
\subsection{Convergence}


When no blame is assigned 
during
a binary game 
in an optimization lab,
the optimum claim will eventually be found.

{\bf Theorem} [Convergence]:
Consider a set $C$ of claims $c(t),$ where $t$ is a real number between 0 and 1.
$c(0)$ is true, and $c(1)$ is false and 
there is an optimal value $t_0$ of $t$ where the truth value of $c(t_0)$ switches
from true to false.
If a sequence of binary games is played using claims in $C$
and binary search without faulty actions, the optimal 
claim $c(t_0)$ will be found.

\subsection{Indeterminate Claims}

SCG can express indeterminate claims that are neither true nor false.
Such claims were studied in Independence Friendly Logic
\cite{sep-if}, an extension of first-order logic.

Consider the following lab:
{\it Instance} = the set of positive real numbers = {\it InstanceSet}.
{\it Solution} = the set of real numbers.
The $valid(i,s)$  function checks that the solution $s$ is the square root of
instance $i$.
The protocol is: \(P:i[0], O:s[1]\) of \( i[0], P:s[2]\) of \( i[0].\)
%\begin{verbatim}
%instance from Proponent
%solution of 0 from Opponent //sO
%solution of 0 from Proponent //sP
%\end{verbatim}
The protocol predicate is: \(s[1]=s[2]\). According to the SCG rules,
$s[1]$ is not known when $s[2]$ is computed.
The lab contains only one claim which
% where the quality is irrelevant.
is neither true nor false: it is indeterminate.
Notice the similarity to the "at least as good as" claim discussed earlier.

{\bf Theorem} [ExistIndeterminateClaims]:
There are indeterminate SCG claims.

%Therefore, we also distinguish between bivalent labs (all claims
%are either true or false) and non-bivalent labs (some claims
%are indeterminate).


