Relation Algebras Can Tile

Undecidability of the equational theory of the class RA of relation algebras can easily be proved using the undecidability of the word-problem for semigroups. With some eeort and ingenuity, one can push this proof through for the larger class SA. We provide another \cause" for undecidability which works for even larger classes than SA. The reason is that we can encode the tiling problem. In doing so we will meet very simple BAO-varieties with undecidable equational theories which might be useful in other undecidability proofs. Our work is part of the research project which tries to establish the border between undecidability and decidability in relational type The ultimate goal of this research is to come up with versions of relational algebra which are still suitable for modern dynamic applications but whose equational theory is decidable or even tractable. (Maddux, 1994): Corollary 16 shows that the equational theory of any variety K such that RRA K SA is undecidable. The ingenious proof goes by showing that for every quasi-equation q in the language of semigroups, there is an eeectively obtainable equation q in the language of the class K such that q is valid in all semigroups if and only if q is valid in K. The theorem follows then immediately from the fact that the quasi-equational theory of semigroups is undecidable. (Maddux, 1980) uses the same problem 1 to prove undecidability of Df 3 and CA 3. (Andr eka et al., 1995) also use (versions of) the semigroup-wordproblem to show undecidability of weakenings of RA (but where composition is associative). We prove a strengthening of Maddux' SA-theorem using a very diierent undecidable problem: N N tiling. We also show that it will be very hard (maybe impossible) to obtain the same result using the quasi-equational theory of semigroups (Theorems 4.1 and 4.3). This covers the rst theme of the paper: nd a \diierent cause" for undecidability in relational type algebras. This answers a question raised in (van Benthem, 1996) p. 174 about the connection between these two styles of undecidability-analysis regarding varieties of the relational algebraic type. We restrict the discussion here to the case in which all algebras are boolean algebras with operators (BAO's). So when we say that an axiom A causes undecidability, we really mean A plus the BAO-axioms causes undecidability. The second theme is to show how easily we get undecidable varieties. If we look at …