|
|
|
@ -619,7 +619,7 @@ structure as follows. |
|
|
|
\begin{eqs} |
|
|
|
(A \eff E)\sigma &\defas& A\sigma \eff E\sigma\\ |
|
|
|
(A \to C)\sigma &\defas& A\sigma \to C\sigma\\ |
|
|
|
(\forall \alpha^K.C)\sigma &\defas& \forall \alpha^K.C\sigma \quad \text{convert if } \alpha \in \dom(\sigma)\\ |
|
|
|
(\forall \alpha^K.C)\sigma &\simdefas& \forall \alpha^K.C\sigma\\ |
|
|
|
\alpha\sigma &\defas& \begin{cases} |
|
|
|
A & \text{if } (\alpha,A) \in \sigma\\ |
|
|
|
\alpha & \text{otherwise} |
|
|
|
@ -976,7 +976,7 @@ cannot be satisfied. |
|
|
|
|
|
|
|
\paragraph{Typing computations} |
|
|
|
The \tylab{App} rule states that an application $V\,W$ has computation |
|
|
|
type $C$ if the abstractor term $V$ has type $A \to C$ and the |
|
|
|
type $C$ if the function-term $V$ has type $A \to C$ and the |
|
|
|
argument term $W$ has type $A$, that is both the argument type and the |
|
|
|
domain type of the abstractor agree. |
|
|
|
% |
|
|
|
@ -1050,87 +1050,83 @@ that the binder $x : A$. |
|
|
|
\label{fig:base-language-small-step} |
|
|
|
\end{figure} |
|
|
|
% |
|
|
|
In this section we present the dynamic semantics of \BCalc{}. We |
|
|
|
choose to use a contextual small-step semantics, since in conjunction |
|
|
|
with fine-grain call-by-value, it yields a considerably simpler |
|
|
|
semantics than the traditional structural operational semantics |
|
|
|
(SOS)~\cite{Plotkin04a}, because only the rule for let bindings admits |
|
|
|
a continuation whereas in ordinary call-by-value SOS each congruence |
|
|
|
rule admits a continuation. |
|
|
|
% |
|
|
|
|
|
|
|
The semantics are based on a substitution model of computation. Thus, |
|
|
|
before presenting the reduction rules, we define an adequate |
|
|
|
substitution function. As usual we work up to |
|
|
|
In this section we present the dynamic semantics of \BCalc{}. We opt |
|
|
|
to use a \citet{Felleisen87}-style contextual small-step semantics, |
|
|
|
since in conjunction with fine-grain call-by-value (FGCBV), it yields |
|
|
|
a considerably simpler semantics than the traditional structural |
|
|
|
operational semantics (SOS)~\cite{Plotkin04a}, because only the rule |
|
|
|
for let bindings admits a continuation wheres in ordinary |
|
|
|
call-by-value SOS each congruence rule admits a continuation. |
|
|
|
% |
|
|
|
The simpler semantics comes at the expense of a more verbose syntax, |
|
|
|
which is not a concern as one can readily convert between fine-grain |
|
|
|
call-by-value and ordinary call-by-value. |
|
|
|
|
|
|
|
The reduction semantics are based on a substitution model of |
|
|
|
computation. Thus, before presenting the reduction rules, we define an |
|
|
|
adequate substitution function. As usual we work up to |
|
|
|
$\alpha$-conversion~\cite{Church32} of terms in $\BCalc{}$. |
|
|
|
% |
|
|
|
\paragraph{Term substitution} |
|
|
|
We write $M[V/x]$ for the substitution of some value $V$ for some |
|
|
|
variable $x$ in some computation term $M$. We use the same notation |
|
|
|
for substitution on values, i.e. $W[V/x]$ denotes the substitution of |
|
|
|
$V$ for $x$ in some value $W$. We define substitution as a ternary |
|
|
|
function, whose signature is given by |
|
|
|
We define a term substitution map, |
|
|
|
$\sigma : (\VarCat \times \ValCat)^\ast$ as list of pairs mapping a |
|
|
|
variable to its value replacement. We denote a single mapping as $V/x$ |
|
|
|
meaning substitute $V$ for the variable $x$. We write multiple |
|
|
|
mappings using list notation, i.e. $[V_0/x_0,\dots,V_n/x_n]$. The |
|
|
|
domain of a substitution map is set generated by projecting the first |
|
|
|
component, i.e. |
|
|
|
% |
|
|
|
\[ |
|
|
|
-[-/-] : \TermCat \times \ValCat \times \VarCat \to \CompCat, |
|
|
|
\bl |
|
|
|
\dom : (\VarCat \times \ValCat)^\ast \to \ValCat\\ |
|
|
|
\dom(\sigma) \defas \{ x \mid (\_/x) \in \sigma \} |
|
|
|
\el |
|
|
|
\] |
|
|
|
% |
|
|
|
and we realise it by pattern matching on the first argument. |
|
|
|
The application of a type substitution map on a term $t \in \TermCat$, |
|
|
|
written $t\sigma$, is defined inductively on the term structure as |
|
|
|
follows. |
|
|
|
% |
|
|
|
\[ |
|
|
|
\ba[t]{@{~}l@{~}c@{~}r} |
|
|
|
\begin{eqs} |
|
|
|
x[V/y] &\defas& \begin{cases} |
|
|
|
V & \text{if } x = y\\ |
|
|
|
x\sigma &\defas& \begin{cases} |
|
|
|
V & \text{if } (x, V) \in \sigma\\ |
|
|
|
x & \text{otherwise} |
|
|
|
\end{cases}\\ |
|
|
|
(\lambda x^A.M)[V/y] &\defas& \lambda x^A.M[V/y] \quad \text{if } x \neq y \text{ and } x \notin \FV(V)\\ |
|
|
|
(\Lambda \alpha^K. M)[V/y] &\defas& \Lambda \alpha^K. M[V/y]\\ |
|
|
|
\Unit[V/y] &\defas& \Unit\\ |
|
|
|
\Record{\ell = W; W'}[V/y] &\defas& \Record{\ell = W[V/y]; W'[V/y]}\\ |
|
|
|
(\ell~W)^R[V/y] &\defas& (\ell~W[V/y])^R\\ |
|
|
|
(W\,W')[V/y] &\defas& W[V/y]\,W'[V/y]\\ |
|
|
|
(W\,T)[V/y] &\defas& W[V/y]~T\\ |
|
|
|
(\ba[t]{@{}l} |
|
|
|
\Let\;\Record{\ell = x; y} = W\\ |
|
|
|
\In\;N)[V/z] |
|
|
|
\ea &\defas& |
|
|
|
\ba[t]{@{~}l} |
|
|
|
\Let\;\Record{\ell = x; y} = W[V/z]\\ |
|
|
|
\In\;N[V/z] |
|
|
|
\ea \quad |
|
|
|
\ba[t]{@{}l} |
|
|
|
\text{if } x \neq z, y \neq z,\\ |
|
|
|
\text{and } x,y\notin \FV(V) |
|
|
|
\ea\\ |
|
|
|
(\Case\;(\ell~W)^R\{ |
|
|
|
\ba[t]{@{}l} |
|
|
|
\ell~x \mapsto M\\ |
|
|
|
; y \mapsto N \})[V/z] |
|
|
|
\ea |
|
|
|
&\defas& |
|
|
|
\Case\;(\ell~W[V/z])^R\{ |
|
|
|
\ba[t]{@{}l} |
|
|
|
\ell~x \mapsto M[V/z]\\ |
|
|
|
; y \mapsto N[V/z] \} |
|
|
|
\ea\quad |
|
|
|
\ba[t]{@{}l} |
|
|
|
\text{if } x \neq z, y \neq z,\\ |
|
|
|
\text{and } x, y \notin \FV(V) |
|
|
|
\ea\\ |
|
|
|
(\Let\;x \revto M \;\In\;N)[V/y] &\defas& \Let\;x \revto M[V/y] \;\In\;N[V/y] \quad\text{if } x \neq y \text{ and } x \notin \FV(V) |
|
|
|
(\lambda x^A.M)\sigma &\simdefas& \lambda x^A.M\sigma\\ |
|
|
|
(\Lambda \alpha^K.M)\sigma &\defas& \Lambda \alpha^K.M\sigma\\ |
|
|
|
(V~T)\sigma &\defas& V\sigma~T |
|
|
|
\end{eqs} |
|
|
|
&~~& |
|
|
|
\begin{eqs} |
|
|
|
(V~W)\sigma &\defas& V\sigma~W\sigma\\ |
|
|
|
\Unit\sigma &\defas& \Unit\\ |
|
|
|
\Record{\ell = V; W}\sigma &\defas& \Record{\ell = V\sigma;W\sigma}\\ |
|
|
|
(\ell~V)^R\sigma &\defas& (\ell~V\sigma)^R\\ |
|
|
|
\end{eqs}\bigskip\\ |
|
|
|
\multicolumn{3}{c}{ |
|
|
|
\begin{eqs} |
|
|
|
(\Let\;\Record{\ell = x; y} = V\;\In\;N)\sigma &\simdefas& \Let\;\Record{\ell = x; y} = V\sigma\;\In\;N\sigma\\ |
|
|
|
(\Case\;(\ell~V)^R\{ |
|
|
|
\ell~x \mapsto M |
|
|
|
; y \mapsto N \})\sigma |
|
|
|
&\simdefas& |
|
|
|
\Case\;(\ell~V\sigma)^R\{ |
|
|
|
\ell~x \mapsto M\sigma |
|
|
|
; y \mapsto N\sigma \}\\ |
|
|
|
(\Let\;x \revto M \;\In\;N)\sigma &\simdefas& \Let\;x \revto M[V/y] \;\In\;N\sigma |
|
|
|
\end{eqs}} |
|
|
|
\ea |
|
|
|
\] |
|
|
|
% |
|
|
|
% \begin{cases} |
|
|
|
% \Let\;x \revto M[V/y]\;\In\;N & \text{if } x = y\\ |
|
|
|
% \Let\;x \revto M[V/y]\;\In\;N[V/y] & \text{otherwise} |
|
|
|
% \end{cases} |
|
|
|
% \end{eqs} |
|
|
|
% |
|
|
|
We write $t[t_0/x_0,\dots,t_n/x_n]$ to mean the simultaneous |
|
|
|
substitution of $t_0$ for $x_0$ up to $t_n$ for $x_n$ in $t$. |
|
|
|
% |
|
|
|
|
|
|
|
The attentive reader will notice that we are using the same notation |
|
|
|
for type and term substitutions. We justify this choice by the fact |
|
|
|
that we can lift type substitution pointwise on the term syntax |
|
|
|
constructors, enabling us to use one uniform notation for |
|
|
|
substitution. Thus we shall generally allow a mix of pairs of |
|
|
|
variables and values and pairs of type variables and types to occur in |
|
|
|
the same substitution map. |
|
|
|
|
|
|
|
\paragraph{Reduction semantics} |
|
|
|
The reduction relation $\reducesto : \CompCat \pto \CompCat$ is defined |
|
|
|
|
