has a sister attribute (or ).

1.  The AVMs [${\left[\begin{array}{ll} {\mbox{\sc v}} &0\\ {\mbox{\sc 0}} &{\mbox{\it +}} \end{array}\right]}$]{} and [${\left[\begin{array}{ll} {\mbox{\sc v}} &1\\ {\mbox{\sc 1}} &{\mbox{\it +}} \end{array}\right]}$]{} encode the natural numbers zero and one.

2.  The AVMs [${\left[\begin{array}{ll} {\mbox{\sc v}} &0\\ {\mbox{\sc 0}} &[F] \end{array}\right]}$]{} and [${\left[\begin{array}{ll} {\mbox{\sc v}} &1\\ {\mbox{\sc 1}} &[F] \end{array}\right]}$]{} encode natural numbers iff the AVM $[F]$ encodes a natural number.

#### Syntactic rules that tests two numbers for equality.

Assume a nonterminal $A$ with some AVM [${\left[\begin{array}{ll} {\mbox{\sc n}} &[F]\\ {\mbox{\sc m}} &[H] \end{array}\right]}$]{}, where $[F]$ and $[H]$ encode natural number $x$ and $y$, respectively. We present one syntactic rule that derives from this nonterminal $A$ a nonterminal $B$ with AVM [${\left[\begin{array}{ll} {\mbox{\sc n}} &[F]\\ {\mbox{\sc m}} &[H] \end{array}\right]}$]{} if $x = y$.

Clearly, this simple test takes one step. A more sophisticated test, which also tests for inequality, would compare $[F]$ and $[G]$ bit-by-bit. Such a test would take $O(\min(\log(x),\log(y)))$ $=O(\min({\mathopen|[F]\mathclose|},{\mathopen|[H]\mathclose|}))$ derivation steps.

#### Syntactic rules that multiply by two.

Assume a nonterminal $A$ with some AVM ${\left[\begin{array}{ll} {\mbox{\sc n}} &[F] \end{array}\right]}$, where $[F]$ encodes natural number $x$. We present one syntactic rule that derives from this nonterminal $A$ a nonterminal $B$ with 