On the Sequential Nature of Unification

The problem of unification of terms is log-spacc complete for P. In deriving this lower bound no use is made of the potentially concise representation of terms by directed acyclic graphs. In addition, the problem remains complete even if infinite substitutions arc allowed. A consequence of this result is that parallelism cannot significantly improve on the best sequential solutions for unification. The duar"probcm of computing the congruence closure of an equivalence relation is *, . 3i,. also log-space complete for P. However. we-show that for the problem of term matching, an important subase of unification, there is a good parallel algorithm using O(log2 ntime and nOQ) processors on a PRAM. For the O(log 2 n) parallel time upper bound w-asswnethat the terms are represented by directed acyclic graphs; if the longer string representation is usedw lnali an * O(log n) parallel time bound I's 1. Introductioin Unification is an important step in resolution theorem proving [R] with applications to a variety of symbolic computation problems. In particular, unification is used in PROLOG interpreters [CM]. ,, ~ type inference algorithms [MI, and term rewriting systems [GKM]. Many symbol manipulation problems are inherently difficult and thus do not have efficient solutions. Theorem provers and PROLOG interpreters do not always give us the answers we want fast enough. One way to combat the difficulty of these problems is by coordinating many processors to solve a single problem instance by working on several subproblems in parallel Although there are a number of ways to introduce parallelism into interpreters [S] and theorem provers, unification is a prime target since it is the most commonly repeated operation in these tasks. However, our analysis suggests that parallel unification algorithms will not perform significantly faster than the best sequential algorithms known .4 (e.g.. [PW] runs in linear time). We show that, unless PKNC, an unlikely twist of complexity theory [C], no parallel algorithm for unification will run in time bounded by a polynomial in the logarithm of the input size, and using a number of processors bounded by a polynomial in the size of the input. We use the PRAM of [FW1 as our model of parallel computation, although we could, just as well, have used any other "reasonable parallel model" [J]. 'uppted by a anreil Fellowship. Suoed pmrly by NSF grat MCS-I210830 and partly by ONR-DARPA rant l... N00014-$3-K-0146. Supponed by an IBM Fellowship. -. ,....._.. , -. , '. ,' -,. ", ..-. , .-. ,-.,-. •.,..: . . TTC.I ,,-7 V11%7--%": 7-i--,.