Laboratoire de l'Informatique et du Parallélisme » Linear Arithmetic

\section{Soundness}

\label{sect:soundness}

The main result of this section is the following:

\begin{theorem}

	\label{thm:soundness}

	If $\arith^i$ proves $\forall \vec u^\normal . \forall \vec x^\safe . \exists y^\safe . A(\vec u ; \vec x , y)$ then there is a $\mubci{i-1}$ program $f(\vec u ; \vec x)$ such that $\Nat \models A(\vec u ; \vec x , f(\vec u ; \vec x))$.

\end{theorem}

The main problem for soundness is that we have predicates, for example equality, that take safe arguments in our theory but do not formally satisfy the polychecking lemma for $\mubc$ functions. 

For this we will use length-bounded witnessing, borrowing a similar idea from Bellantoni's previous work \cite{Bellantoni95}.

\begin{definition}

[Length bounded basic functions]

We define \emph{length-bounded equality}, $\eq(l;x,y)$ as the characteristic function of the predicate:

\[

x \mode l = y \mode l

\]

which is definable by safe recursion on $l$:

\[

\begin{array}{rcl}

\eq (0 ; x,y) & \dfn & \equivfn (;\bit (0;x),\bit(0;y) ) \\

\eq (\succ i l; x,y) & \dfn & \cond (; \eq ( u;x,y ) , 0, \equivfn (; \bit (\succ i u ; x ) , \bit (\succ i l ; y ))  )

\end{array}

\]

We also define length-bounded inequality as:

\[

\begin{array}{rcl}

\leqfn (0 ; x ,y) & \dfn & \cimp (; \bit (0;x), \bit (0;y) ) \\

\leqfn (\succ i l ; x,y) & \dfn & \orfn ( ; <(\bit (\succ i l ; x) , \bit(\succ i l ; y) ) , \andfn (; \equivfn (\bit (\succ i l ; x) , \bit(\succ i l ; y)) , \leqfn (l;x,y ) ) )

\end{array}

\]

\end{definition}

\anupam{Do we need the general form of length-boundedness? E.g. the $*$ functions from Bellantoni's paper? Put above if necessary. Otherwise just add sequence manipulation functions as necessary.}

Notice that $\eq$ is a polymax bounded polyomial checking function on its normal input, and so can be added to $\mubc$ without problems.

\begin{definition}

	[Length bounded characteristic functions]

	We define $\mubci{}$ programs $\charfn{}{A} (l , \vec u; \vec x)$, parametrised by a formula $A(\vec u ; \vec x)$, as follows.

%	If $A$ is a $\Pi_{i}$ formula then:

	\[

	\begin{array}{rcl}

	\charfn{}{s\leq t} (l, \vec u ; \vec x) & \dfn & \leqfn(l;s,t) \\

	\smallskip

	\charfn{}{s=t} (l, \vec u ; \vec x) & \dfn & \eq(l;s,t) \\

	\smallskip

	\charfn{}{\neg A} (l, \vec u ; \vec x) & \dfn & \neg (;\charfn{}{A}(l , \vec u ; \vec x)) \\

	\smallskip

	\charfn{}{A\cor B} (l, \vec u ; \vec x ) & \dfn & \cor (; \charfn{}{A} (l, \vec u ; \vec x , w), \charfn{}{B} (\vec u ;\vec x) ) \\

	\smallskip

	\charfn{}{A\cand B} (l, \vec u ; \vec x ) & \dfn & \cand(; \charfn{}{A} (l, \vec u ; \vec x , w), \charfn{}{B} (\vec u ;\vec x) ) \\

	\smallskip

	\charfn{}{\exists x^\safe . A(x)} (l, \vec u ;\vec x) & \dfn & \begin{cases}

	1 & \exists x^\safe . \charfn{}{A(x)} (l, \vec u ;\vec x , x) = 1 \\

	0 & \text{otherwise} 

	\end{cases} \\

	\smallskip

	\charfn{}{\forall x^\safe . A(x)} (l, \vec u ;\vec x ) & \dfn & 

	\begin{cases}

	0 & \exists x^\sigma. \charfn{}{ A(x)} (l, \vec u; \vec x , x) = 0 \\

	1 & \text{otherwise}

	\end{cases}

	\end{array}

	\]

\end{definition}

\begin{proposition}

	$\charfn{}{A} (l, \vec u ; \vec x)$ is the characteristic function of $A (\vec u \mode l ; \vec x \mode l)$.

\end{proposition}

\begin{definition}

	[Length bounded witness function]

	We now define $\Wit{\vec u ; \vec x}{A} (l , \vec u ; \vec x)$ for a $\Sigma_{i+1}$-formula $A$ with free variables amongst $\vec u; \vec x$.

	\[

	\begin{array}{rcl}

	\Wit{\vec u ; \vec x}{A} (l, \vec u ; \vec x , w) & \dfn & \charfn{}{A} (l, \vec u ; \vec x)  \text{ if $A$ is $\Pi_i$} \\

	\smallskip

	\Wit{\vec u ; \vec x}{A \cor B} (l,\vec u ; \vec x , \vec w^A , \vec w^B) & \dfn & \cor ( ; \Wit{\vec u ; \vec x}{A} (l,\vec u ; \vec x , \vec w^A) ,\Wit{\vec u ; \vec x}{B} (l,\vec u ; \vec x , \vec w^B)  )  \\

	\smallskip

	\Wit{\vec u ; \vec x}{A \cand B} (l,\vec u ; \vec x , \vec w^A , \vec w^B) & \dfn & \cand ( ; \Wit{\vec u ; \vec x}{A} (l,\vec u ; \vec x , \vec w^A) ,\Wit{\vec u ; \vec x}{B} (l,\vec u ; \vec x , \vec w^B)  )  \\

	\smallskip

	\Wit{\vec u ; \vec x}{\exists x^\safe . A(x)} (l,\vec u ; \vec x , \vec w , w) & \dfn & \Wit{\vec u ; \vec x , x}{A(x)} ( l,\vec u ; \vec x , w , \vec w )

	\\

	\smallskip

	\Wit{\vec u ; \vec x}{\forall u \leq |t(\vec u;)| . A(x)} (l , \vec u ; \vec x, w) & \dfn & 

	\forall u \leq |t(\vec u;)| . \Wit{u , \vec u ; \vec x}{A(u)} (l, u , \vec u ; \vec x, \beta(u;w) )

	\end{array}

	\]

\end{definition}

\anupam{may as well use a single witness variable since need it for sharply bounded quantifiers anyway.}

\anupam{sharply bounded case obtained by sharply bounded lemma}

\begin{proposition}

	If, for some $w$, $\wit{\vec u ; \vec x}{A} (l, \vec u ; \vec x,w) =1$, then $A (\vec u \mode l ; \vec x \mode l)$ is true.

	\anupam{check statement, need proof-theoretic version?}

\end{proposition}

In order to prove Thm.~\ref{thm:soundness} we need the following lemma:

\begin{lemma}

	[Proof interpretation]

	\label{lem:proof-interp}

	For any $\arith^i$ proof of a $\Sigma^\safe_i$ sequent $\Gamma \seqar \Delta$, there is a $\mubci{i-1}$ function $f$ such that, for any $l, \vec u , \vec x  , w$, we have:

	\[

	\wit{\vec u ; \vec x}{ \wedge \Gamma } (l, \vec u ; \vec x , w) 

	\quad \leq \quad

	\wit{\vec u ; \vec x}{\vee \Delta} (l, \vec u ; \vec x , f(l, \vec u ; \vec x , w))

	\]

	\anupam{maybe want $f(\vec u \mode l ; \vec x \mode l , w)$}

	\anupam{Also, perhaps split for formulae of $\Gamma$, to avoid lots of (de)coding}

\end{lemma}

\begin{proof}

	We assume the proof, say $\pi$, is in integer positive free-cut free form, by the results from the previous section.

	This means that the predicate $\charfn{\vec u ; \vec x}{A}$ is defined for each formula $A(\vec u ; \vec x)$ occurring in a proof, so the theorem is well-stated.

	We define the function $f$ inductively, by considering the various final rules of $\pi$.

	\paragraph*{Negation}

	Can assume only on atomic formulae, so no effect.	

	\paragraph*{Quantifiers}

	\anupam{Do $\exists$-right and $\forall$-right, left rules are symmetric.}

	\paragraph*{Contraction}

	Left contraction simply duplicates an argument, whereas right contraction requires a conditional on a $\Sigma^\safe_i$ formula.

	\paragraph*{Induction}

	Corresponds to safe recursion on notation.

	Suppose final step is:

	\[

	\vlinf{\pind}{}{\Gamma , \normal (t) , A(0) \seqar A(t) , \Delta}{ \left\{\Gamma , \normal (u) , A(u) \seqar A(\succ i u ) , \Delta \right\}_{i=0,1} }

	\]

	For simplicity we will assume $\Delta $ is empty, which we can always do by Prop.~\todo{DO THIS!}

		Now, by the inductive hypothesis, we have functions $h_i$ such that:

		\[

		\wit{u , \vec u ; \vec x}{LHS} (l , u , \vec u ; \vec x ,  w) =1

		\quad \implies \quad

		\wit{u , \vec u ; \vec x}{RHS} (l , u , \vec u ; \vec x ,  h_i (u \mode l , \vec u \mode l ; \vec x \mode l) ) =1

		\]

	We define $ f$ as follows:

	\[

	\begin{array}{rcl}

	 f (0 , \vec u ; \vec x, \vec w^\Gamma , w^{\normal (t)} , w^{A(0)}) & \dfn &  w^{A(0)} \\

	 f( \succ i u , \vec u ; \vec x , \vec w^\Gamma , w^{\normal (t)} , w^{A(0)}) & \dfn & 

	 h_i (u , \vec u ; \vec x , \vec w^\Gamma w^{\normal (?)}, f(u , \vec u ; \vec x , \vec w ))

	\end{array}

	\]

	\anupam{Must check above, could be problems in recursive case.}

	\anupam{Wait, should $\normal (t)$ have a witness? Also there is a problem like Patrick said for formulae like: $\forall x^\safe . \exists y^\safe. (\normal (z) \cor \cnot \normal (z))$, where $z$ is $y$ or otherwise.}

\end{proof}

We are now ready to prove the soundness theorem.

\begin{proof}

	[Proof of Thm.~\ref{thm:soundness} from Lemma~\ref{lem:proof-interp}]

	(watch out for dependence on $l$, try do without)

	Suppose $\arith^i \proves \forall \vec u^\normal . \exists x^\normal . A(\vec u ; x)$. Then by raising, inversion, free-variable normal forms, we have a proof of,

	\[

	\normal (\vec u ) \seqar \exists x^\normal . A(\vec u , x ;)

	\]

	whence, by Lemma~\ref{lem:proof-interp}, we have a $\mubci{i-1}$ function $f$ such that:

	\[

	\vec u \mode l = \vec a \mode l

	\quad \implies \quad

	\wit{\vec u ; }{A} ( l , \vec u , f(\vec u \mode l;) ) =1

	\]

	Now it suffices to choose an $l$ bigger that both all the $\vec u$ and $f(\vec u)$, which is a polynomial in $\vec u$ by the polymax bounding lemma.

\end{proof}