THE COMPLEXITY OF DECISION PROCEDURES IN RELEVANCE LOGIC Alasdair Urquhart Philosophy and Computer Science University of Toronto Abstract Lower and upper bounds are proved for the problem of whether a given pure implicational formula is derivable in the system R of relevant implication (equivalently, in Church’s theory of weak implication). The problem is shown to be exponential space hard under log-lin reducibility by adapting a method used by Mayr and Meyer to prove the same lower bound for the word problem for commutative semigroups. Results of McAloon are adapted to show that the problem is primitive recursive in the Ackermann function. 1 Introduction Relevance logic is distinguished among non-classical logics by the richness of its mathematical as well as philosophical structure. This richness is nowhere more evident than in the difficulty of the decision problem for the main rele- vant propositional logics. These logics are in general undecidable [22]; however, there are important decidable subsystems. The decision procedures for these subsystems convey the impression of great complexity. The present paper gives a mathematically precise form to this impression by showing that the decision problem for R → , the implication fragment of R, is exponential space hard. This means that any Turing machine which solves this decision problem must use an exponential amount of space (relative to the input size) on infinitely many inputs. The first-degree fragment common to E, R and most other well-known rele- vance logics has a simple decision procedure. The details of the Anderson-Belnap decision procedure are to be found in [4, Chapter III] A four-valued matrix is characteristic for this fragment; since the decision problem for the two-valued propositional calculus can be reduced efficiently to the decision problem for this fragment, the complexity of the decision problem is the same as that for the classical two-valued propositional calculus (that is to say, the problem is NP-complete). 1