Abstract
We investigate the proof complexity of Salomaa’s axiom system \(F_1\) for regular expression equivalence. We show that for two regular expression E and F over the alphabet \(\varSigma \) with \(L(E)=L(F)\) an equivalence proof of length \(O\left( |\varSigma |^4\cdot \textsc {Tower}(\max \{h(E),h(F)\}+4)\right) \) can be derived within \(F_1\), where h(E) (h(F), respectively) refers to the height of E (F, respectively) and the tower function is defined as \(\textsc {Tower}(1)=2\) and \(\textsc {Tower}(k+1)=2^{\textsc {Tower}(k)}\), for \(k\ge 1\). In other words
This is in sharp contrast to the fact, that regular expression equivalence admisses exponential proof length if not restricted to the axiom system \(F_1\). From the theoretical point of view the exponential proof length seems to be best possible, because we show that regular expression equivalence admits a polynomial bounded proof if and only if \(\textsf {NP}=\textsf {PSPACE}\).
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Notes
- 1.
For convenience, parentheses in regular expressions are sometimes omitted and the concatenation is simply written as juxtaposition. The priority of operators is specified in the usual fashion: concatenation is performed before union, and star before both product and union.
- 2.
The notation \((E_1,E_2)\equiv (F_1,F_2)\), for regular expressions \(E_1\), \(E_2\), \(F_1\), and \(F_2\), stands for \(E_1\equiv F_1\) and \(E_2\equiv F_2\). The equation \((E_1,E_2)=(F_1,F_2)\) is a shorthand notation for the system of the two equations \(E_1=F_1\) and \(E_2=F_2\). Furthermore, the expressions \((E_1,E_2)+(F_1,F_2)\) and \((E_1,E_2)\cdot F_1\) define \((E_1+F_1,E_2+F_2)\) and \((E_1\cdot F_1,E_2\cdot F_1)\), respectively.
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Beier, S., Holzer, M. (2017). On Regular Expression Proof Complexity. In: Charlier, É., Leroy, J., Rigo, M. (eds) Developments in Language Theory. DLT 2017. Lecture Notes in Computer Science(), vol 10396. Springer, Cham. https://doi.org/10.1007/978-3-319-62809-7_5
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DOI: https://doi.org/10.1007/978-3-319-62809-7_5
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