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 $f:\{0, 1\}^*\rightarrow \{0, 1\}^*$; then there exists a computable function $g: \{0, 1\}^*\rightarrow {\mathbb {N}}$ such that $\{ f(x) \#^{g(x)} x \mid x \in \{0, 1\}^*\}$ 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 $n_0$ such that for every string $x \in {\mathcal {L}}(G)$ of size $|x| \ge n_0$, the output $y = A(G, x)$ is in ${\mathcal {L}}(G)$ and has $|y| > |x|$.

• 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 $o(n/\log n)$ steps after reading each input symbol.

Bruno Loff is the recipient of FCT postdoc grant number SFRH/BPD/116010/2016. This work is partially funded by the ERDF through the COMPETE 2020 Programme within project POCI-01-0145-FEDER-006961, and by National Funds through the FCT as part of project UID/EEA/50014/2013. This work was partially supported by CMUP (UID/MAT/00144/2013), FCT (Portugal), FEDER and PT2020. We thank M. Kutrib and M. Holzer for the fruitful discussions on this subject.