The Computational Complexity ofTaxonomic

McAllester, Given, and Fatima 10] have developed a procedure for infering tax-onomic relationships between classes deened by predicates and relations. Their decision procedure runs in O(n 3) time and O(n 2) space on a sequential random-access machine (RAM). I have investigated the computational complexity of this inference problem with a view to seeing whether faster sequential algorithms or good parallel algorithms might be found. A restricted form of the taxonomic inference task of McAllester, et al, which I will call taxonomic closure, can be seen as a generalization of congruence closure, which has been investigated by Kozen 6], Nelson and Oppen 11], and Downey, Sethi, and Tarjan 4]. I will show that the decision problems corresponding to both taxonomic closure and congruence closure are P-complete, even if terms are restricted to contain only monadic function applications. Thus these problems probably cannot be eeciently parallelized. I also show that the monadic taxonomic closure decision problem is complete for two-way non-deterministic pushdown automata (2NPDA) | a problem class for which the best known algorithm takes O(n 3) time on a sequential RAM. The negative implications of these results for taxonomic and congruence closure apply to the more powerful taxonomic inference system of McAllester, et al, and to the later extension of this system to \Montague literals" by McAllester and Givan 9]. I will discuss the signiicance of these results for engineering applications and for McAllester and Givan's speculation that the decision procedure for Montague literals might explain some aspects of natural language. The system for taxonomic inference that I will discuss is a subset of a more general system of inference for \taxonomic literals" described by McAllester, et 1