# Logics for Word Transductions with Synthesis

## TL;DR

This paper introduces LT, a logic for expressing properties of transductions with a focus on synthesis, showing that any LT-definable transduction can be realized by a regular functional transduction, bridging logic and automata.

## Contribution

The paper presents a new logic LT for transductions, proves its expressive power and decidability, and establishes a synthesis method linking LT-definable transductions to regular functions.

## Key findings

- LT can express all regular functional transductions.
- Decidable satisfiability and equivalence for LT with specific complexities.
- Any LT-definable transduction can be synthesized into a regular functional transduction.

## Abstract

We introduce a logic, called LT, to express properties of transductions, i.e. binary relations from input to output (finite) words. In LT, the input/output dependencies are modelled via an origin function which associates to any position of the output word, the input position from which it originates. LT is well-suited to express relations (which are not necessarily functional), and can express all regular functional transductions, i.e. transductions definable for instance by deterministic two-way transducers. Despite its high expressive power, LT has decidable satisfiability and equivalence problems, with tight non-elementary and elementary complexities, depending on specific representation of LT-formulas. Our main contribution is a synthesis result: from any transduction R defined in LT , it is possible to synthesise a regular functional transduction f such that for all input words u in the domain of R, f is defined and (u,f(u)) belongs to R. As a consequence, we obtain that any functional transduction is regular iff it is LT-definable. We also investigate the algorithmic and expressiveness properties of several extensions of LT, and explicit a correspondence between transductions and data words. As a side-result, we obtain a new decidable logic for data words.