The Computational Power of Parsing Expression Grammars

Abstract

We propose a new computational model, the scaffolding automaton, which exactly characterises the computational power of parsing expression grammars (PEGs). Using this characterisation we show that: PEGs have unexpected power and semantics. We present several PEGs with surprising behaviour, and languages which, unexpectedly, have PEGs, including a PEG for the language of palindromes whose length is a power of two.PEGs are computationally “universal”, in the following sense: take any computable function; then there exists a computable function such that has a PEG.There can be no pumping lemma for PEGs. There is no total computable function A with the following property: for every well-formed PEG G, there exists such that for every string of size the output is in and has |x|.PEGs are strongly non real-time for Turing machines. There exists a language with a PEG, such that neither it nor its reverse can be recognised by any multi-tape online Turing machine which is allowed to do only steps after reading each input symbol.