Rational transductions were one of the subjects of Maurice Nivat’s thesis, published in 1968 . They are functions from a free monoid into the set of subsets of another free monoid, whose graph is a rational subset of the product monoid; this graph is then called a rational relation. See the books of Eilenberg , Berstel  and Sakarovitch  for further reading.
Rational transductions are very useful tools in many domains. For example, decoding a finite code is a rational transduction which is moreover functional: these functions were studied first by Schützenberger and are a special case of rational functions
which are functions whose graph is a rational relation. As another example, rational transductions serve to classify context-free languages, a point of view initiated by Nivat, using the concept of abstract family of languages; see also Berstel’s book .
Discrete group theory is certainly one of the areas where rational transductions showed their significance and relevance. We claim however that they were not yet employed with their full strength. We explain why.
A Kleene-like result shows the equivalence between rational relations and relations recognized by two-tape automata. Automatic groups and semigroups, together with their asynchronous versions are defined via formal structures whose components are relations recognized by special types of two-tapes automata, thus of special rational relations. The main idea of our contribution is to substitute arbitrary rational relations for the restricted type of rational relations of the literature.
This leads to the definition of a quasi-automatic semigroup (or group). Such a semigroup has a finite set of generators, and there exist rational subsets , , , such that, being the canonical homomorphism , one has properties (1), (2) and (3) given in Section 3.1.
We give an brief outline of the contents of our work.
We begin by verifying that the monoid version of this definition is compatible with its semigroup version. We compare quasi-automatic semigroups to previouly considered classes. We show that they contain strictly the rational semigroups of Sakarovitch. They contain also strictly the automatic semigroups of Campbell et al. They contain the asynchronously automatic semigroups of Wei et al., and we conjecture that this inclusion is strict, although we have no example proving it.
Automatic groups are defined using a set of generators; it is then shown that the definition is independent of the chosen set of generators; the same holds for the asynchronous groups. For semigroups however, automaticity depends on the set of generators (and it is unknown for asynchronous automatic semigroups). We show that our notion of quasi-automatic semigroups is independent of the generators.
We show that one may compute in exponential time a representative for each word. We show that the word problem for a quasi-automatic semigroup is decidable in exponential time. We prove a weak Lipschitz property: roughly speaking, if two words are at distance at most 1 when viewed in the semigroup, then their prefixes, viewed in the semigroup, are at bounded distance; the strong form of this property, which is true for automatic semigroups, is when each prefix of is close to each prefix of of the same length: for groups, this property characterizes automaticity.
We show that if a quasi-automatic semigroup is graded, then it is automatic. Finally, we give two results on quasi-automatic groups. We show that a quasi-automatic group is finitely presented and has an exponential isoperimetric inequality: this means that the group is the quotient of a free group by a normal subgroup which is finitely generated (as normal subgroup) and that each element of it, of reduced length , is a product of at most conjugates of generators of the normal subgroup. The last result is that the weak Lipschitz property for groups implies quasi-automaticity.
Several open questions are given at the end of the article.
A word about the proofs: they are all based on the theory of rational relations (or transductions), as one may find it in Berstel’s book. We use several times Nivat’s bimorphism theorem111The third author remembers very well lectures on transductions and in particular on this result, by Maurice Nivat, in 1974 at the University of Paris 7., and the composition theorem of Elgot and Mezei (which asserts that the composition of rational transductions is a rational tranduction). For complexity matters, we use a construction of Arnold and Latteux which is an effective version of the combination of two results: the fact that each rational relation contains a rational function with the same domain, due to Eilenberg, and the fact that a rational is equal to the product of a left and of a right sequential function, due to Elgot and Mezei.
Let be semigroup. A subset of is rational if it is obtained from finite subsets of by applying the operations of union , product and subsemigroup generation .
If turns out to be a monoid, then in the previous definition, one may replace subsemigroup generation by submonoid generation . This is because and .
In the sequel, we are mainly interested in the case where is a finitely generated free semigoup, a finitely generated free monoid or a direct product of such semigroups.
Rationality of a subset is preserved under direct image by a semigroup homomorphism, see  Proposition III.2.2.
A homomorphism from a free monoid to another one is called alphabetic if it sends each generator onto a generator or onto the empty word.
A rational subset of a free monoid is also called a rational language, whereas a rational subset of a product of two free monoids is called a rational relation, or a rational transduction. The word “transduction” refers to the fact that a relation , subset of , may be seen as a function from into the set of subsets of . More precisely, the transduction associated to a relation is the function . A transduction extends naturally to a function from the subsets of to the subsets of , which preserves arbitrary union.
Each rational subset of is a rational subset of , and similarly for rational subsets of .
Moreover, the intersection of and of any rational subset of is a rational subset of . Similarly, the intersection of and of any rational subset of is a rational subset of .
A subset of a semigroup is called recognizable if it is the inverse image under some homomorphism of into a finite semigroup of a subset of the latter. Recognizable subsets of are closed under Boolean operations.
By Kleene’s theorem, recognizable subsets of a free monoid ( finite) coincide with rational subsets. It follows that the set of rational languages is closed under Boolean operations. Another consequence is that in a finitely generated semigroup, the intersection of a recognizable subset and of a rational subset is rational, see  Proposition III.2.6; we apply this in the sequel for the monoid .
For each rational relation , there exists a rational subset of some finitely generated free monoid and alphabetic homomorphisms , , such that . One may even assume that for any letter , if and only if . Thus, for any word in , .
The inverse of the relation is the relation .
The composition of two relations and is the relation . Note that we follow the conventions of : if are the transductions associated to , then the transduction is associated to the relation .
Note that there is a canonical monoid embedding . Its image is the set of pairs such that . Each subset composed of such pairs may be identified with a subset of . We shall use the following result given in  Theorem IX.6.1: let be a rational relation such that if , then have same length (it is called length-preserving). Then is rational as subset of the monoid .
3. Quasi-automatic semigroups
In this section we introduce a new family of semigroups which we call quasi-automatic and show that it contains previously defined families such as the rational semigroup and the synchronous and the asynchronous automatic semigroups.
Let be a semigroup. A quasi-automatic semigroup structure on is a -tuple , where is a finite generating set of , is the natural semigroup homomorphism (sending each onto itself), where is a rational language, and where , for each letter , are rational relations such that:
for each , .
We say that is a quasi-automatic semigroup structure on , with respect to .
The previous definition of quasi-automatic structure on a semigroup has a natural analogue for monoids. We give it now and show that for monoids, the two definitions are equivalent.
Let be a monoid. A quasi-automatic monoid structure on is is a 5-tuple , where the finite set generates as monoid, where is the natural monoid homomorphism , where is a rational language and where , for each letter , are rational relations, such that
for each , .
Let be a monoid.
1. If is generated as semigroup by the finite set and has a quasi-automatic semigroup structure with respect to , then has a quasi-automatic monoid structure with respect to .
2. If is generated as monoid by the finite set and has a quasi-automatic monoid structure with respect to , then has a quasi-automatic semigroup structure with respect to or .
1. Let be a quasi-automatic semigroup structure of . We may extend to by letting and still denote it by . Then clearly, (a), (b), (c) are satisfied (details are left to the reader): is a quasi-automatic monoid structure on .
2. Let be a quasi-automatic monoid structure of . Suppose first that . Then generates as semigroup. Moreover, let , , , . Then satisfy (1), (2), (3) (details are left to the reader), so that they define a quasi-automatic semigroup structure on .
Suppose on the contrary that . This implies that and that . We take a new letter and the new alphabet . Define the semigroup homomorphism by if and .
Let , if and . In other words, is the identity on and maps the empty word onto . Denote also by the mapping , .
Let , , if , and . We show that is a quasi-automatic semigroup structure on .
Equalities of points (1), (2) and (3) of the new structure follow from the identity .
It remains to prove that preserves rationality. Indeed, since rational languages are closed under Boolean operations, the language is rational. Since is a rational language, is rational since rational languages are closed under boolean operations. If is any subset of , one has , where if and otherwise. If is rational, so is ; indeed, the set is rational, since it is , where is the rational language image of under the rational transduction whose graph is the inverse of ; the third set is rational for a similar reason, and is rational since finite; moreover, the last two sets are recognizable, hence their union too, as well as its complement, and rationality is preserved by intersection by a recognizable set.
Suppose that a monoid has a quasi-automatic structure . Suppose that we know a word such that . Then it is decidable if is a group.
We verify first that is left-invertible if and only if there exists such that is in . Indeed, if this holds, then , so that has the left inverse . Conversely, if is left-invertible, then for some , and then .
It is decidable to know if there exists such that is in . Indeed, this is equivalent to . This intersection is rational, in an effective way, since is recognizable (see  Proposition 2.6). Now it is decidable to know if a rational relation is nonempty (see  Proposition 8.2).
In order to conclude, note that if is a group, then each is left invertible. Conversely, if this holds then, since generates as monoid, each element of is left invertible; this in turn implies that each element in is invertible; thus is a group. ∎
3.3. Comparison with the rational semigroups of Sakarovitch
Following [18, 16], a semigroup is called rational if it has a finite generating set with the following properties: there exists a rational language such that the natural homomorphism induces a bijection and that the function , is rational (that is, its graph is a rational subset of ). Rational monoids are defined similarly, and the two definitions are compatible  p.22.
We show that if is rational, then is a quasi-automatic semigroup. Indeed, we have by assumption; moreover, the relation is rational, because it is equal to , which is the image of under the diagonal homomorphism sending each letter onto . Moreover, for , is equal to ; this is equal to the intersection of (which is recognizable) with the graph of the rational function sending onto , which is the composition of (clearly a rational function) followed by . Rational functions are closed under composition, by the theorem of Elgot and Mezei, and intersection with a recognizable relation preserves rationality. Hence is rational.
Thus (1), (2), (3) are satisfied. It follows that each rational semigroup is quasi-automatic. The converse is not true, since an infinite group cannot be rational, by  Example 4.2; but there are infinite groups that are automatic (for example ; see also  Theorems 3.4.1 and 3.4.5), hence quasi-automatic, as is shown in Section 3.4.
3.4. Comparison with automatic groups and semigroups
In  (Definition 2.3.1) are defined (synchronous) automatic groups and in  are defined automatic semigroups extending the first notion to semigroups. Note that automaticity for semigroups depends on the choice of generators, see  Example 4.5, although it does not depend on the choice of generators for groups,  Theorem 2.4.1, nor for automatic monoids in some restrictive sense (the generators must be semigroup generators) .
Furthermore, in  (Definition 7.2.1) are defined asynchronous automatic groups (which is independent of the set of generators Theorem 7.3.3) and in  this notion is extended to semigroups (Definition 2.3). The class of asynchronous semigroups is strictly larger than the class of automatic semigroups, as follows from the example given in  (Example 7.4.1).
We show that synchronous and asynchronous automatic semigroups are quasi-automatic. For this we must define synchronous and asynchronous automata.
Synchronous automata have already been considered in . Let be a new symbol. For in , define , where the natural integers are chosen to be the smallest possible so that have the same length. Then an automatic structure is defined by a rational language such that ; moreover, being defined as in Section 3.1, one asks that the sets and , which may be identified with subsets of (see Section 2, last paragraph, for this identification), be rational subsets of this free monoid.
Since erasing a symbol preserves rationality (it is performed by a homomorphism), an automatic structure is also a quasi- automatic structure.
It follows that synchronous automatic semigroups are quasi-automatic semigroups.
For an asynchronous automatic structure, one considers two-tape automata on the alphabet , which have a double determinism: in any state, the automaton can read only on one of the tapes, depending on the state; moreover, for each letter there is at most one transition with this letter. These automata are called deterministic 2-tape automata; see [17, 10].
Relations that are recognized by such automata are rational relations. Indeed, these automata are special cases of transducers, and the latter recognize rational relations, see , Theorem III.6.1.
Now let be an asynchronous automatic semigroup. This means that is generated by a finite set , and that being defined as before, the relations are recognized by such automata; here and similarly define .
In this case, the relations are rational, as seen above. Since and are the image of and under the homomorphism defined by the identity on and , they are rational relations.
It follows that asynchronous automatic semigroups are quasi-automatic.
4. Properties of quasi-automatic semigroups
4.1. Change of representatives
Let be a quasi-automatic semigroup structure on the semigroup . Let be a rational language such that .Then induces a quasi-automatic structure of .
The set is a recognizable subset of , see  Theorem III.1.5. It follows that its intersection with and with each is a rational subset of . This implies the result. ∎
4.2. Change of generators
Let be a quasi-automatic structure on the semigroup . If is another finite set of generators of , then there exists a quasi-automatic structure on with respect to .
The theorem allows us to say that a semigroup is quasi-automatic: this definition depends only on , and not on the chosen generating set.
With the previous notations, let . Define . This notation is consistent with the notation when . Moreover .
is a rational relation.
This is clear when or . We show that for any words , is the composition of the relations and . By the theorem of Elgot and Mezei, this will imply that any is rational, by induction on the length of .
Let . There exists by (1) a word in such that . Then . Moreover , since . Thus , which implies that .
Conversely, let . There exists such that and . Then , which shows that . ∎
Proof of Theorem 4.1.
Consider the natural homomorphism , which is the identity on ; it is surjective.
Each is a product in of elements of ; we may therefore define a homomorphism such that , , where in . We then have .
Define . It is a rational language in . We have .
Let . We show that (where ). Let ; then , hence there exist such that ; moreover, , hence and therefore , so that . Conversely, if , then , and ; thus and , so that .
Let and . We show that , where has been chosen in such a way that ( is surjective). Let ; then , hence there exist such that ; moreover, , hence and therefore , so that . Conversely, if , then , and ; thus and , so that .
Since is a homomorphism, it preserves rationality and are therefore rational. Thus (1), (2) and (3) are proved and there exists a quasi-automatic structure on with respect to . ∎
4.3. Computing representatives
The term “representative” suggests the choice of a unique element in an equivalence class. This is not quite the meaning here. The idea is, given an arbitrary word to associate a word with the same image: . If does not map on bijectively, uniqueness of such a word is not guaranteed. Thus, in our context, since the computations in the semigroup (monoid or group) are done via , by “representative” of an arbitrary word, we mean a word in that has the same image by .
Let be a quasi-automatic semigroup structure on the semigroup . There exists a function such that for any word and any letter , , and . Moreover, for some , the length of is and may be computed in exponential time with respect to the length of .
Let be a rational function such that for any word . Such a function exists by Eilenberg’s cross-section theorem,  Proposition IX.8.2. By a theorem of Elgot and Mezei, each rational function is the product of a left and of a right subsequential function (see also  Theorem 5.2). Note that one may use also the theorem in , that shows directly that each rational transduction contains a function, with the same domain, and which is the composition of a left and of a right sequential function. Furthermore, this result is effective in the sense that it actually constructs the two sequential functions.
The image of a word by a sequential function may be computed in linear time in , and its length is not more than linear in . This follows since such a function is computed by a deterministic automata with output.
Thus we may find such that , and the computing time of is .
There exist words , such that , and we may assume that .
We define for any word , .
By construction, we have . Hence . This implies that .
The length of is clearly .
Denote by the time needed to compute . We show that it is , which is exponential. This is true for since . Assume that the inequality is true for . Then is the time to compute , plus the time needed to compute from (which is ); thus .
To show that is a representative of we proceed by induction. By construction, and . Now assume and . Since , we have and since , we have . ∎
Recall that a semigroup is rationally presented if it has a finite generating set and a presentation where is a rational subset of .
If is a quasi-automatic semigroup, then it is rationally presented.
Consider a quasi-automatic structure on .
We use the function of Theorem 4.2. Consider the semigroup congruence generated by the relations determined by the pairs in . Since is rational, is a finite union of rational relations, hence is rational.
We contend that for any words , if and only if . This will imply the theorem.
By construction, if , then . Since generates , we obtain that implies .
Conversely, let be such that . We claim that for any word , .The claim implies that , . By what we have already proved, we have and . Thus . Since both words are in , we obtain by definition of that , hence and therefore . It follows from the claim that .
It remains to prove the claim. It is true if , since . Suppose now that . We show that . Since by Theorem 4.2, we have , and therefore . Thus . This proves the claim by induction. ∎
Each quasi-automatic semigroup has a rational presentation such that for any words , in if and only if, for and for some words , one has , and each is obtained from by replacing some prefix of by some word , with or . Moreover the lengths of the words are exponentially bounded with respect to .
We take the same as in the previous proof. Then the ”if” part is evident. In order to prove the ”only if” part, we follow the previous proof.
We take , , (), , , of length , , of length , , , , .
We have and .
Moreover, for , and since .
Note that and . Thus .
The rest of the argument is similar. ∎
4.5. Word problem
By definition, is isomorphic to the quotient of the free semigroup by the congruence generated by the pairs . A similar definition holds for monoid presentations where and are substituted for and .
We recall that the word problem for a presentation consists of determining whether or not two words and are equivalent.
If is a quasi-automatic semigroup, then the word problem in is decidable in exponential time.
The algorithm for the word problem goes as follows: let be two words; compute and ; check if is in . If yes, then in ; if no, in .
Regarding complexity, we may by Theorem 4.2 compute and in exponential time with respect to . Moreover their lengths are at most exponential in . In order to conclude, we apply the following result: given a rational relation , and two words , one may check if , in quadratic time with respect to , see  Theorem 3.3. ∎
4.6. Weak Lipschitz property
Let be a semigroup with generating set . The distance between two elements in is the distance between them in the corresponding Cayley graph, viewed as an undirected graph. Moreover, let as before and some language satisfying . Suppose that for some , and for any words in , such that the distance of and is at most 1, one has: there exist and , such that in one has:
for any , the distance between and is at most .
In this case, we say that the triple has the weak Lipschitz property.
The first condition is useful for the proof and for applications. It is however not essential: it may be skipped and then the new property is equivalent to the previous one.
The weak Lipschitz property implies the undirected asynchronous fellow traveler’s property of , Definition 2.7.
The weak Lipschitz property is a weak form of the Lipschitz property ( Lemma 2.3.2), or fellow traveler property ( Definition 3.11): in the latter, the prefixes of the same length of and are at bounded distance in .
Note that if has a zero, then the Lipschitz property (weak or not) is vacuous, since, as observed in  p. 375, the distance between any two elements in is bounded: it is at most twice the length of the zero. This implies that in general, the converse of Theorem 4.5 does not hold, see  p. 375. However for groups, automacity is equivalent to the Lipschitz property, see  Theorem 2.3.5. Also, for automatic semigroups, there is a geometric characterization, see  Theorem 4.8; see also  for a geometric characterization of a stronger version of automatic monoids.
Let be a semigroup with a finite generating set . Let be a quasi-automatic structure on . Then has the weak Lipschitz property.
Let be a rational subset of . Let be the number of states of some automaton recognizing . Then for any and for any prefix of , there exists a word of length at most such that .
The proof of this lemma is a straightforward exercise in automata theory.
Proof of Theorem 4.5.
By Nivat’s theorem, for each rational relation , there exists a rational subset of some finitely generated free monoid and alphabetic homomorphisms such that . Moreover, for any , if and only if . Note that this ensures that for any word in .
We apply this theorem to and , , or the inverses of them; we then take to be the maximum of the corresponding constants given in the previous lemma. We take .
Let be such that are at distance at most 1. Then , where is one of the relations above. There exists such that . Let , . By the properties of , . Let , ; then .
Let . By the previous lemma, for some word of length at most , we have and therefore . Then the distance of the images under of and is at most 1. Since , and , the distance between and is at most . This is equal to . ∎
4.7. Graded quasi-automatic semigroups
We say that a semigroup is graded if it has a degree, that is a semigroup homomorphism into , and if it is generated as semigroup by elements of degree 1. There are many graded semigroups: , , plactic monoids, braid monoids …, and more generally each semigroup having a homogeneous presentation.
Let be a graded semigroup. If is quasi-automatic, then it is automatic.
Recall from Section 2 the identification of any subset of pairs of words of equal length in with a subset of . Recall also the following: let be a rational relation such that if , then have same length. Then is rational as subset of .
Since is graded, it has a generating subset , each element of which is of degree 1; since is finitely generated, some finite subset of generates . Let be a quasi-automatic structure on . Since the generators are of degree 1, preserves the degree.
By the property (2) of , we must have for any ; thus, by Eilenberg’s theorem, is a rational subset of .
Similarly, by property (3), for each , ; let be a new symbol. The set is clearly rational. Thus by Eilenberg’s theorem, it is a rational subset of .
We have seen that a quasi-automatic semigroup is rationally presented. By a theorem of Anisimov and Seifert  (see also  Theorem III.2.7), every subgroup of the free group which happens to be a rational subset of the free group is actually finitely generated. Hence every rationally presented group is actually finitely presented. For a quasi-automatic group, this will be proved below, with the further property that the group has an exponential isoperimetric inequality.
An isoperimetric inequality for a group means that the group is of the form , where is finite, is the free group on , is a normal subgroup of generated, as normal subgroup, by a finite set and that for some function and any in , is a product of no more than elements of the form , , (here is the length of the reduced word representing ). Of course, the inequality is called exponential if grows exponentially.
Isoperimetric inequalities are important since they characterize finitely presented groups having a decidable word problem: the function must be a recursive function, see , Theorem 2.2.5. The terminology comes from the fact that the length of is the perimeter of a Dehn diagram for , whereas the number of cells is , see Section 2.2 and in particular p. 44 in .
Let be a group which is a quasi-automatic semigroup. Then has a finite presentation, as group, with an exponential isoperimetric inequality.
In order to prove this result, we follow as much as possible the proof of Theorem 2.3.12 (due to Thurston) in , which asserts that an automatic group is finitely presented, with a quadratic isoperimetric inequality.
We need some notations. We may assume that has a quasi-automatic semigroup structure, with respect to a finite set of generators closed under inversion: this means that there is an anti-automorphism of the free monoid , denoted , such that for any word (we extend to by ).
The homomorphism factorizes as , with the natural homomorphism commuting with inversion, and a surjective group homomorphism .
We begin by a lemma.
Let , and be elements of a group. Let (). Then
If , this is clear since . Suppose that it is true for and prove it for . We have . This is equal by induction to