The Emptiness Problem for Valence Automata over Graph Monoids

The Emptiness Problem for Valence Automata over Graph Monoids

Georg Zetzsche zetzsche@lsv.fr LSV, CNRS & ENS Paris-Saclay, France
Abstract.

This work studies which storage mechanisms in automata permit decidability of the emptiness problem. The question is formalized using valence automata, an abstract model of automata in which the storage mechanism is given by a monoid. For each of a variety of storage mechanisms, one can choose a (typically infinite) monoid such that valence automata over are equivalent to (one-way) automata with this type of storage. In fact, many important storage mechanisms can be realized by monoids defined by finite graphs, called graph monoids. Examples include pushdown stacks, partially blind counters (which behave like Petri net places), blind counters (which may attain negative values), and combinations thereof.

Hence, we study for which graph monoids the emptiness problem for valence automata is decidable. A particular model realized by graph monoids is that of Petri nets with a pushdown stack. For these, decidability is a long-standing open question and we do not answer it here.

However, if one excludes subgraphs corresponding to this model, a characterization can be achieved. Moreover, we provide a description of those storage mechanisms for which decidability remains open. This leads to a model that naturally generalizes both pushdown Petri nets and the priority multicounter machines introduced by Reinhardt.

The cases that are proven decidable constitute a natural and apparently new extension of Petri nets with decidable reachability. It is finally shown that this model can be combined with another such extension by Atig and Ganty: We present a further decidability result that subsumes both of these Petri net extensions.

The author is supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD) and by Labex DigiCosme, Univ. Paris-Saclay, project VERICONISS.

1. Introduction

For each storage mechanism in one-way automata, it is an important question whether the emptiness problem is decidable. It therefore seems prudent to aim for general insights into which properties of storage mechanisms are responsible for decidability or undecidability.

Our approach to obtain such insights is the model of valence automata. These feature a finite-state control and a (typically infinite) monoid that represents a storage mechanism. The edge inscriptions consist of an input word and an element of the monoid. Then, a computation is accepting if it arrives in a final state and composing the encountered monoid elements yields the neutral element. This way, by choosing a suitable monoid, one can realize a variety of storage mechanisms. Hence, our question becomes: For which monoids is the emptiness problem for valence automata over decidable?

We address this question for a class of monoids that was introduced in [19] and accommodates a number of storage mechanisms that have been studied in automata theory. Examples include pushdown stacks, partially blind counters (which behave like Petri net places), and blind counters (which may attain negative values; these are in most situations interchangeable with reversal-bounded counters), and combinations thereof. See [22, 23] for an overview. These monoids are defined by graphs and thus called graph monoids111They are not to be confused with the closely related, but different concept of trace monoids [5], i.e. monoids of Mazurkiewicz traces, which some authors also call graph monoids..

A particular type of storage mechanism that can be realized by graph monoids are partially blind counters that can be used simultaneously with a pushdown stack. Automata with such a storage are equivalent to pushdown Petri nets (PPN), i.e. Petri nets where the transitions can also operate on a pushdown stack. This means, a complete characterization of graph monoids with a decidable emptiness problem would entail an answer to the long-standing open question of whether reachability is decidable for this Petri net extension [15]. Partial solutions have recently been obtained by Atig and Ganty [2] and by Leroux, Sutre, and Totzke [12].

Contribution

While this work does not answer this open question concerning PPN, it does provide a characterization among all graph monoids that avoid this elusive storage type. More precisely, we identify a set of graphs, ‘PPN-graphs’, each of which corresponds precisely to PPN with one Petri net place. Then, among all graphs avoiding PPN-graphs as induced subgraphs, we characterize those for which the graph monoid results in a decidable emptiness problem. Furthermore, we provide a simple, more mechanical (as opposed to algebraic) description of

  1. the storage mechanism emerging as the most general decidable case and

  2. a type of mechanism equivalent to the cases we leave open.

The model 1 is a new extension of partially blind counter automata (i.e. Petri nets). While the decidability proof employs a reduction to Reinhardt’s priority multicounter machines [15], the model 1 seems to be expressively incomparable to Reinhardt’s model. The model 2 is a class of mechanisms whose simplest instance are the pushdown Petri nets and which also naturally subsumes priority multicounter machines (see also Section 3).

Another recent extension of the decidability of reachability of Petri nets has been obtained by Atig and Ganty [2]. In fact, it is a partial solution to the reachability problem for PPN. Their proof also relies on priority multicounter machines. They show that given a finite-index context-free language and a language generated by a Petri net, it is decidable whether the intersection is empty. Note that without the finite-index requirement, this would be equivalent to the reachability problem for PPN. Our final contribution is a decidability result that subsumes both the decidability of model 1 and the result of Atig and Ganty. We present a natural language class that contains both the intersections considered by Atig and Ganty and the languages of model 1 and still has a decidable emptiness problem. To this end, we employ a slightly stronger (and perhaps simpler) version of Atig and Ganty’s reduction.

Hence, the perspective of valence automata allows us to identify natural storage mechanisms that {enumerate*}[label=()]

push the frontier of decidable emptiness (and hence reachability) and

let us naturally interpret PPN and priority multicounter machines as special cases of a more powerful model that might enjoy decidability, respectively.

The paper is structured as follows. We present the main results in Section 3 and prove them in Sections 6, 5 and 4. Section 4 presents the undecidability part, Section 5 treats the decidable cases, and Section 6 shows the expressive equivalence with the more mechanical descriptions. In Section 7, we present the enhanced decidability result that also subsumes the one by Atig and Ganty.

This work is an extended version of the paper [21]. This version provides proofs of the results of [21] and the enhanced decidability result. Moreover, it contains proofs of some results that first appeared in [19, 20], but have not yet undergone journal peer review.

2. Preliminaries

A monoid is a set together with a binary associative operation such that contains a neutral element. Unless the monoid at hand warrants a different notation, we will denote the neutral element by and the product of by . If is a set of symbols, denoted the set of words over . The length of the word is denoted . An alphabet is a finite set of symbols. The empty word is denoted by . Let is a set of pairs of symbols, then the semi-Dyck language over , denoted is the smallest subset of such that and whenever , then also for every . If , then we also write instead of . Moreover, if , then the words in are called semi-Dyck words over . If is a word with for , then denotes in reverse, i.e. .

For an alphabet and languages , the shuffle product is the set of all words where , , and . For a subset , we define the projection morphism by for and for . Moreover, we define and for , we set .

Valence automata

As a framework for studying which storage mechanisms permit decidability of the emptiness problem, we employ valence automata. They feature a monoid that dictates which computations are valid. Hence, by an appropriate choice of the monoid, valence automata can be instantiated to be equivalent to a concrete automata model with storage. For the purposes of this work, equivalent is meant with respect to accepted languages. Therefore, we regard valence automata as language accepting devices.

Let be a monoid and an alphabet. A valence automaton over is a tuple , in which {enumerate*}

is a finite set of states,

is a finite subset of , called the set of edges,

is the initial state, and

is the set of final states. For , , and , we write if there is an edge such that and . The language accepted by is then

The class of languages accepted by valence automata over is denoted by . If is a class of monoids, we write for .

Graphs

A graph is a pair where is a finite set and is a subset of . The elements of are called vertices and those of are called edges. Vertices are adjacent if . If for some , then is called a looped vertex, otherwise it is unlooped. A subgraph of is a graph with and . Such a subgraph is called induced (by ) if , i.e. contains all edges from incident to vertices in . By , for , we denote the subgraph of induced by . By (), we denote a graph that is a cycle (path) on four vertices; see Fig. 1. Moreover, denotes the graph obtained from by deleting all loops: We have , where . The graph is loop-free if . Finally, a clique is a loop-free graph in which any two distinct vertices are adjacent.

(a)

(b)
Figure 1. Graphs and .

Products and presentations

If , are monoids, then denotes their direct product, whose set of elements is the cartesian product of and and composition is defined component-wise. By , we denote the -fold direct product, i.e. with factors.

Let be a (not necessarily finite) set of symbols and be a subset of . The pair is called a (monoid) presentation. The smallest congruence of the free monoid containing is denoted by and we will write for the congruence class of . The monoid presented by is defined as . Note that since we did not impose a finiteness restriction on , up to isomorphism, every monoid has a presentation. If and , we also use the shorthand to denote the monoid presented by .

Furthermore, for monoids , we can find presentations and such that . We define the free product to be presented by . Note that is well-defined up to isomorphism. In analogy to the -fold direct product, we write for the -fold free product of .

Graph monoids

A presentation in which is a finite alphabet is a Thue system. To each graph , we associate the Thue system over the alphabet . is defined as

In particular, we have whenever . To simplify notation, the congruence is then also denoted by . We are now ready to define graph monoids. To each graph , we associate the monoid

The monoids of the form are called graph monoids.

Storage mechanisms as graph monoids

Let us briefly discuss how to realize storage mechanisms by graph monoids. First, suppose and are disjoint graphs. If is the union of and , then by definition. Moreover, if is obtained from and by drawing an edge between each vertex of and each vertex of , then .

If consists of one vertex and has no edges, the only rule in the Thue system is . In this case, is also denoted as and we will refer to it as the bicyclic monoid. The generators and are then also written and , respectively. It is not hard to see that corresponds to a partially blind counter, i.e. one that attains only non-negative values and has to be zero at the end of the computation. Moreover, if consists of one looped vertex, then is isomorphic to and thus realizes a blind counter, which can go below zero and is zero-tested in the end.

If one storage mechanism is realized by a monoid , then the monoid corresponds to the mechanism that builds stacks: A configuration of this new mechanism consists of a sequence , where are configurations of the mechanism realized by . We interpret this as a stack with the entries . One can open a new stack entry on top (by multiplying ), remove the topmost entry if empty (by multiplying ) and operate on the topmost entry using the old mechanism (by multiplying elements from ). In particular, describes a pushdown stack with two stack symbols. See [22] for more examples and [23] for more details.

As a final example, suppose is one edge short of being a clique, then , where is the number of vertices in . Then, by the observations above, valence automata over are equivalent to Petri nets with unbounded places and access to a pushdown stack. Hence, for our purposes, a pushdown Petri net is a valence automaton over for some .

3. Results

As a first step, we exhibit graphs for which includes the recursively enumerable languages.

Theorem 3.1.

Let be a graph such that contains or as an induced subgraph. Then is the class of recursively enumerable languages. In particular, the emptiness problem is undecidable for valence automata over .

This unifies and slightly strengthens a few undecidability results concerning valence automata over graph monoids. The case that all vertices are looped was shown by Lohrey and Steinberg [14] (see also the discussion of Theorem 3.3). Another case appeared in [19]. We prove Theorem 3.1 in Section 4.

It is not clear whether Theorem 3.1 describes all for which exhausts the recursively enumerable languages. For example, as mentioned above, if is one edge short of being a clique, then valence automata over are pushdown Petri nets. In particular, the emptiness problem for valence automata is equivalent to the reachability problem of this model, for which decidability is a long-standing open question [15]. In fact, it is already open whether reachability is decidable in the case of , although Leroux, Sutre, and Totzke have recently made progress on this case [12]. Therefore, characterizing those with a decidable emptiness problem for valence automata over would very likely settle these open questions222Strictly speaking, it is conceivable that there is a decision procedure for each , but no uniform one that works for all . However, this seems unlikely..

However, we will show that if we steer clear of pushdown Petri nets, we can achieve a characterization. More precisely, we will present a set of graphs that entail the behavior of pushdown Petri nets. Then, we show that among those graphs that do not contain these as induced subgraphs, the absence of and already characterizes decidability.

PPN-graphs

A graph is said to be a PPN-graph if it is isomorphic to one of the following three graphs:

We say that the graph is PPN-free if it has no PPN-graph as an induced subgraph. Observe that a graph is PPN-free if and only if in the neighborhood of each unlooped vertex, any two vertices are adjacent.

Of course, the abbreviation ‘PPN’ refers to ‘pushdown Petri nets’. This is justified by the following fact. It is proven in Section 5 (page 5).

Proposition 3.1.

If is a PPN-graph, then .

Transitive forests

In order to exploit the absence of and as induced subgraphs, we will employ a characterization of such graphs as transitive forests. The comparability graph of a tree is a simple graph with the same vertices as , but has an edge between two vertices whenever one is a descendant of the other in . A graph is a transitive forest if the simple graph is a disjoint union of comparability graphs of trees. For an example of a transitive forest, see Fig. 2.

Figure 2. Example of a transitive forest. The solid edges are part of the trees whose comparability graphs make up the graph. The transitive forest consists of both the solid and the dashed edges.

Let denote the smallest isomorphism-closed class of monoids such that

  1. for each , we have and

  2. for , we also have and .

Our main result characterizes those PPN-free for which valence automata over have a decidable emptiness problem.

Theorem 3.2.

Let be PPN-free. Then the following conditions are equivalent:

  1. Emptiness is decidable for valence automata over .

  2. contains neither nor as an induced subgraph.

  3. is a transitive forest.

  4. .

We present the proof in Section 5. Note that this generalizes the fact that emptiness is decidable for pushdown automata (i.e. graphs with no edges) and partially blind multicounter automata (i.e. cliques), or equivalently, reachability in Petri nets.

Note that if has a loop on every vertex, then is a group. Groups that arise in this way are called graph groups. In general, if a monoid is a group, then emptiness for valence automata over is decidable if and only if the rational subset membership problem is decidable for  [11]. The latter problem asks, given a rational set over and an element , whether ; see [13] for more information. Therefore, Theorem 3.2 extends the following result of Lohrey and Steinberg [14], which characterizes those graph groups for which the rational subset membership problem is decidable.

Theorem 3.3 (Lohrey and Steinberg [14]).

Let be a graph in which every vertex is looped. Then the rational subset membership problem for the group is decidable if and only if is a transitive forest.

Lohrey and Steinberg show decidability by essentially proving that in their case, the languages in have semilinear Parikh images (although they use different terminology). Here, we extend this argument by showing that in the equivalent cases of Theorem 3.2, the Parikh images of are those of languages accepted by priority multicounter machines. The latter were introduced and shown to have a decidable reachability problem by Reinhardt [15].

Intuition for decidable cases

In order to provide an intuition for those storage mechanisms (not containing a pushdown Petri net) with a decidable emptiness problem, we present an equally expressive class of monoids for which the corresponding storage mechanisms are easier to grasp. Let be the smallest isomorphism-closed class of monoids with

  1. for each , we have ,

  2. for each , we also have and .

Thus, realizes those storage mechanisms that can be constructed from a finite set of partially blind counters () by building stacks () and adding blind counters (). Then, in fact, the monoids in produce the same languages as those in .

Proposition 3.3.

.

Section 3 is proven in Section 6. While our decidability proof for will be a reduction to priority multicounter machines (see Section 5 for a definition), it seems likely that these two models are incomparable in terms of expressiveness (see the remarks after Theorem 5.3).

Intersections with finite-index languages

This work exhibits valence automata over as an extension of Petri nets that features a type of stack but retains decidability of the emptiness problem. Another recent result of this kind has been obtained by Atig and Ganty [2]. They showed that given a finite-index context-free language and a Petri net language , it is decidable whether is empty. Moreover, they also employ a reduction to priority multicounter machines. This raises the question of how the two results relate to each other. In Section 7, we present a natural language class that subsumes both the languages of Atig and Ganty and those of and prove that emptiness is still decidable. Intuitively, this class is obtained by taking languages of Atig and Ganty and then applying operators corresponding to building stacks and adding blind counters. The precise definition and the result can be found in Section 7.

Intuition for open cases

We also want to provide an intuition for the remaining storage mechanisms, i.e. those defined by monoids about which Theorems 3.2 and 3.1 make no statement. To this end, we describe a class of monoids that are expressively equivalent to these remaining cases. The remaining cases are given by those graphs where does not contain or , but contains a PPN-graph. Let denote the class of monoids , where is such a graph. Let be the smallest isomorphism-closed class of monoids with

  1. and

  2. for each , we also have and .

This means, realizes those storage mechanisms that are obtained from a pushdown stack, together with one partially blind counter () by the transformations of building stacks () and adding partially blind counters ().

Proposition 3.3.

.

We prove Section 3 in Section 6. Of course, generalizes pushdown Petri nets, which correspond to monoids for . Moreover, also subsumes priority multicounter machines (see p. 5 for a definition) in a straightforward way: Every time we build stacks, we can use the new pop operation to realize a zero test on all the counters we have added so far. Let and . Then, priority -counter machines correspond to valence automata over where the stack heights never exceed .

Remark 3.3.

Priority multicounter machines are already subsumed by pushdown Petri nets alone: Atig and Ganty [2, Lemma 7] show implicitly that for each priority multicounter machine, one can construct a pushdown Petri net that accepts the same language. Hence, valence automata over are not the first perhaps-decidable generalization of both pushdown Petri nets and priority multicounter machines, but they generalize both in a natural way.

4. Undecidability

In this section, we prove Theorem 3.1. It should be mentioned that a result similar to Theorem 3.1 was shown by Lohrey and Steinberg [14]: They proved that if every vertex in is looped and contains or as an induced subgraph, then the rational subset membership problem is undecidable for . Their proof adapts a construction of Aalbersberg and Hoogeboom [1], which shows that the disjointness problem for rational sets of traces is undecidable when the independence relation has or as an induced subgraph. An inspection of the proof presented here, together with its prerequisites (Theorems 4.2 and 4.1), reveals that the employed ideas are very similar to the combination of Lohrey and Steinberg’s and Aalbersberg and Hoogeboom’s proof.

A language class is a collection of languages that contains at least one non-empty language. In this work, for each language class, there is a way to finitely represent each member of the class. Moreover, an inclusion between language classes and is always meant to be effective, in other words: Given a representation of a language in , we can compute a representation of that language in . The same holds for equalities between language classes.

Let and be alphabets. A relation is called a rational transduction if there is an alphabet , a regular language , and morphisms and such that (see [3]). For a language , we define . A language class is a full trio if for every language in , the language is effectively contained in as well. Here, “effectively” means again that given a representation of a language from and a description of , one can effectively compute a representation of . For a language , we denote by the smallest full trio containing . Note that if , the class contains precisely the languages for rational transductions . For example, it is well-known that for every monoid , the class is a full trio [6]. A full AFL is a full trio that is also closed under Kleene iteration, i.e. for each member , the language is effectively a member as well.

Here, we use the following fact. We denote the recursively enumerable languages by .

Lemma 4.0.

Let and let be defined as

Then equals , the smallest full trio containing .

Section 4 is essentially due to Hartmanis and Hopcroft, who stated it in slightly different terms:

Theorem 4.1 (Hartmanis and Hopcroft [9]).

Let be the smallest full AFL containing . Every recursively enumerable language is the homomorphic image of the intersection of two languages in .

By the following auxiliary result of Ginsburg and Greibach [8, Theorem 3.2a], Section 4 will follow from Theorem 4.1.

Theorem 4.2 (Ginsburg and Greibach [8]).

Let and . The smallest full AFL containing equals .

As announced, Section 4 now follows.

Section 4.
author=R1author=R1todo: author=R1I am not fully convinced of the interest of sharing the symbols and between and , since the proof uses homomorphisms liberally.

Since clearly , it suffices to show . According to Theorem 4.1, this amounts to showing that for any and in , where is the smallest full AFL containing the language . Hence, let . By Theorem 4.2, and belong to . This means we have for some rational transduction for . Using a product construction, it is now easy to obtain a rational transduction with .∎

The proof of Theorem 3.1 will require one more auxiliary lemma. In the following, denotes the congruence class of with respect to .

Lemma 4.2.

Let be a graph, let be a subset of vertices, and let be defined as . Then implies for .

Proof.

An inspection of the rules in the Thue system reveals that if , then either or . In any case, . Since is a congruence and a morphism, this implies the Section.∎

Note that the foregoing Section does not hold for arbitrary alphabets . For example, if , , and , then , but .

We are now ready to prove Theorem 3.1.

Theorem 3.1.

Observe that if and only if can be transformed into by finitely many times replacing an infix with an infix for some . Since is finite, this implies that the set of all with is recursively enumerable. (In fact, whether can be decided in polynomial time [19, 23].) In particular, one can recursively enumerate runs of valence automata over and hence . For the other inclusion, recall that is a full trio for any monoid . Furthermore, if is an induced subgraph of , then embeds into , meaning . Hence, according to Section 4, it suffices to show that if equals or .

Figure 3. Graphs where is or . Dotted lines represent edges that may or may not exist in .

Let . and . If equals or , then with and . See Fig. 3. We construct a valence automaton over for as follows. First, reads a word in . Here, when reading or , it multiplies or , respectively, to the storage monoid. When reading or , it multiplies or , respectively. After this, switches to another state and nondeterministically multiplies an element from . Then it changes into an accepting state. We shall prove that accepts . Let the morphism be defined by and for and and .

Suppose . Then and there is a with . Let . If we can show for , then clearly . For symmetry reasons, it suffices to prove this for . Let . Since , we have in particular by Section 4. Moreover,

for some . Again, by projecting to , we obtain and hence . If , then it is easy to see that cannot be reduced to , since there is no edge in . Therefore, we have . It follows inductively that for all . Since , this implies .

We shall now prove . Let be the morphism defined by and and . We show by induction on that implies . Since for each , clearly has a run that puts into the storage, this establishes . Suppose ends in . Then for some , with and . Note that then . Since there are edges in , we have . Moreover, since deletes and , we have . Therefore,

By induction, we have and hence . If ends in , then one can show completely analogously. This proves and hence the Theorem.∎

5. Decidability

In this Section, we prove Theorem 3.2 and Section 3. First, we mention existing results that are ingredients to our proofs.

Let be a class of languages. A -grammar is a quadruple where and are disjoint alphabets and . is a finite set of pairs with and , . A pair is called a production of . We write if and for some and with . Moreover, means that there are with for and and . Furthermore, we have if for some . The language generated by is . The class of all languages that are generated by -grammars is called the algebraic extension of and is denoted . Of course, if , then . Moreover, it is easy to see that if , then .

The following is easy to show in the same way one shows that the context-free languages constitute a full trio [3]. A proof can be found in [23].

Lemma 5.0.

If is a full trio, then is a full trio as well.

A monoid is called finitely generated if there is a finite subset such that every element of can be written as a product of elements of . A language is called an identity language for if there is a surjective morphism with . We will also use the following well-known fact about valence automata. A proof can be found, e.g., in [23, 10].

Proposition 5.0.

Let be a finitely generated monoid. Then:

  1. is the smallest full trio containing all identity languages of .

  2. If is any identity language of , then is the smallest full trio containing .

The well-known theorem of Chomsky and Schützenberger [3], expressed in terms of valence automata, states that is the class of context-free languages. This formulation, along with a new proof, is due to Kambites [10]. Let and denote the class of regular and context-free languages, respectively. Then we have and . Here, denotes the trivial monoid . Moreover, notice that for every language class . Since furthermore valence automata over are equivalent to pushdown automata, we have in summary:

(1)

In order to work with general free products, we use the following result, which expresses the languages in in terms of and . It was first shown in [19]. In [4], it was extended to more general products. For the convenience of the reader, we include a proof.

Proposition 5.0 ([19]).

Let and be monoids. Then is included in .

Proof.

For every monoid , we have , where ranges over the finitely generated submonoids of . Moreover, every finitely generated submonoid of is included in some , where is a finitely generated submonoid of , for . Therefore, we have , where ranges over the finitely generated submonoids of , for . Thus, it suffices to show the Section in the case that and are finitely generated.

For , let be a presentation of such that is finite. Then is presented by . Consider the languages for . Then is an identity language of and hence contained in . Moreover, by definition of , the language is an identity language of .

According to Section 5, the class is a full trio. Thus, Section 5 tells us that it suffices to show that the identity language of is contained in .

Consider the binary relation on where if and only if for some , there are , , such that , , and . It is now easy to see that if and only if .

This allows us to construct a -grammar for . Let , where . In order to describe the productions, we need to define two languages. For , let

Then can be obtained from using full trio operations and is thus contained in . Our grammar contains only three productions: , , and (recall that as a regular language, belongs to each ). Then, it is immediate that if and only if and hence .∎

Section 5 tells us that the languages in are confined to the algebraic extension of . Our next ingredient, Section 5, will complement Section 5 by describing monoids such that the algebraic extension of is confined to . We need two auxiliary lemmas, for which the following notation will be convenient. We write for monoids if there is a morphism such that . Clearly, if , then : Replacing in a valence automaton over all elements with yields a valence automaton over that accepts the same language.

Lemma 5.0.

If and , then we have .

Proof.

Let and be morphisms with and . Then defining as the morphism with and clearly yields .∎

For a monoid , we define . Observe that the set can be thought of as the storage contents that can occur in a valid run of a valence automaton over . The following result appeared first in [20]. We include a proof for the convenience of the reader.

Lemma 5.0 ([20]).

Let be a monoid with . Then we have for every . In particular, for every .

Proof.

Observe that if and , then

Therefore, it suffices to prove .

Let for . We show . Suppose is presented by . We regard the monoids and as embedded into