We characterize regular string transductions as programs in a linear
λ-calculus with additives. One direction of this equivalence is proved
by encoding copyless streaming string transducers (SSTs), which compute regular
functions, into our λ-calculus. For the converse, we consider a
categorical framework for defining automata and transducers over words, which
allows us to relate register updates in SSTs to the semantics of the linear
λ-calculus in a suitable monoidal closed category. To illustrate the
relevance of monoidal closure to automata theory, we also leverage this notion
to give abstract generalizations of the arguments showing that copyless SSTs
may be determinized and that the composition of two regular functions may be
implemented by a copyless SST. Our main result is then generalized from strings
to trees using a similar approach. In doing so, we exhibit a connection between
a feature of streaming tree transducers and the multiplicative/additive
distinction of linear logic.
Keywords: MSO transductions, implicit complexity, Dialectica categories,
Church encodingsComment: 105 pages, 24 figure