Categorical semantics and composition of tree transducers

ion from Y yields T∆+AI(QX) (r̂)X ←−−−− AI ( Q(ΣX)). Now we need the coproduct of monads: With T∆ = |∆ | and Lemma 6.2.4.1 we can write the rule as |∆ + ( AI(QX) )? | (r̂)X ←−−−− AI ( Q(ΣX)). Abstracting from X gives using from X gives us | | · (∆ + ) · ( ) · AI · Q r̂ ←−− AI · Q · Σ. We have the adjunctions Q a U and AI a ΛI as in Subsection 6.4.3 and ( ) ? a | | from Corollary 4.4.4.6. Now we use ( ) w.r.t. ( ) · AI · Q a U · ΛI · | |, Definition and Lemma 6.2.4.2 (i), and Definition 6.2.3.4 to write the rule as ( )? · AI · Q (∆ ) r̂ ←−− Σ. The latter is the rule of a monadic transducer with pattern ( )? · AI · Q · ( ) +. The observation function is defined just as in Subsection 6.4.3. Altogether we have: 6.4.5.1 Proposition. The macro tree transducers are equivalent7 to the monadic transducers M = ( ( )? · AI · Q · ( ) , %, ω ) : ∆← Σ on Setא0 where I is a finite set, Q is a cocartesian and Σ and ∆ are bicartesian. 6.4.5.2 Example. Let us now illustrate the monadic operations of the monad AI(( )?(∆ ?)+). It is helpful to have a look at Example 4.4.1.4 and Example 6.4.2.3 before. For every context variable y and terms t1, . . . tk we draw t1 . . . tk y for the applicative term ’y t1 · · · tk. The unit is simple: X ΛIT∆+AIX x λ 1 · · · k. 1 · · · k x ηX