs to ^Pk(A) if and only if there existk polynomials p1, :::, pk and a polynomial (k+ 1)-ary relationR such that

x²L ^{() 9}y12A such that^jy1j p1(^jx^j)

8y22A such that^jy2j p2(^jx^j)

9y32A such that^jy3j p3(^jx^j) ... Qyk2A such that^jykj pk(^jx^j)

R(x;y1;:::;yk)

where Q is an universal quantier if k is even and an existential quantier if k is odd.

In particular, ^P0 = P, the class of sets recognized by deterministic Turing machines in polynomial time, and ^P1 =NP, the class of sets recognized by non deterministic Turing machines in polynomial time. Finally, the set ^P =^Sk0^Pk is the polynomial hierarchy (sometimes also denoted byPH).

Theorem 3.2.

(Stockmeyer 1977) A subset ofA⁺ is denable in F2(<) if and only if it belongs to the classPH.

Restricting to the set 1F2(<) of existential (second order) formul denes the class NP.

Theorem 3.3.

(Fagin 1974) A subset ofA⁺is denable in 1F2(<) if and only if it belongs to the classNP.

The question whether the class NP is strictly contained in PH is a famous open problem of complexity theory. Therefore it is not known whether existential second order formul are less expressive than full second order formul. At this point it is tempting to give a logical description of other complexity classes. The four results below give the avor of this theory. The reader is referred to the original article of Immerman [31] for more information on this topic. LetL (resp. NL, PSPACE) be the class of sets accepted by a deterministic Turing machine in logarithmic space (resp. by a non deterministic Turing machine in logarithmic space, by a deterministic Turing machine in polynomial space). The following inclusions are well-known:

LNLP NP PHPSPACE

Denote byRTCthe operator that computes the reexive and transitive closure of a relation.

For instance, ifS(x;y) is the successor relation,RTC[S(x;y)](u;v) is the relationuvand RTC[S(x1;x⁰1)^{^}S(x2;x⁰2)](0;y;x;z) is the relationx+y =z. Given a set of formulF, we denote byF+RTC the set of formul expressible usingF plus the operatorRTC.

Theorem 3.4.

(Immerman 1987) A subset ofA⁺ is denable inF2(<) +RTC if and only if it belongs to the classPSPACE.

The characterization of the class NL, also due to Immerman, was originally stated in a slightly more complicated way, but can be simplied sinceNLis closed under complement [32,72].

Theorem 3.5.

(Immerman 1987) A subset ofA⁺ is denable inF1(S) +RTC if and only if it belongs to the classNL.

To obtain the classL, we need a deterministic version of the operatorRTC. Given a rst order binary relationR, the deterministic part ofRis the relationRd dened by

Rd(x;y)R(x;y)^{^}(⁸z R(x;z) =⁾z=y)

Now dene DRTC(R) as the reexive transitive closure of Rd. For instance, S = Sd on words since every position has at most one successor, so that<=DRTC[S(x;y)]. Given a set of formulF, we denote byF+DRTC the set of formul expressible usingF plus the operatorDRTC.

Theorem 3.6.

(Immerman 1987, Vardi) A subset ofA⁺ is denable inF1(S) +DRTC if and only if it belongs to the classL.

We conclude this section with the classP, for which we introduce the least xpoint operator LFP. Given a monotone operator'on relations (that is, such thatRS implies'(R) '(S)), dene

LFP(') = ^\

'(R)=R

R

For instance, if '(R) = (x=y)^_(⁹z(S(x;z)^{^}R(z;y)), then LFP(')[R] is the reexive and transitive closure ofR. In particular, LFP is more powerful than RTC. As for RTC andDRTC, given a set of formulF, we denote byF+LFP the set of formul expressible usingF plus the operatorLFP.

Theorem 3.7.

(Immerman 1987) A subset ofA⁺ is denable inF1(S) +LFP if and only if it belongs to the classP.

These results are stated when formul are interpreted on nite words only. For an extension of these results to nite structures see Immerman [31] and the beautiful survey of Fagin [25].

3.3. Monadic second order

Considering only monadic second order formul on words is a much more drastic restriction.

Theorem 3.8.

(Buchi [12], Elgot [22]) Let L be a subset of A⁺ (resp. A^N, A^Z). The following conditions are equivalent:

(1) Lis denable inWMF2(<), (2) Lis denable inMF2(<),

(3) Lis a rational (resp. ^N-rational,^Z-rational) set.

Furthermore there exists an eective algorithm to pass from a formula to a rational (resp.^N- rational,^Z-rational) expression and vice-versa.

It is fair to say that this eective algorithm is not very ecient. Indeed, it is shown in [41] that there is no elementary time bounded decision procedure for deciding whether a given sentence ofWMF2(S) is true or not.

Corollary 3.9.

The theories MF2(^N;S), MF2(^Z;S), MF2(^N;<), MF2(^Z;<), and the corresponding weak monadic second order theories are decidable.

Proof.

We give the proof forMF2(^N;<), but the other cases are similar. Let'be a sentence ofMF2(^N;<). Take an empty alphabet. Then theorem 3.8 gives an eective algorithm to computeL^N('). Then'is true on^Nif and only ifL^N(')⁶=^;.

It is amusing to obtain Presburger's result from Buchi's theorem. The nice paper of Hodgson [28] contains other interesting decidability results derived from Buchi's theorem.

Corollary 3.10.

(Presburger) The theoryF1(^N;+;0) is decidable.

Proof.

The idea is to code the integers by nite subsets of ^N and then to interpret the addition within weak monadic second order logic. If

n= 2ⁱ¹+ 2ⁱ²+ + 2^ik withi1< i2< ::: < ik

is the binary expansion of an integern, then nis coded by the set ^fi1;i2; ::: ;ikg. Thus the coding of an integer n is the set of positions of the bits 1 in the binary expansion of n. For instance, 13, whose binary expansion is 1101, is coded by the set ^f3;2;0^g, and 20, whose binary expansion is 10100 is coded by^f4;2^g.

Now, let x, y, andz be three integers, coded by the setsX,Y, andZ, respectively.

Then one can code the equalityx+y=z by introducing a second order variableR, which

codes the positions of carries in the addition. For instance, the addition of the binary expansions of 13 and 20 is represented below.

R=^f5;4;3^g 1 1 1 0 0 0 X=^f4;2^g 1 0 1 0 0 Y=^f3;2;0^g + 0 1 1 0 1 Z=^f5;0^g 1 0 0 0 0 1

Now,ibelongs toZ if and only if one or three of the setsX,Y, andRcontainsi. Similarly, i+ 1²Rif and only ifibelongs to at least two of the three setsX,Y, andR. This can be easily coded into the following formula:

9R(0²= R)^{^ 8}x⁸y S(x;y)^!

h(y²R)^$ (x²X)^{^}(x²Y)^_ (x²X)^{^}(x²R)^_ (x²R)^{^}(x²Y)^{i^}

(x²Z)^$h (x²R)^{^}(x²X)^{^}(x²Y)^_ (x²R)^{^}(x =²X)^{^}(x =²Y)^_ (x =²R)^{^}(x²X)^{^}(x =²Y)^_ (x =²R)^{^}(x =²X)^{^}(x²Y)ⁱ We now arrive to rst order theory. There, the theory branches into two dierent directions.

We rst consider the rst order theory of the linear ordering and later (section 3.6) the rst order theory of successor.

3.4. First order theory of the linear ordering

In this section, we present the results which connect rst order logic, star-free sets and aperiodic semigroups. These statements, given the proper denitions, hold for nite, innite and bilateral words. They summarize a series of deep results of Schutzenberger [58], McNaughton and Papert [40], Ladner [34], Thomas [74], Perrin [47] and Perrin-Pin [50]. We rst dene the key concepts of this statement : star-free sets and aperiodic semigroups.

Boolean operations comprise union, intersection, complementation and set dierence.

It can be shown that the rational subsets ofA are closed under nite boolean operations.

The set of star-free subsets ofA is the smallest set of subsets ofA containing the nite sets and closed under nite boolean operations and product. For instance, ifX^c denotes the complement of a setX, one hasA=^;c, showing that the set A is star-free, since the empty set is a nite set. More generally, ifB is a subset of the alphabet A, the setB is also star-free sinceB is the complement of the set of words that contain at least one letter ofB⁰=AⁿB. This leads to the following star-free expression

B=AⁿA(AⁿB)A= (^;c(AⁿB)^;c)^c

IfA =^fa;b^g, the set (ab) is star-free, since (ab) is the set of words not beginning with b, not nishing byaand containing neither the factoraa, nor the factorbb. This gives the star-free expression

(ab)=Aⁿ bA^[Aa^[AaaA^[AbbA=^;cn b^{;c[ ;c}a^{[ ;c}aa^{;c[ ;c}bb^;c Readers may convince themselves that the sets ^fab;ba^g and a(ab)b are also star-free but may also wonder whether there exist any non star-free rational sets. In fact, there are

1 2 b

a

some, for instance the sets (aa)and^fb;aba^g, or similar examples that can be derived from the algebraic approach presented below.

The star-free subsets of A^N (resp A^Z) are dened in a way similar to the rational subsets ofA^N (respA^Z). The set of star-free subsets of A^N is the smallest set^S of subsets ofA^Nclosed under nite boolean operations and such that ifX is a star-free subset ofA⁺ andY ^{2 S}, thenXY ^{2 S}. The set of star-free subsets ofA^Zis the smallest set^Sof subsets ofA^Z closed under nite boolean operations and such that ifY is a star-free subset of A⁺ andX,Z are star-free subsets ofA^N, then ~XY Z^{2 S}.

A nite semigroup S is aperiodic if and only if it ultimately satises the equations xⁿ=xⁿ⁺¹. This notion is in some sense \orthogonal" to the notion of groups. Indeed, one can show that a semigroup is aperiodic if and only if it is^H-trivial, or, equivalently, if it contains no non-trivial subgroup. Note that any nite monoidMsatises an equation of the formx^n+p=xⁿ. The two extremal cases are n= 0 and p= 1. The rst case corresponds to groups (ifM is a group, then M satises the equation x^Card(M) = 1), the second case to aperiodic monoids. The connection between aperiodic semigroups and star-free sets was established by Schutzenberger [58] for nite words and by Perrin [47,48] for innite and bilateral words:

Theorem 3.11.

A recognizable subset ofA⁺ (resp. A^N,A^Z) is star-free if and only if its syntactic semigroup (resp. !-semigroup,-semigroup) is aperiodic.

Example 3.4.

Let A=^fa;b^g and consider the set L= (ab)⁺. Its minimal automaton is represented below:

Figure 3.1.

The minimal automaton of (ab)⁺

The transitions and the relations dening the syntactic semigroupS ofL are given in the following tables

a 1 22

b 1

aaab 1 ba 2

a²=b²= 0 aba=a bab=b

Sincea² =a³, b² =b³, (ab)² =ab and (ba)² =ba, S is aperiodic and thus L is star-free.

Consider now the setL⁰ = (aa)⁺. Its minimal automaton is represented below: