Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
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WALTER TAYLOR 25<br />
greatest example <strong>of</strong> an undecidable set is given by GOdel's incompleteness theorem<br />
[op. cit.], about which we will say no more. Here we will discuss decidability<br />
properties <strong>of</strong> equations. When we say that a property P <strong>of</strong> finite sets E <strong>of</strong> equations is<br />
(un)decidable, we mean that<br />
is (un)decidable.<br />
w0=<br />
We may first ask when a theory E itself is a decidable set <strong>of</strong> equations, or, as it is<br />
frequently put, "the word problem for free E-algebras is solvable." (See the discussion<br />
<strong>of</strong> word problems below.) There are two common methods for showing that E is a<br />
decidable equational theory, the first being to find a recursive procedure to convert<br />
every term o to a unique ("normal form") term o' with (o = o') E 2 and such that if o<br />
and r are distinct normal forms then (o = r) 2. (The decision procedure then<br />
reduces to comparison <strong>of</strong> normal forms - and conversely, a decision procedure for E<br />
obviously implies the existence <strong>of</strong> normal forms.) E.g. every group term reduces<br />
uniquely to either 1 or<br />
.nl .n2 n k<br />
Xl x --' Xik,<br />
where xil : xi2 :'" : Xik. Several <strong>of</strong> the best known equational theories are<br />
decidable, as may be seen similarly. See e.g. Margaris [284] for implicative<br />
semilattices, following work <strong>of</strong> McKay and Diego. Eq(2,+) is decidable (in fact its full<br />
first order theory is decidable, by Ehrenfeucht and Bfichi [70])- a simple method<br />
given by Selman and Zimbarg-Sobrinho [unpublished] is closely related to<br />
Karn<strong>of</strong>sky's identities 8.12 above. Martin [ 286] gave a decision procedure for (2,4-,.)<br />
with normal forms (cf. 9.20 above). Richardson [377] gave normal forms for<br />
(co, l,x+y,xy,xY) (cf. Problem 2 in õ8). Finally, we remark that the Birkh<strong>of</strong>f-Witt<br />
theorem yields a procedure for finding normal forms for (free) Lie algebras and rings,<br />
as observed by P. Hall [175]. See Bergman [43] for some detailed methods for<br />
finding normal forms, mainly in ring theory; also see [ 158] and [248].<br />
Notice that representing free algebras uniquely via terms (as we did for FK(X ) in<br />
õ2) really requires a normal form. Often a normal form is required for finding the<br />
cardinality <strong>of</strong> Fv(X), a topic we will come to in 14.5 below. For some other results<br />
related to normal forms, see Hule [194].