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Overview

Comment: | Added material to the report on the handling of explicit quantifiers. |
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Downloads: | Tarball | ZIP archive | SQL archive |

Timelines: | family | ancestors | descendants | both | forall-exists |

Files: | files | file ages | folders |

SHA1: | 8a438a8abdfb67b6d72fac2b284dc0fc06ce5190 |

User & Date: | andy 2015-04-22 10:30:20 |

Context

2015-04-22
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20:49 | Added poster for appreciation reception. check-in: 8078001da1 user: andy tags: trunk | |

10:30 | Added material to the report on the handling of explicit quantifiers. Leaf check-in: 8a438a8abd user: andy tags: forall-exists | |

10:29 | Presentation work. Added PDF of presentation. Leaf check-in: a24ff57f47 user: andy tags: trunk | |

Changes

Changes to report/arend-report.pdf.

cannot compute difference between binary files

Changes to report/arend-report.tex.

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is implicitly $\exists$-quantified. \end{itemize} Note that, if \texttt{:-} is regarded as $\leftarrow$, this is the opposite of the implicit quantification in the specification. However, these rules match the normal expectation for proof-style statements. For example: \begin{itemize} \item $\mathsf{nat}(X) -> \mathsf{add}(X,a,X)$ is equivalent to $\forall X: \mathsf{nat}(X) -> \mathsf{add}(X,a,X)$. \item $\mathsf{nat}(X) \land \mathsf{nat}(Y) -> \mathsf{add}(X,Y,Z) \land \mathsf{nat}(Z)$ is equivalent to $\forall X,Y: \mathsf{nat}(X) \land \mathsf{nat}(Y) -> \exists Z: \mathsf{add}(X,Y,Z) \land \mathsf{nat}(Z)$. \end{itemize} Internally, when processing a proof tree, Arend maintains two sets of in-scope variables: \emph{universals} and \emph{existentials}. Universals are ................................................................................ (This presentation is based in part on that in \mcitet{pfenning2006logic}.) It is interesting to directly compare the two implementations of unification (one of the few modules which is semantically similar in both implementations). Both systems implement the same algorithm. The Javascript implementation consists of 136 lines of non-comment code and is quite difficult to follow; the Prolog implementation requires only 69 lines, an almost 50\% reduction! This despite the fact that the Prolog implementation does \emph{not} use the built-in unification to simplify the algorithm at all; it would be largely the same if only traditional assignment were used. \subsection{Proof Checking} The proof checker is implemented in approximately 450 lines of Prolog code. It essentially implements the rules of the reasoning logic presented in figure \ref{reasoning-rules}, by recursively checking the proof tree. That is, it first checks the root node to ensure that the proof object is consistent with the conclusion (antecedents and consequent) by ensuring that it has the correct number and type of subproofs. The subproofs are then recursively checked, working through the tree until the axiomatic rules are reached (or, in the case of an incomplete proof, a \texttt{hole} is found, indicating an unproved ................................................................................ \] \[ \inference[By induction] {\inference[$->$R] {\inference[Case] {\inference[Add-$0$] {} {x = 0 |- \mathsf{add}(0,0,0)} & \inference[Backchain] {\inference[IH] {\assume{x = s(x'), \mathsf{nat}^\downarrow(x') |- \mathsf{nat}^\downarrow(x')}} {x = s(x'), \mathsf{nat}^\downarrow(x') |- \mathsf{add}(x',0,x')}} {x = s(x'), \mathsf{nat}^\downarrow(x') |- \mathsf{add}(s(x'),0,s(x'))}} {\mathsf{nat}^\uparrow(x) |- \mathsf{add}(x,0,x)}} {\mathsf{nat}^\uparrow(x) -> \mathsf{add}(x,0,x)}} {\mathsf{nat}(x) -> \mathsf{add}(x,0,x)} \] (This example also demonstrates one of the flaws of the derivation proof format, its insatiable appetite for horizontal space.) |
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is implicitly $\exists$-quantified. \end{itemize} Note that, if \texttt{:-} is regarded as $\leftarrow$, this is the opposite of the implicit quantification in the specification. However, these rules match the normal expectation for proof-style statements. For example: \begin{itemize} \item $\mathsf{nat}(X) -> \mathsf{add}(X,z,X)$ is equivalent to $\forall X: \mathsf{nat}(X) -> \mathsf{add}(X,z,X)$. \item $\mathsf{nat}(X) \land \mathsf{nat}(Y) -> \mathsf{add}(X,Y,Z) \land \mathsf{nat}(Z)$ is equivalent to $\forall X,Y: \mathsf{nat}(X) \land \mathsf{nat}(Y) -> \exists Z: \mathsf{add}(X,Y,Z) \land \mathsf{nat}(Z)$. \end{itemize} Internally, when processing a proof tree, Arend maintains two sets of in-scope variables: \emph{universals} and \emph{existentials}. Universals are ................................................................................ (This presentation is based in part on that in \mcitet{pfenning2006logic}.) It is interesting to directly compare the two implementations of unification (one of the few modules which is semantically similar in both implementations). Both systems implement the same algorithm. The Javascript implementation consists of \lnum{136} lines of non-comment code and is quite difficult to follow; the Prolog implementation requires only \lnum{69} lines, an almost \lnum{50\%} reduction! This despite the fact that the Prolog implementation does \emph{not} use the built-in unification to simplify the algorithm at all; it would be largely the same if only traditional assignment were used. \subsection{Proof Checking} The proof checker is implemented in approximately \lnum{450} lines of Prolog code. It essentially implements the rules of the reasoning logic presented in figure \ref{reasoning-rules}, by recursively checking the proof tree. That is, it first checks the root node to ensure that the proof object is consistent with the conclusion (antecedents and consequent) by ensuring that it has the correct number and type of subproofs. The subproofs are then recursively checked, working through the tree until the axiomatic rules are reached (or, in the case of an incomplete proof, a \texttt{hole} is found, indicating an unproved ................................................................................ \] \[ \inference[By induction] {\inference[$->$R] {\inference[Case] {\inference[Add-$0$] {} {\mathsf{add}(0,0,0)} & \inference[Backchain] {\inference[IH] {\assume{\mathsf{nat}^\downarrow(x') |- \mathsf{nat}^\downarrow(x')}} {\mathsf{nat}^\downarrow(x') |- \mathsf{add}(x',0,x')}} {\mathsf{nat}^\downarrow(x') |- \mathsf{add}(s(x'),0,s(x'))}} {\mathsf{nat}^\uparrow(x) |- \mathsf{add}(x,0,x)}} {\mathsf{nat}^\uparrow(x) -> \mathsf{add}(x,0,x)}} {\mathsf{nat}(x) -> \mathsf{add}(x,0,x)} \] (This example also demonstrates one of the flaws of the derivation proof format, its insatiable appetite for horizontal space.) |