Aperiodic String Transducers

Regular string-to-string functions enjoy a nice triple characterization through deterministic two-way transducers (2DFT), streaming string transducers (SST) and MSO definable functions. This result has recently been lifted to FO definable functions, with equivalent representations by means of aperiodic 2DFT and aperiodic 1-bounded SST, extending a well-known result on regular languages. In this paper, we give three direct transformations: i) from 1-bounded SST to 2DFT, ii) from 2DFT to copyless SST, and iii) from k-bounded to 1-bounded SST. We give the complexity of each construction and also prove that they preserve the aperiodicity of transducers. As corollaries, we obtain that FO definable string-to-string functions are equivalent to SST whose transition monoid is finite and aperiodic, and to aperiodic copyless SST

On the decomposition of finite-valued streaming string transducers

We prove the following decomposition theorem: every 1-register streaming string transducer that associates a uniformly bounded number of outputs with each input can be effectively decomposed as a finite union of functional 1-register streaming string transducers. This theorem relies on a combinatorial result by Kortelainen concerning word equations with iterated factors. Our result implies the decidability of the equivalence problem for the considered class of transducers. This can be seen as a first step towards proving a more general decomposition theorem for streaming string transducers with multiple registers

Composing Copyless Streaming String Transducers

Streaming string transducers (SSTs) implement string-to-string transformations by reading each input word in a single left-to-right pass while maintaining fragments of potential outputs in a finite set of string variables. These variables get updated on transitions of the transducer, where they can be assigned new values described by concatenations of variables and output symbols. An SST is called copyless if every update is such that no variable occurs more than once amongst all of the assigned expressions. The transformations realized by copyless SSTs coincide with Courcelle's monadic second-order logic graph transducers (MSOTs) when restricted to string graphs. Copyless SSTs with nondeterminism are known to be equivalent to nondeterministic MSOTs as well. MSOTs, both deterministic and nondeterministic, are closed under composition. Given the equivalence of MSOTs and copyless SSTs, it is easy to see that copyless SSTs are also closed under composition. The original proof of this fact, however, was based on a direct construction to produce a composite copyless SST from two given copyless SSTs. A counterexample discovered by Joost Englefriet showed that this construction may produce copyful transducers. We revisit the original composition constructions for both deterministic and nondeterministic SSTs and show that, although they can introduce copyful updates, the resulting copyful behavior they exhibit is superficial. To characterize this mild copyful behavior, we define a subclass of copyful SSTs, called diamond-free SSTs, in which two copies of a common variable are never combined in any subsequent assignment. In order to recover a modified version of the original construction, we provide a method for producing an equivalent copyless SST from any diamond-free copyful SST

Verification of graph programs with monadic second-order logic

In this thesis, we consider Hoare-style verification for the graph programming language GP 2. In literature, Hoare-style verification for graph programs has been studied by using extensions of nested conditions called E-conditions and M-conditions as assertions. However, E-conditions are only able to express first-order properties of GP 2 graphs, while M-conditions can only express properties of a non-attributed graph. Hence, there is still no logic that can express monadic second-order properties of GP 2 graphs. Moreover, both E-conditions and M-conditions may not be easy to comprehend by programmers used to formal specifications expressed in standard first-order logic. Here, we present an approach to verify GP 2 graph programs with a standard monadic second-order logic. We show how to construct a strongest liberal postcondition with respect to a rule schema and a precondition. We then extend this construction to obtain a strongest liberal postcondition for arbitrary loop-free programs. Also, we show how to construct a precondition expressing successful execution of a loop-free program, and failing execution of a so-called iteration command. These constructions allow us to define a partial proof calculus that can handle a larger class of graph programs than what can be verified by the calculus that uses E-conditions and M-conditions as assertions. Other than partial proof calculus whose assertions are monadic second-order logic, we also define semantic partial proof calculus. Similar calculus has been introduced in literature, but here we update the calculus by considering a GP 2 command that was not considered in existing work