THE JOURNAL OF SYMBOLIC LOGIC Volume 64. Number 4. Dec. 1999 THE COMPLEXITY OF DECISION PROCEDURES IN RELEVANCE LOGIC II ALASDAIR URQUHART ?1. Introduction. In this paper, we show that there is no primitive recursive decision procedure for the implication-conjunction fragments of the relevant logics R, E and T, as well as for a family of related logics. The lower bound on the complexity is proved by combining the techniques of an earlier paper on the same subject [20] with a method used by Lincoln, Mitchell, Scedrov and Shankar in proving that propositional linear logic is undecidable. The decision problem for the pure implicational fragments of E and R were solved by Saul Kripke in a tour deforce of combinatorial reasoning, published only as an abstract [9]. Belnap and Wallace extended Kripke's decision procedure to the implication-negation fragment of E in [3]; an account of their decision method is to be found in [1, pp. 124-139]. The decision method extends immediately to the implication/negation fragment of R. In fact, in the case of R we can go farther; Meyer in his thesis [13] showed how to translate the logic LR, which results from R by omitting the distribution axiom, into RA, so that the decision procedure can be extended to all of LR. This decision procedure has been implemented as a program KRIPKE by Thistlewaite, McRobbie and Meyer [17]. The program is not simply a straightforwardimplementation of the decision procedure;finite matrices are used extensively to prune invalid nodes from the search tree. The decision methods of Kripke, Belnap, Wallace and Meyer are of a truly marvelous complexity. In fact, they are so complex that it is not immediately clear how to compute an upper bound on the number of steps requiredby the procedures for an input formula of a given length. The key combinatorial lemma of Kripke that forms the basis for all these decision procedures (see [1, pp. 138-139] or Lemma 11.1 below) simply asserts the finiteness of the search tree without giving an explicit bound on its size. Here we provide a lower bound on the complexity of these decision problems by showing that there is no primitive recursive decision procedure for them. This confirms (in fact, goes beyond) a conjecture of Saul Kripke (in a letter of 1981 to Michael McRobbie [17, p. 40]) that the decidability proof for LR is unprovable in elementary recursive arithmetic. Received August 26, 1997; revised April 4, 1999. The author gratefully acknowledges the support of the National Sciences and Engineering Research Council of Canada. ? 1999. Association for Symbolic Logic 0022-48 12/99/6404-0028/$3.90 1774