Survey 1979: Equational Logic - Department of Mathematics ...

WALTER TAYLOR 25
greatest example of an undecidable set is given by GOdel's incompleteness theorem
[op. cit.], about which we will say no more. Here we will discuss decidability
properties of equations. When we say that a property P of finite sets E of equations is
(un)decidable, we mean that
is (un)decidable.
w0=
We may first ask when a theory E itself is a decidable set of equations, or, as it is
frequently put, "the word problem for free E-algebras is solvable." (See the discussion
of word problems below.) There are two common methods for showing that E is a
decidable equational theory, the first being to find a recursive procedure to convert
every term o to a unique ("normal form") term o' with (o = o') E 2 and such that if o
and r are distinct normal forms then (o = r) 2. (The decision procedure then
reduces to comparison of normal forms - and conversely, a decision procedure for E
obviously implies the existence of normal forms.) E.g. every group term reduces
uniquely to either 1 or
.nl .n2 n k
Xl x --' Xik,
where xil : xi2 :'" : Xik. Several of the best known equational theories are
decidable, as may be seen similarly. See e.g. Margaris [284] for implicative
semilattices, following work of McKay and Diego. Eq(2,+) is decidable (in fact its full
first order theory is decidable, by Ehrenfeucht and Bfichi [70])- a simple method
given by Selman and Zimbarg-Sobrinho [unpublished] is closely related to
Karnofsky's identities 8.12 above. Martin [ 286] gave a decision procedure for (2,4-,.)
with normal forms (cf. 9.20 above). Richardson [377] gave normal forms for
(co, l,x+y,xy,xY) (cf. Problem 2 in õ8). Finally, we remark that the Birkhoff-Witt
theorem yields a procedure for finding normal forms for (free) Lie algebras and rings,
as observed by P. Hall [175]. See Bergman [43] for some detailed methods for
finding normal forms, mainly in ring theory; also see [ 158] and [248].
Notice that representing free algebras uniquely via terms (as we did for FK(X ) in
õ2) really requires a normal form. Often a normal form is required for finding the
cardinality of Fv(X), a topic we will come to in 14.5 below. For some other results
related to normal forms, see Hule [194].