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Rewrote section on tactics and Hoare logic

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Wen Kokke 2019-07-14 16:09:07 +01:00
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37
papers/scp/PLFA.bib
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@ -374,3 +374,40 @@
Title = {A filter lambda model and the completeness of type assignment},
Volume = {48},
Year = {1983}}
@inproceedings{Kokke-2015,
abstract = {As proofs in type theory become increasingly
complex, there is a growing need to provide better
proof automation. This paper shows how to implement
a Prolog-style resolution procedure in the
dependently typed programming language
Agda. Connecting this resolution procedure to Agda’s
reflection mechanism provides a first-class proof
search tactic for first-order Agda terms. As a
result, writing proof automation tactics need not be
different from writing any other program. },
year = {2015},
month = jun,
isbn = {978-3-319-19796-8},
booktitle = {Mathematics of Program Construction},
volume = {9129},
series = {Lecture Notes in Computer Science},
editor = {Hinze, Ralf and Voigtländer, Janis},
doi = {10.1007/978-3-319-19797-5_14},
title = {Auto in Agda: Programming Proof Search Using Reflection},
url = {https://wenkokke.github.io/pubs/mpc2015.pdf},
publisher = {Springer International Publishing},
author = {Wen Kokke and Wouter Swierstra},
pages = {276-301}}
@thesis{Kidney-2019,
address = {Cork, Ireland},
type = {Bachelor thesis},
title = {Automatically and {Efficiently} {Illustrating} {Polynomial} {Equalities} in {Agda}},
url = {https://doisinkidney.com/pdfs/bsc-thesis.pdf},
abstract = {We present a new library which automates the construction of equivalence proofs between polynomials over commutative rings and semirings in the programming language Agda [20]. It is signi cantly faster than Agda’s existing solver. We use re ection to provide a sim- ple interface to the solver, and demonstrate how to use the constructed proofs to provide step-by-step solutions.},
language = {en},
school = {University College Cork},
author = {Kidney, Donnacha Oisín},
month = apr,
year = {2019}}

82
papers/scp/PLFA.tex
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@ -371,28 +371,27 @@ PLFA and SF differ in several particulars. PLFA begins with a computationally
complete language, PCF, while SF begins with a minimal language, simply-typed
lambda calculus with booleans. PLFA does not include type annotations in terms,
and uses bidirectional type inference, while SF has terms with unique types and
uses type checking. SF also covers a simple imperative language with Hoare logic,
and for lambda calculus covers subtyping, record types, mutable references, and normalisation
---none of which are treated by PLFA. PLFA covers an inherently-typed de Bruijn
representation, bidirectional type inference, bisimulation, and an
untyped call-by-name language with full normalisation---none of which
are treated by SF.
uses type checking. SF also covers a simple imperative language with Hoare
logic, and for lambda calculus covers subtyping, record types, mutable
references, and normalisation---none of which are treated by PLFA. PLFA covers
an inherently-typed de Bruijn representation, bidirectional type inference,
bisimulation, and an untyped call-by-name language with full
normalisation---none of which are treated by SF.
SF has a third volume, written by Andrew Appel, on Verified Functional
Algorithms. I'm not sufficiently familiar with that volume to have a view
on whether it would be easy or hard to cover that material in Agda.
And SF recently added a fourth volume on random testing of Coq specifications
using QuickChick. There is currently no tool equivalent to QuickChick
available for Agda.
Algorithms. I'm not sufficiently familiar with that volume to have a view on
whether it would be easy or hard to cover that material in Agda. And SF recently
added a fourth volume on random testing of Coq specifications using QuickChick.
There is currently no tool equivalent to QuickChick available for Agda.
There is more material that would be desirable to include in PLFA
which was not due to limits of time. In future years, PLFA may be
extended to cover additional material, including mutable references,
normalisation, System F, and pure type systems.
We'd especially like to include pure type systems as they
provide the readers with a formal model close to the dependent types
used in the book. Our attempts so far to formalise pure type systems
have proved challenging, to say the least.
There is more material that would be desirable to include in PLFA which was not
due to limits of time. In future years, PLFA may be extended to cover
additional material, including mutable references, normalisation, System F, and
pure type systems. We'd especially like to include pure type systems as they
provide the readers with a formal model close to the dependent types used in the
book. Our attempts so far to formalise pure type systems have proved
challenging, to say the least.
\section{Proofs in Agda and Coq}
@ -457,16 +456,25 @@ structurally smaller arguments. We suspect tactics could cut down the
proof significantly. We tried to compare with SF's proof that reduction
is deterministic, but failed to find that proof.
SF covers an imperative language with Hoare logic, culminating in
code that takes an imperative programme suitably decorated with preconditions
and postconditions and generates the necessary
verification conditions. The conditions are then verified by a custom
tactic, where any questions of arithmetic are resolved by the
``omega'' tactic invoking a decision procedure. The entire
exercise would be easy to repeat in Agda, save for the last step: we
suspect Agda's automation would not be up to verifying the generated
conditions, requiring tedious proofs by hand. However, we had already
decided to omit Hoare logic in order to focus on lambda calculus.
SF covers an imperative language with Hoare logic, culminating in code that
takes an imperative programme suitably decorated with preconditions and
postconditions and generates the necessary verification conditions. The
conditions are then verified by a custom tactic, where any questions of
arithmetic are resolved by the ``omega'' tactic invoking a decision procedure.
The entire exercise would be easy to repeat in Agda, save for the last step, as
Agda does not offer support for proof automation out of the box. It is
certainly possible to implement proof automation in Agda---see, e.g., the auto
tactic by \citet{Kokke-2015}, and the collection of tactics in Ulf Norell's
\texttt{agda-prelude}\footnote{\url{https://github.com/UlfNorell/agda-prelude}}.
The standard library comes equipped with solvers for equations on monoids and
rings\footnote{\url{https://agda.github.io/agda-stdlib/Algebra.Solver.Ring.html}},
and a much improved solver for equalities on rings was recently contributed by
\citet{Kidney-2019}.
We suspect that, while Agda's automation would be up to verifying the generated
conditions, some effort would be require to implement the required custom
tactic, and a section would need to be added to the book to cover proof
automation. For the time being, we have decided to omit Hoare logic in order to
focus on lambda calculus.
To give a flavour of how the texts compare, we show the
proof of progress for simply-typed lambda calculus from both texts.
@ -770,15 +778,13 @@ and encourage others to use it too. Please comment!
\paragraph{Acknowledgments}
A special thank you to my coauthor, Wen Kokke. For inventing ideas on
which PLFA is based, and for hand-holding, many thanks to Conor
McBride, James McKinna, Ulf Norell, and Andreas Abel. For showing me
how much more compact it is to avoid raw terms, thanks to David
Darais. For inspiring my work by writing SF, thanks to Benjamin
Pierce and his coauthors. For comments on a draft of this paper, an
extra thank you to James McKinna, Ulf Norell, Andreas Abel, and
Benjamin Pierce. This research was supported by EPSRC Programme Grant
EP/K034413/1.
For inventing ideas on which PLFA is based, and for hand-holding, many thanks to
Conor McBride, James McKinna, Ulf Norell, and Andreas Abel. For showing me how
much more compact it is to avoid raw terms, thanks to David Darais. For
inspiring my work by writing SF, thanks to Benjamin Pierce and his coauthors.
For comments on a draft of this paper, an extra thank you to James McKinna, Ulf
Norell, Andreas Abel, and Benjamin Pierce. This research was supported by EPSRC
Programme Grant EP/K034413/1.
\section*{References}