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additions to section 2, and spell check

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Jeremy Siek 2019-07-12 11:38:12 -04:00
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papers/scp/PLFA.tex
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@ -101,7 +101,7 @@ foundation of proof is on proud display.
Alas, there is no textbook for programming language theory in Alas, there is no textbook for programming language theory in
Agda. \emph{Verified Functional Programming in Agda} by \citep{Stump-2016} Agda. \emph{Verified Functional Programming in Agda} by \citep{Stump-2016}
covers related ground, but focusses more on programming with dependent covers related ground, but focuses more on programming with dependent
types than on the theory of programming languages. types than on the theory of programming languages.
The original goal was to simply adapt \emph{Software Foundations}, The original goal was to simply adapt \emph{Software Foundations},
@ -172,7 +172,7 @@ central example. The textbook is written as a literate script in Agda.
As with SF, the hope is that using As with SF, the hope is that using
a proof assistant will make the development more concrete a proof assistant will make the development more concrete
and accessible to students, and give them rapid feedback to find and accessible to students, and give them rapid feedback to find
and correct misaprehensions. and correct misapprehensions.
The book is broken into two parts. The first part, Logical Foundations, The book is broken into two parts. The first part, Logical Foundations,
develops the needed formalisms. The second part, Programming Language develops the needed formalisms. The second part, Programming Language
@ -301,33 +301,68 @@ Agda is streamlined by the use of ideas from
\citet{Schafer-Tebbi-Smolka-2015} and \citet{Pfenning-1992}. \citet{Schafer-Tebbi-Smolka-2015} and \citet{Pfenning-1992}.
\paragraph{Big-step: evaluation for call-by-name} \paragraph{Big-step: evaluation for call-by-name}
Introduces the notion of big-step evaluation to develop a Introduces the notion of big-step evaluation, written $\gamma \vdash M
deterministic call-by-name reduction strategy. The main result of this \Downarrow V$, to develop a deterministic call-by-name reduction
chapter is a proof that big-step evaluation implies the existence of a strategy. The main result of this chapter is a proof that big-step
reduction sequence that terminates with a lambda abstraction. evaluation implies the existence of a reduction sequence that
terminates with a lambda abstraction.
\subsection*{Part III, Denotational Semantics} \subsection*{Part III, Denotational Semantics}
\paragraph{Denotational: semantics of the untyped lambda calculus} \paragraph{Denotational: semantics of the untyped lambda calculus}
The early denotational semantics of the lambda calculus based on graph The early denotational semantics of the lambda calculus based on graph
models~\citep{Scott-1976,Engeler-1981,Plotkin-1993} and filter models~\citep{Scott-1976,Engeler-1981,Plotkin-1993} and filter
models~\citep{Barendregt-1983} were particularly simple and elegant. models~\citep{Barendregt-1983} were particularly simple and elegant: a
This chapter presents such a semantics in a big-step style that is function is modeled as a lookup table. This chapter presents such a
approachable to readers familiar with operational semantics. semantics using a big-step notation that is approachable to readers
familiar with operational semantics, writing $\gamma \vdash M
\downarrow d$ for the evaluation of a term $M$ to denotation $d$ in
environment $\gamma$.
\paragraph{Compositional: the denotational semantics is compositional} \paragraph{Compositional: the denotational semantics is compositional}
TODO The hallmark of denotational semantics is that they are compositional:
the meaning of each language form is a function of the meanings of its
parts.
%
We define two functions, named \texttt{curry} and \texttt{apply} that
serve this purpose for lambda abstraction and application.
%
The results in this chapter include congruences for \texttt{curry} and
\texttt{apply} and the compositionality property for the denotational
semantics.
%
The chapter concludes with a functional definition of the denotational
semantics and a proof that it is equivalent to the big-step version.
\paragraph{Soundness: of reduction with respect to denotational semantics} \paragraph{Soundness: of reduction with respect to denotational semantics}
TODO Reduction implies denotational equality. We prove each direction of
the equality, first showing the reduction preserves denotations
(subject reduction), and then showing that reduction reflects
denotations (subject expansion). The first proof is similar to the
type preservation proofs in Part II. The second goes in reverse,
showing that if $M \longrightarrow N$ and $\gamma \vdash N
\downarrow d$, then $\gamma \vdash M \downarrow d$ .
\paragraph{Adequacy: of denotational semantics with respect to reduction} \paragraph{Adequacy: of denotational semantics with respect to reduction}
TODO If a term is denotationally equal to a lambda expression, then it
reduces to a lambda expression. The main lemma shows that if a term's
denotation is functional, i.e., $\gamma \vdash M \downarrow (d \mapsto
d')$, then $M$ terminates according to the call-by-name big-step
semantics, i.e., $\gamma' \vdash M \Downarrow V$. A logical relation
$\mathbb{V}$ is used to relate denotations and values (i.e. closures).
The implication from the big-step semantics to reduction is proved in
the Big-step chapter of Part II.
\paragraph{Contextual Equivalence: is implied by denotational equality} \paragraph{Contextual Equivalence: is implied by denotational equality}
TODO The main criteria for behavior-preserving code transformation (such as
compiler optimization or programmer refactoring) is contextual
equivalence. Two terms are contextually equivalent when they can both
be placed into an arbitrary context (a program with a hole) and the
resulting programs behave the same (e.g., they both terminate or they
both diverge). This chapter ties together the previous results
(Compositionality, Soundness, and Adequacy) to show that denotational
equality implies contextual equivalence. Thus, it is safe to use
denotational equality to justify code transformations.
\subsection*{Discussion} \subsection*{Discussion}
@ -654,7 +689,7 @@ expressions'' \citep{Kiselyov-2009}.)
\end{figure} \end{figure}
The second part of PLFA first discusses two different approaches to The second part of PLFA first discusses two different approaches to
modelling simply-typed lambda calculus. It first presents raw modeling simply-typed lambda calculus. It first presents raw
terms with named variables and a separate typing relation and terms with named variables and a separate typing relation and
then shifts to inherently-typed terms with de Bruijn indices. then shifts to inherently-typed terms with de Bruijn indices.
Before writing the text, I had thought the two approaches Before writing the text, I had thought the two approaches
@ -665,7 +700,7 @@ Figure~\ref{fig:raw} presents the raw approach.
It first defines $\texttt{Id}$, $\texttt{Term}$, It first defines $\texttt{Id}$, $\texttt{Term}$,
$\texttt{Type}$, and $\texttt{Context}$, the abstract syntax $\texttt{Type}$, and $\texttt{Context}$, the abstract syntax
of identifiers, raw terms, types, and contexts. of identifiers, raw terms, types, and contexts.
It then defines two judgements, It then defines two judgments,
$\GG\:{\ni}\:\ex\:{\typecolon}\:\AY$ and $\GG\:{\ni}\:\ex\:{\typecolon}\:\AY$ and
$\GG\:{\vdash}\:\EM\:{\typecolon}\:\AY$, $\GG\:{\vdash}\:\EM\:{\typecolon}\:\AY$,
which hold when under context $\GG$ the variable $\ex$ which hold when under context $\GG$ the variable $\ex$
@ -674,11 +709,11 @@ and the term $\EM$ have type $\AY$, respectively.
Figure~\ref{fig:inherent} presents the inherent approach. Figure~\ref{fig:inherent} presents the inherent approach.
It first defines $\texttt{Type}$ and $\texttt{Context}$, the abstract syntax It first defines $\texttt{Type}$ and $\texttt{Context}$, the abstract syntax
of types and contexts, of which the first is as before and the second is of types and contexts, of which the first is as before and the second is
as before with identifiers dropped. In place of the two judgements, as before with identifiers dropped. In place of the two judgments,
the types of variables and terms are indexed by a context and a type, the types of variables and terms are indexed by a context and a type,
so that $\GG\:{\ni}\:\AY$ and $\GG\:{\vdash}\:\AY$ denote so that $\GG\:{\ni}\:\AY$ and $\GG\:{\vdash}\:\AY$ denote
variables and terms, respectively, that under context $\GG$ hae type $\AY$. variables and terms, respectively, that under context $\GG$ have type $\AY$.
The indexed types closely resemble the previous judgements: The indexed types closely resemble the previous judgments:
we now represent a variable or a term by the proof that it is well-typed. we now represent a variable or a term by the proof that it is well-typed.
In particular, the proof that a variable is well-typed in the raw approach In particular, the proof that a variable is well-typed in the raw approach
corresponds to a de Bruijn index in the inherent approach. corresponds to a de Bruijn index in the inherent approach.
@ -686,8 +721,8 @@ corresponds to a de Bruijn index in the inherent approach.
The raw approach requires more lines of code than The raw approach requires more lines of code than
the inherent approach. The separate definition of raw the inherent approach. The separate definition of raw
terms is not needed in the inherent terms is not needed in the inherent
approach; and one judgement in the raw approach needs to check that approach; and one judgment in the raw approach needs to check that
$\ex \not\equiv \why$, while the corresponding judgement in the $\ex \not\equiv \why$, while the corresponding judgment in the
inherent approach does not. The difference becomes more pronounced inherent approach does not. The difference becomes more pronounced
when including the code for substitution, reductions, and proofs of when including the code for substitution, reductions, and proofs of
progress and preservation. In particular, where the raw approach progress and preservation. In particular, where the raw approach
@ -724,7 +759,7 @@ advantages of de Bruijn indices and inherent types.
There are actually four possible designs, as the choice of named There are actually four possible designs, as the choice of named
variables vs de Bruijn indices, and the choice of raw vs variables vs de Bruijn indices, and the choice of raw vs
inherently-typed terms may be made independently. There are synergies inherently-typed terms may be made independently. There are synergies
beween the two. Manipulation of de Bruijn indices can be notoriously between the two. Manipulation of de Bruijn indices can be notoriously
error-prone without inherent-typing to give assurance of correctness. error-prone without inherent-typing to give assurance of correctness.
In inherent typing with named variables, simultaneous substitution by In inherent typing with named variables, simultaneous substitution by
open terms remains difficult. open terms remains difficult.
@ -742,7 +777,7 @@ I look forward to experience teaching from the new text,
and encourage others to use it too. Please comment! and encourage others to use it too. Please comment!
\paragraph{Acknowledgements} \paragraph{Acknowledgments}
A special thank you to my coauthor, Wen Kokke. For inventing ideas on 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 which PLFA is based, and for hand-holding, many thanks to Conor
@ -763,3 +798,25 @@ EP/K034413/1.
\end{document} \end{document}
% LocalWords: Agda adr Kokke Siek Wadler Woodlawn Bloomington Coq et
% LocalWords: PLFA evaluator normalise programme programmes al Huet
% LocalWords: ICFP subst Bove Mixfix de Janeiro summarising favour
% LocalWords: organised formalising Colour Gonthier CompCert Kastner
% LocalWords: SEL honours formalisms monus associativity reflexivity
% LocalWords: commutativity equational of's Isomorphism isomorphism
% LocalWords: extensionality disjunction intuitionistic existentials
% LocalWords: Booleans booleans monoid specialisation PCF Plotkin EP
% LocalWords: fixpoint DeBruijn Bruijn indices Bisimulation Untyped
% LocalWords: bisimulation normalisation untyped Abadi Barendregt
% LocalWords: Rosser Tait Schafer Tebbi Smolka Pfenning Denotational
% LocalWords: denotational Engeler lookup Compositional congruences
% LocalWords: compositional compositionality denotationally Hoare
% LocalWords: refactoring subtyping Appel QuickChick formalise SF's
% LocalWords: Stlc StlcProp Coq's reinvoke Agda's postconditions Ulf
% LocalWords: flavour layed emphasises tvar tapp CoqIDE sublanguage
% LocalWords: submenus Danas monospaced FreeMono Felleisen Redex
% LocalWords: Rosu Serbanuta realise Capretta cryptocurrencies Oleg
% LocalWords: evaluators Dagand Scherer Kiselyov eval Goguen McKinna
% LocalWords: Altenkirch Reus Allais Darais Conor Norell Andreas
% LocalWords: EPSRC