In theoretical computer science and formal language theory, a regular tree grammar is a formal grammar that describes a set of directed trees, or terms.^[1] A regular word grammar can be seen as a special kind of regular tree grammar, describing a set of single-path trees.

Definition

A regular tree grammar G is defined by the tuple

G = (N, Σ, Z, P),

where

N is a finite set of nonterminals,
Σ is a ranked alphabet (i.e., an alphabet whose symbols have an associated arity) disjoint from N,
Z is the starting nonterminal, with $Z \in N$ , and
P is a finite set of productions of the form A → t, with $A \in N$ , and $t \in T Σ (N)$ , where T_Σ(N) is the associated term algebra, i.e. the set of all trees composed from symbols in $Σ \cup N$ according to their arities, where nonterminals are considered nullary.

Derivation of trees

The grammar G implicitly defines a set of trees: any tree that can be derived from Z using the rule set P is said to be described by G. This set of trees is known as the language of G. More formally, the relation ⇒_G on the set T_Σ(N) is defined as follows:

A tree $t 1 \in T Σ (N)$ can be derived in a single step into a tree $t 2 \in T Σ (N)$ (in short: t₁ ⇒_G t₂), if there is a context S and a production $(A \to t) \in P$ such that:

t₁ = S[A], and
t₂ = S[t].

Here, a context means a tree with exactly one hole in it; if S is such a context, S[t] denotes the result of filling the tree t into the hole of S.

The tree language generated by G is the language $L (G) = {t \in T Σ | Z \Rightarrow G * t}$ .

Here, T_Σ denotes the set of all trees composed from symbols of Σ, while ⇒_G* denotes successive applications of ⇒_G.

A language generated by some regular tree grammar is called a regular tree language.

Examples

Let G₁ = (N₁,Σ₁,Z₁,P₁), where

N₁ = {Bool, BList } is our set of nonterminals,
Σ₁ = { true, false, nil, cons(.,.) } is our ranked alphabet, arities indicated by dummy arguments (i.e. the symbol cons has arity 2),
Z₁ = BList is our starting nonterminal, and
the set P₁ consists of the following productions:
- Bool → false
- Bool → true
- BList → nil
- BList → cons(Bool,BList)

An example derivation from the grammar G₁ is

BList ⇒ cons(Bool,BList) ⇒ cons(false,cons(Bool,BList)) ⇒ cons(false,cons(true,nil)).

The image shows the corresponding derivation tree; it is a tree of trees (main picture), whereas a derivation tree in word grammars is a tree of strings (upper left table).

The tree language generated by G₁ is the set of all finite lists of boolean values, that is, L(G₁) happens to equal T_Σ1. The grammar G₁ corresponds to the algebraic data type declarations (in the Standard ML programming language):

  datatype Bool
    = false
    | true
  datatype BList
    = nil
    | cons of Bool * BList

Every member of L(G₁) corresponds to a Standard-ML value of type BList.

For another example, let $G 2 = (N 1, Σ 1, BList 1, P 1 \cup P 2)$ , using the nonterminal set and the alphabet from above, but extending the production set by P₂, consisting of the following productions:

BList₁ → cons(true,BList)
BList₁ → cons(false,BList₁)

The language L(G₂) is the set of all finite lists of boolean values that contain true at least once. The set L(G₂) has no datatype counterpart in Standard ML, nor in any other functional language. It is a proper subset of L(G₁). The above example term happens to be in L(G₂), too, as the following derivation shows:

BList₁ ⇒ cons(false,BList₁) ⇒ cons(false,cons(true,BList)) ⇒ cons(false,cons(true,nil)).

Language properties

If L₁, L₂ both are regular tree languages, then the tree sets $L 1 \cap L 2, L 1 \cup L 2$ , and L₁ \ L₂ are also regular tree languages, and it is decidable whether $L 1 \subseteq L 2$ , and whether L₁ = L₂.

Alternative characterizations and relation to other formal languages

Regular tree grammars are a generalization of regular word grammars.
The regular tree languages are also the languages recognized by bottom-up tree automata and nondeterministic top-down tree automata.^[2]
Rajeev Alur and Parthasarathy Madhusudan related a subclass of regular binary tree languages to nested words and visibly pushdown languages.^[3]^[4]

Applications

Applications of regular tree grammars include:

Instruction selection in compiler code generation^[5]
A decision procedure for the first-order logic theory of formulas over equality (=) and set membership (∈) as the only predicates^[6]
Solving constraints about mathematical sets^[7]
The set of all truths expressible in first-order logic about a finite algebra (which is always a regular tree language)^[8]
Graph-search ^[9]

References

Regular tree grammars were already described in 1968 by:
- Brainerd, W.S. (1968). "The Minimalization of Tree Automata". Information and Control. 13 (5): 484–491. doi:10.1016/s0019-9958(68)90917-0.
- Thatcher, J.W.; Wright, J.B. (1968). "Generalized Finite Automata Theory with an Application to a Decision Problem of Second-Order Logic". Mathematical Systems Theory. 2 (1): 57–81. doi:10.1007/BF01691346. S2CID 31513761.
A book devoted to tree grammars is: Nivat, Maurice; Podelski, Andreas (1992). Tree Automata and Languages. Studies in Computer Science and Artificial Intelligence. Vol. 10. North-Holland.
Algorithms on regular tree grammars are discussed from an efficiency-oriented view in: Aiken, A.; Murphy, B. (1991). "Implementing Regular Tree Expressions". ACM Conference on Functional Programming Languages and Computer Architecture. pp. 427–447. CiteSeerX 10.1.1.39.3766.
Given a mapping from trees to weights, Donald Knuth's generalization of Dijkstra's shortest-path algorithm can be applied to a regular tree grammar to compute for each nonterminal the minimum weight of a derivable tree. Based on this information, it is straightforward to enumerate its language in increasing weight order. In particular, any nonterminal with infinite minimum weight produces the empty language. See: Knuth, D.E. (1977). "A Generalization of Dijkstra's Algorithm". Information Processing Letters. 6 (1): 1–5. doi:10.1016/0020-0190(77)90002-3.
Regular tree automata have been generalized to admit equality tests between sibling nodes in trees. See: Bogaert, B.; Tison, Sophie (1992). "Equality and Disequality Constraints on Direct Subterms in Tree Automata". Proc. 9th STACS. LNCS. Vol. 577. Springer. pp. 161–172.
Allowing equality tests between deeper nodes leads to undecidability. See: Tommasi, M. (1991). Automates d'Arbres avec Tests d'Égalités entre Cousins Germains. LIFL-IT.

Automata theory: formal languages and formal grammars

Chomsky hierarchy	Grammars	Languages	Abstract machines
Type-0 — Type-1 — — — — — Type-2 — — Type-3 — —	Unrestricted (no common name) Context-sensitive Positive range concatenation Indexed — Linear context-free rewriting systems Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular — Non-recursive	Recursively enumerable Decidable Context-sensitive Positive range concatenation^* Indexed^* — Linear context-free rewriting language Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular Star-free Finite	Turing machine Decider Linear-bounded PTIME Turing Machine Nested stack Thread automaton restricted Tree stack automaton Embedded pushdown Nondeterministic pushdown Deterministic pushdown Visibly pushdown Finite Counter-free (with aperiodic finite monoid) Acyclic finite

Each category of languages, except those marked by a ^*, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.