1: The parse tree of the arithmetic expression a * (b + c). 

Contexts in source publication

Context 1

... us consider now the parsing of an input program. If the program is syntactically correct, then the parsing yields a tree, called the derivation tree (or parsing tree) of the program, which shows the structure of it according to the definition of the programming language (cf. Figure 1.1). A derivation tree is a finite, directed, ordered, acyclic, labeled graph such that each node has at most one incoming edge, and there is exactly one node with no incoming edges, which is called the root node (or just root). Clearly, every node has zero or more, so-called, child nodes. Moreover, every node different from the root has exactly one parent ...

Context 2

... that the graph representation of a term will be a tree in the above sense, see Figure 1.3, with the additional property that different nodes can have the same label only if they have the same number of outgoing edges. However, this latter is just a technical matter. Now we abstract from the syntax of both the source language and of the target language and we allow trees over ranked alphabets to be the input tree and the output tree. Moreover, we replace the machine-oriented computation paradigm of the compiler with a more abstract computation paradigm which is called term rewriting. As a result of these big abstractions steps, we obtain an abstract computation device which takes an input tree over a ranked alphabet, then, by manipulating on the input tree as a term rewrite system, computes an output tree over another ranked alphabet. Such an abstract device is called a tree transducer. A more detailed description of this abstraction can be found in Section 1 of ...

Context 3

... that u is an intermediate tree over Q ∪ Σ ∪ ∆ such that above the states in u there are only symbols from Σ while below them only symbols from ∆. Assume also that there is a subtree σ(q 1 (v 1 ), . . . , q k (v k )) of u, where k ≥ 0, σ ∈ Σ with rank k, q 1 , . . . , q k ∈ Q, and v 1 , . . . , v k ∈ T ∆ (see Figure 1.5, where the gray and white parts of u are consisting of symbols from ∆ and Σ respectively, and between the two parts states are from Q). If M has a rule of the form σ(q 1 (x 1 ), . . . , q k (x k )) → q(r), then M can apply it to rewrite u into a tree v, which we denote by u ⇒ M v. In fact, M replaces the subtree σ(q 1 Figure 1.5: Application of a rewriting rule of a bottom-up tree ...

Context 4

... that u is an intermediate tree over Q ∪ Σ ∪ ∆ such that above the states in u there are only symbols from Σ while below them only symbols from ∆. Assume also that there is a subtree σ(q 1 (v 1 ), . . . , q k (v k )) of u, where k ≥ 0, σ ∈ Σ with rank k, q 1 , . . . , q k ∈ Q, and v 1 , . . . , v k ∈ T ∆ (see Figure 1.5, where the gray and white parts of u are consisting of symbols from ∆ and Σ respectively, and between the two parts states are from Q). If M has a rule of the form σ(q 1 (x 1 ), . . . , q k (x k )) → q(r), then M can apply it to rewrite u into a tree v, which we denote by u ⇒ M v. In fact, M replaces the subtree σ(q 1 Figure 1.5: Application of a rewriting rule of a bottom-up tree ...

Context 5

... fact M is a term rewrite system which rewrites an input tree starting at the root of the tree and proceeding towards its leafs. In more detail, a general step of the rewriting by M can be described as follows. Assume that u is an intermediate tree over Q ∪ Σ ∪ ∆ such that above the states in u there are only symbols from ∆ while below them only symbols from Σ. Assume also that there is a subtree q(σ(u 1 , . . . , u k )) of u, where q ∈ Q, k ≥ 0, σ ∈ Σ with rank k, and u 1 , . . . , u k ∈ T Σ (see Figure 1.4, where the gray and white parts of u are consisting of symbols from ∆ and Σ respectively, and in-between the two parts states are from Q). If there is a rule in R of the form q(σ(x 1 , . . . , x k )) → r, then M rewrites u into another tree v, which fact we denote by u ⇒ M v, where v is the tree obtained by replacing the subtree q(σ(u 1 If r does not contain a variable x i for some 1 ≤ i ≤ k, then the tree u i will not show up in the tree v. In this case we say that M deletes u i ...

Context 6

... In case γ = ε the statement is clear because we can construct M as the disjoint union of the relabelings M j which compute the relations {(s j , t j )}. Hence, in this particular case M is a relabeling. Now let us assume that γ ∈ (∆ (1) ) + with length(γ) = m. Then, for every j ∈ [n], there are γ j ∈ (Σ (1) ) + and u j ∈ T Σ such that length(γ j ) = m and γ j u j = s j . Obviously {u 1 , t 1 , . . . , u n , t n } is uniform, so, by the discussion of the case γ = ε, there is a relabeling t 1 ), . . . , (u n , t n )}. Let q 0 and, for every j ∈ [n], p j1 , . . . , p j(m−1) be new states. Moreover, construct the ...

Context 7

... length(occ(u, x 1 )) ≤ ||Q||, and v ∈ T ∆ (X 1 ), we have that length(occ(u, x 1 )) = length(occ (v, x 1 )). Now we show that ≺ is computable for every periodic permutation ...