Valence Automata over Graph Monoids

• Valence automata over graph monoids are finite automata enhanced with a monoid-valued register that is updated multiplicatively and must return to the identity for acceptance.
• The framework models various storage systems—such as pushdown storage, blind and partially blind counters—by encoding operations via graphs that dictate commutation and cancellation rules.
• It yields classification results on expressiveness, decidability, and semilinearity, with algebraic normal forms and equivalence algorithms providing insights into computational complexity.

Valence automata over graph monoids are finite automata equipped with a monoid-valued register, initialized with the identity and updated multiplicatively along transitions, with acceptance requiring that the accumulated monoid value return to the identity. In this setting, the storage discipline is encoded algebraically by a graph: depending on the construction, the graph may specify partial commutation, cancellation, or a graph product of vertex monoids. The framework subsumes pushdown storage, blind counters, partially blind counters, and several concurrent or partially commutative memories, and it has developed into a unifying language for studying expressiveness, decidability, semilinearity, and algorithmic complexity in automata with storage (Buckheister et al., 2013, Zetzsche, 2013, Zetzsche, 2017, Arvind et al., 2020).

1. Formal model and monoids-as-memory

For a monoid $M$, a valence automaton over $M$ can be presented as a finite automaton whose transitions are labeled by input symbols together with elements of $M$. Operationally, the machine maintains a register containing an element of $M$, initially $1$, and a transition labeled by $m\in M$ updates register content by right multiplication. A run is accepting precisely when it ends in an accepting state with register content $1$. In the notation of the monoid-automata literature, the accepted language family is written either as $\mathfrak{L}_1(M)$ or as $\mathsf{VA}(M)$, depending on the source, and regular valence grammars over $M$ generate the same languages as valence automata over $M$0 (Salehi et al., 2017, Buckheister et al., 2013).

A useful generalization is the rational monoid automaton with targets. Here the register is initialized in some element of a rational subset $M$1, and acceptance requires the final value to lie in relation with a rational terminal subset $M$2. Ordinary valence automata are recovered as the special case $M$3. This extension is important because graph-based storage monoids are often not groups, so rational target sets provide a natural way to formulate more general acceptance conditions than strict return to the identity (Salehi et al., 2017).

The same monoid-control principle extends to pushdown automata and context-free grammars. A valence pushdown automaton assigns an element of $M$4 to each pushdown transition and accepts when the ordinary pushdown condition holds and the product of transition valences is $M$5. A context-free valence grammar labels productions by elements of $M$6, and a derivation is valid when the ordered product of rule valences is $M$7. A notable structural result is that valence pushdown automata are only as powerful as finite valence automata. Thus, attaching an ordinary pushdown to monoid control does not increase expressive power beyond the monoid-controlled finite-state model itself (Salehi et al., 2017).

2. Graph monoids and graph products as storage structures

The term “graph monoid” is used in several related senses. In the broadest construction, one fixes a graph $M$8 and a family of vertex monoids $M$9, and forms the graph product

$M$0

obtained from the free product of the $M$1 by imposing commutation between elements attached to adjacent vertices. This interpolates between free product and direct product: if $M$2, one gets $M$3, and if $M$4 is a clique, one gets $M$5 (Buckheister et al., 2013).

A second standard meaning is the partially commutative monoid, also called a trace monoid or pc monoid. Given a finite alphabet $M$6 and a symmetric reflexive independence relation $M$7, the monoid is

$M$8

where $M$9 is the congruence generated by $M$0 whenever $M$1. The associated non-commutation graph has an edge exactly when two generators do not commute. In the usual graph-monoid viewpoint, this is the complement of the commutation graph (Arvind et al., 2020).

A third construction, central in valence-automata work on storage, starts from a graph $M$2 and introduces paired generators $M$3 for each vertex. The defining relations are $M$4 for every $M$5, together with commutations between all letters attached to adjacent vertices. The resulting monoid is written $M$6. In this scheme, an unlooped single vertex yields the bicyclic monoid $M$7, which models a partially blind counter, while a looped single vertex yields $M$8, which models a blind counter (Zetzsche, 2013, Zetzsche, 2017).

These constructions recover standard storage mechanisms. A clique of unlooped vertices gives $M$9, corresponding to $1$0 partially blind counters; a looped clique gives $1$1, corresponding to $1$2 blind counters; and an anti-clique on at least two unlooped vertices gives a free product of copies of $1$3, which realizes pushdown storage. More generally, graph products over the basis $1$4 model pushdowns, blind multicounters, partially blind multicounters, and mixed systems of these kinds (Buckheister et al., 2013, Zetzsche, 2013, Zetzsche, 2017).

3. Expressive power, regularity, context-freeness, and semilinearity

A central classification theorem identifies when monoid control actually increases expressive power. For a monoid $1$5, the following are equivalent: valence grammars over $1$6 generate only context-free languages; valence automata over $1$7 accept only regular languages; valence automata over $1$8 are determinizable; valence transducers over $1$9 perform only rational transductions; and every finitely generated submonoid $m\in M$0 has finite $m\in M$1, equivalently finite $m\in M$2 and finite $m\in M$3. As an immediate consequence, finite monoids do not increase the power of finite automata, context-free grammars, or finite-state transducers in the valence setting (Zetzsche, 2011).

This theorem has a sharp implication for trace monoids. Because a free partially commutative monoid has no nontrivial right-invertible elements, every finitely generated submonoid $m\in M$4 satisfies $m\in M$5. Hence valence automata over trace monoids accept only regular languages, valence grammars over trace monoids generate only context-free languages, and deterministic and nondeterministic variants have the same expressive power. A common misconception is therefore false: pure partial commutation, without additional inverse-like structure, does not by itself produce nonregular behavior in valence automata (Zetzsche, 2011).

For general graph products, the context-free boundary is more subtle. A monoid $m\in M$6 is an FRI-monoid if every finitely generated submonoid has only finitely many right-invertible elements, and this property is exactly equivalent to $m\in M$7. For a graph product $m\in M$8 with nontrivial right-invertible part at each vertex, valence automata accept only context-free languages exactly when four structural conditions hold: each vertex monoid is itself context-free; no two non-FRI vertex monoids are adjacent; if a non-FRI vertex is adjacent to two FRI vertices then those FRI vertices are adjacent to each other; and the graph $m\in M$9 is chordal (Buckheister et al., 2013).

Semilinearity forms a parallel line of investigation. For graph products of copies of the bicyclic monoid and the integers, there is a necessary and sufficient condition for all accepted languages to have semilinear Parikh image, and all languages accepted by valence automata over torsion groups have semilinear Parikh image. More generally, the monoids-as-memory framework also yields negative and positive results for rational monoid automata beyond the graph-product setting, including the existence of a context-free language not recognized by any rational monoid automaton over a finitely generated permutable monoid and the fact that rational monoid automata over finitely generated completely simple or completely $1$0-simple permutable monoids form a semilinear full trio (Buckheister et al., 2013, Salehi et al., 2017).

4. Silent transitions, emptiness, and the decidability frontier

The role of $1$1-transitions in valence automata over graph monoids is highly nonuniform. For the class $1$2 generated from the trivial monoid by the operations $1$3 and $1$4, one obtains storages built from blind counters and stack formation. When a graph $1$5 satisfies the two basic shape conditions that looped vertices are pairwise adjacent and unlooped vertices are pairwise nonadjacent, the following are equivalent: $1$6-transitions can be eliminated from valence automata over $1$7; every language in $1$8 is context-sensitive; membership for every such language is in $1$9; every such language is decidable; and $\mathfrak{L}_1(M)$0 avoids a specific induced looped path on four vertices (Zetzsche, 2013).

For clique graphs, the situation is especially crisp. If $\mathfrak{L}_1(M)$1 is complete on distinct vertices, then $\mathfrak{L}_1(M)$2, where $\mathfrak{L}_1(M)$3 is the number of unlooped vertices and $\mathfrak{L}_1(M)$4 the number of looped ones. In this case,

$\mathfrak{L}_1(M)$5

Thus one partially blind counter together with any number of blind counters still admits elimination of silent transitions, whereas two or more partially blind counters do not (Zetzsche, 2013).

The emptiness problem for valence automata over graph monoids has a similarly sharp graph-theoretic boundary outside the pushdown-Petri-net region. If the loop-free part $\mathfrak{L}_1(M)$6 contains an induced $\mathfrak{L}_1(M)$7 or $\mathfrak{L}_1(M)$8, then $\mathfrak{L}_1(M)$9 is the class of recursively enumerable languages; in particular, emptiness is undecidable. For PPN-free graphs, emptiness is decidable exactly when $\mathsf{VA}(M)$0 contains neither $\mathsf{VA}(M)$1 nor $\mathsf{VA}(M)$2, equivalently exactly when $\mathsf{VA}(M)$3 is a transitive forest, equivalently exactly when the monoid $\mathsf{VA}(M)$4 lies in the class $\mathsf{VA}(M)$5 generated from the $\mathsf{VA}(M)$6 by free product and multiplication by $\mathsf{VA}(M)$7 (Zetzsche, 2017).

The unresolved zone is the one corresponding to pushdown Petri nets. Valence automata over $\mathsf{VA}(M)$8 realize pushdown Petri nets, and decidability of emptiness or reachability for these systems remains open. The same work isolates a broader class $\mathsf{VA}(M)$9 that naturally generalizes both pushdown Petri nets and Reinhardt’s priority multicounter machines, and treats it as the principal remaining frontier (Zetzsche, 2017).

5. Bounded context switching over graph monoids

Valence systems over graph monoids support a graph-internal notion of bounded context switching. Fix a graph $M$0 and operations $M$1 for $M$2. The graph monoid $M$3 is the quotient of $M$4 by commuting independent operations and canceling $M$5. A context is then defined as a maximal segment of a computation using only a dependent set of symbols, where dependence means that no two distinct underlying vertices are independent. This definition conservatively generalizes existing notions: for a single pushdown it imposes no restriction, for multi-pushdowns it becomes the usual number of stack switches, and for Petri nets or blind counters it counts counter switches (Meyer et al., 2018).

The main complexity result is uniform in the storage graph: reachability within a bounded number of context switches is in $M$6 for every graph monoid. The proof is algebraic rather than model-specific. A computation with $M$7 contexts and total effect $M$8 admits a decomposition into at most $M$9 blocks, and these blocks are freely reducible by high-level cancellation and commutation rules. This block normal form is the key finite witness behind the $M$00 upper bound (Meyer et al., 2018).

For fixed graphs, finer complexity bounds are known. If the loop-free reduct $M$01 is a clique, bounded-context reachability is $M$02-complete; if $M$03 is not a clique, the problem is $M$04-hard. If $M$05 is a transitive forest, bounded-context reachability is in $M$06. If $M$07 is an induced subgraph of $M$08, the problem is $M$09-complete. The remaining gap concerns graphs whose loop-free reduct contains an induced $M$10 but no $M$11; the paper conjectures $M$12-hardness there without proving it (Meyer et al., 2018).

6. Equivalence problems and algebraic normal forms

Partially commutative monoids also support a quantitative analogue of valence-automaton analysis through weighted automata. In this setting, transitions are labeled by generators of a pc monoid and weighted over a field, and runs are evaluated modulo the commutation congruence. Multiplicity equivalence asks whether two automata assign the same coefficient to every monoid element. A Schützenberger-type theorem holds: for an automaton of size $M$13 over a pc monoid with alphabet size $M$14, the recognized series is nonzero if and only if some word of length $M$15 has nonzero coefficient (Arvind et al., 2020).

Graph structure then yields concrete algorithms. If the non-commutation graph has a clique edge cover of size $M$16, zero testing, and hence multiplicity equivalence, is decidable in deterministic time

$M$17

If the non-commutation graph is a $M$18-monoid, meaning it can be covered by at most $M$19 cliques and stars, then zero testing is decidable in randomized time

$M$20

As corollaries, one obtains the first deterministic quasi-polynomial-time algorithms for multiplicity equivalence of $M$21-tape automata and for equivalence of deterministic $M$22-tape automata when $M$23 is constant (Arvind et al., 2020).

At a more algebraic level, arbitrary graph products of monoids admit unique left Foata normal forms. If an element has two left Foata normal forms, then they have the same number of blocks and corresponding blocks represent the same element. In parallel, graph products preserve several one-sided structural properties: graph products of left abundant monoids are left abundant, graph products of left Fountain monoids are left Fountain, and graph products of regular monoids are abundant. As a special case, the graph product of right cancellative monoids is right cancellative (Dandan et al., 2021).

For valence automata, these results provide the algebraic infrastructure for storage analysis over graph-product monoids. Unique Foata normal forms give canonical representatives of storage effects, while preserved abundance and cancellativity properties constrain how products can be reduced, compared, and factored. This suggests a division of labor within the subject: language-theoretic classifications determine what families of languages are possible, while graph-product normal forms and equivalence algorithms determine how those storage effects can be analyzed effectively (Dandan et al., 2021, Arvind et al., 2020).