Check-in [c98b0d6d4e]

Not logged in

Overview
Comment: Commented out tentative section of universal/existential quantification. Tarball | ZIP archive | SQL archive family | ancestors | descendants | both | trunk files | file ages | folders c98b0d6d4ee03e9397bb88734a9b4fe0bc990b7d andy 2015-04-22 20:50:02
Context
 2015-04-22 20:50 Added jsonterm module, eventually to replace term_to_json (which is terrible). check-in: 0784891462 user: andy tags: trunk 20:50 Commented out tentative section of universal/existential quantification. check-in: c98b0d6d4e user: andy tags: trunk 20:49 Added poster for appreciation reception. check-in: 8078001da1 user: andy tags: trunk
Changes

Changes to report/arend-report.pdf.

cannot compute difference between binary files



Changes to report/arend-report.tex.

  1325   1325   apply a substitution $X \mapsto 1$ to $\forall X, P(X)$ it is \emph{not}
1326   1326   correct to give $\forall 1, P(1)$. In addition, consider the problem of
1327   1327   applying the substitution
1328   1328   $X \mapsto Y$ to $\forall Y, P(X,Y)$; clearly, one of the $Y$'s will need to
1329   1329   be renamed. Since the $Y$ in the substitution is most likely bound somewhere
1330   1330   else in the derivation, we choose to rename the $Y$ bound within the $\forall$.
1331   1331
1332         -\subsection{Universal and Existential Quantification}
1333         -
1334         -In the specification language, variables are implicitly quantified: variables
1335         -occurring in the head of a clause are universally quantified, while
1336         -variables that occur \emph{only} in the body of a clause are existentially
1337         -quantified.\marginnote{This is the same as the implicit quantifications rules
1338         -for Prolog.} However, the reasoning language supports both implicit and
1339         -explicit quantification. We wish to preserve compatibility between the two
1340         -languages, so that queries which are provable in the specification language,
1341         -are also provable in the reasoning language. For this reason, we adopt the
1342         -following scheme with regard to implicitly quantified variables in
1343         -reasoning queries:
1344         -\begin{itemize}
1345         -\item An unquantified variable occurring \emph{to the left} of an implication
1346         -    is implicitly $\forall$-quantified.
1347         -\item An unqualified variable occurring to the \emph{right} of an implication
1348         -    is implicitly $\exists$-quantified.
1349         -\end{itemize}
1350         -
1351         -Note that, if \texttt{:-} is regarded as $\leftarrow$, this is the opposite of
1352         -the implicit quantification in the specification. However, these rules match
1353         -the normal expectation for proof-style statements. For example:
1354         -\begin{itemize}
1355         -\item $\mathsf{nat}(X) -> \mathsf{add}(X,z,X)$ is equivalent to
1356         -    $\forall X: \mathsf{nat}(X) -> \mathsf{add}(X,z,X)$.
1357         -\item $\mathsf{nat}(X) \land \mathsf{nat}(Y) -> \mathsf{add}(X,Y,Z) \land 1358 - \mathsf{nat}(Z)$ is equivalent to $\forall X,Y: \mathsf{nat}(X) \land 1359 - \mathsf{nat}(Y) -> \exists Z: \mathsf{add}(X,Y,Z) \land \mathsf{nat}(Z)$.
1360         -\end{itemize}
1361         -
1362         -Internally, when processing a proof tree, Arend maintains two sets of
1363         -in-scope variables: \emph{universals} and \emph{existentials}. Universals are
1364         -those introduced by the $\forall$-R and $\exists$-L rules; they implicitly
1365         -represent fresh \emph{constants} and cannot be bound by unification.
1366         -Existentials are introduced by $\exists$-R and $\forall$-L rules; they are
1367         -true variables which have simply not been instantiated yet. In fact, the
1368         -behavior of unification is to supply the instantiation of one or more
1369         -existential variables, possibly in terms of the in-scope universal variables.
1332  +%\subsection{Universal and Existential Quantification}
1333  +%
1334  +%In the specification language, variables are implicitly quantified: variables
1335  +%occurring in the head of a clause are universally quantified, while
1336  +%variables that occur \emph{only} in the body of a clause are existentially
1337  +%quantified.\marginnote{This is the same as the implicit quantifications rules
1338  +%for Prolog.} However, the reasoning language supports both implicit and
1339  +%explicit quantification. We wish to preserve compatibility between the two
1340  +%languages, so that queries which are provable in the specification language,
1341  +%are also provable in the reasoning language. For this reason, we adopt the
1342  +%following scheme with regard to implicitly quantified variables in
1343  +%reasoning queries:
1344  +%\begin{itemize}
1345  +%\item An unquantified variable occurring \emph{to the left} of an implication
1346  +%    is implicitly $\forall$-quantified.
1347  +%\item An unqualified variable occurring to the \emph{right} of an implication
1348  +%    is implicitly $\exists$-quantified.
1349  +%\end{itemize}
1350  +%
1351  +%Note that, if \texttt{:-} is regarded as $\leftarrow$, this is the opposite of
1352  +%the implicit quantification in the specification. However, these rules match
1353  +%the normal expectation for proof-style statements. For example:
1354  +%\begin{itemize}
1355  +%\item $\mathsf{nat}(X) -> \mathsf{add}(X,z,X)$ is equivalent to
1356  +%    $\forall X: \mathsf{nat}(X) -> \mathsf{add}(X,z,X)$.
1357  +%\item $\mathsf{nat}(X) \land \mathsf{nat}(Y) -> \mathsf{add}(X,Y,Z) \land 1358 +% \mathsf{nat}(Z)$ is equivalent to $\forall X,Y: \mathsf{nat}(X) \land 1359 +% \mathsf{nat}(Y) -> \exists Z: \mathsf{add}(X,Y,Z) \land \mathsf{nat}(Z)$.
1360  +%\end{itemize}
1361  +%
1362  +%Internally, when processing a proof tree, Arend maintains two sets of
1363  +%in-scope variables: \emph{universals} and \emph{existentials}. Universals are
1364  +%those introduced by the $\forall$-R and $\exists$-L rules; they implicitly
1365  +%represent fresh \emph{constants} and cannot be bound by unification.
1366  +%Existentials are introduced by $\exists$-R and $\forall$-L rules; they are
1367  +%true variables which have simply not been instantiated yet. In fact, the
1368  +%behavior of unification is to supply the instantiation of one or more
1369  +%existential variables, possibly in terms of the in-scope universal variables.
1370   1370
1371   1371
1372   1372
1373   1373   \subsection{Unification}
1374   1374
1375   1375   The Robinson unification algorithm is presented, in natural deduction style,
1376   1376    in figure~\ref{unification-rules}.