The complexity of β-reduction in low orders
This paper presents the complexity of β-reduction for redexes of order 1; 2 and 3. It concludes with the following results — evaluation of Boolean expressions can be reduced to β-reduction of order 1 and β-reduction of order 1 is in O(n log n), β-reduction of order 2 is complete for PTIME, and β-reduction of order 3 is complete for PSPACE.
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