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Adjust grammar and proofs in Rewriting chapter

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Jip J. Dekker 2021-04-07 10:10:11 +10:00
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2 changed files with 63 additions and 68 deletions
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.chktexrc
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@ -1,5 +1,5 @@
# Exclude these environments from syntax checking
VerbEnvir { pgfpicture tikzpicture mzn nzn grammar proof }
MathCmd { hypo infer1 infer2 infer3 }
MathCmd { \hypo }
WipeArg { \mzninline:{} \nzninline:{} \Sem:{} \texttt:{} }

123
chapters/4_rewriting.tex
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@ -68,15 +68,17 @@ and where expressions have \mzninline{par} type.
\begin{figure}
\begin{grammar}
<model> ::= <fun-decl>*
<model> ::= <pred-decl>*<fun-decl>*
<fun-decl> ::= "function" <type-inst>":" <ident>"(" <param> ["," <param>]* ")" [ "=" <exp> ]";"
<pred-decl> ::= "predicate" <ident>"(" <param> ["," <param>]* ");"
<fun-decl> ::= "function" <type-inst>":" <ident>"(" <param> ["," <param>]* ")" "=" <exp>";"
<literal> ::= <constant> | <ident>
<exp> ::= <literal>
\alt <ident>"(" <literal> ["," <literal>]* ")"
\alt "let" "{" <item>* "}" "in" <exp>
\alt <ident>"(" <exp> ["," <exp>]* ")"
\alt "let" "{" <item>* "}" "in" <ident>
\alt "if" <exp> "then" <exp> "else" <exp> "endif"
\alt "[" <exp> "|" <gen-exp> "where" <exp> "]"
\alt "(" <exp> ["," <exp>]* ")"
@ -164,22 +166,23 @@ The translation from \minizinc\ to \microzinc\ involves the following steps:
constraint for each element of the array.
\begin{mzn}
predicate all_different(array[int] of var int: x) =
function var bool: all_different(array[int] of var int: x) =
forall([
int_neq(element(x,i),element(x,j))
| i in range(1,length(x)),
j in range(1,length(x))
int_neq(element(x, i),element(x, j))
| i in range(1, length(x)),
j in range(1, length(x))
where int_lt(i,j)
]);
predicate main(int: n) = let {
array[range(1,n)] of var int: pigeon;
function var bool: main(int: n) = let {
var bool: root_ctx;
array[range(1, n)] of var int: pigeon;
constraint forall([
set_in(element(pigeon,i), range(1,int_minus(n,1)))
| i in range(1,n)
set_in(element(pigeon, i), range(1, int_minus(n,1)))
| i in range(1, n)
])
constraint all_different(pigeon);
} in true;
} in root_ctx;
\end{mzn}
For a concrete instance of the model, we can then create a simple \nanozinc\
@ -308,24 +311,20 @@ suitable alpha renaming.
\begin{figure*}
\centering
\begin{prooftree}
\hypo{\ptinline{F(\(p_1, \ldots, p_k\)) = E;} \in \Prog \text{~where the~} p_i \text{~are fresh}}
\infer[no rule]1{\Sem{\(a_i\)}{\Prog, \Env_{i-1}} \Rightarrow \tuple{v_i,
\Env_i'}, \ \Env_0=\Env, \Env_i=\Env_i'\cup\{p_i\mapsto v_i\sep[]\}, \forall 1 \leq i \leq k}
\infer[no rule]1{\Sem{E}{\Prog, \Env_k} \Rightarrow \tuple{v, \Env'}}
\infer1[(Call)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env_{0}} \Rightarrow \tuple{v, \Env'}}
\hypo{\ptinline{F(\(p_1, \ldots{}, p_k\)) = E;} \in{} \Prog{} \text{~where the~} p_i \text{~are fresh}}
\infer[no rule]1{\Sem{E}{\Prog, \Env_{[p_i \mapsto a_i, \forall{} 1 \leq{} i \leq{} k]}} \Rightarrow \tuple{v, \Env'}}
\infer1[(Call)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env} \Rightarrow{} \tuple{v, \Env'_{[a_i \mapsto p_i, \forall{} 1 \leq{} i \leq{} k]}}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\ptinline{F} \in \text{Builtins}}
\hypo{\Sem{\(a_i\)}{\Prog, \Env} \Rightarrow \tuple{v_i, \Env}, \forall{} 1 \leq{} i \leq{} k}
\infer2[(CallBuiltin)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env} \Rightarrow \tuple{ \mathit{eval}(\ptinline{F}(v_1,\ldots, v_k)), \Env}}
\infer1[(CallBuiltin)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env} \Rightarrow{} \tuple{ \mathit{eval}(\ptinline{F}(a_1,\ldots, a_k)), \Env}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\ptinline{F(\(p_1, \ldots, p_k\));} \in \Prog}
\hypo{\Sem{\(a_i\)}{\Prog, \Env_{i-1}} \Rightarrow \tuple{v_i, \Env_i}, \forall 1 \leq i \leq k}
\infer2[(CallPrimitive)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env_0} \Rightarrow
\tuple{ x, \{ x \mapsto \ptinline{F}(v_1,\ldots, v_k) \sep [] \} \cup \Env_k}}
\infer1[(CallPredicate)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env_0} \Rightarrow
\tuple{\texttt{constraint~} \text{F}(v_1, \ldots, v_k), \Env_k}}
\end{prooftree}
\caption{\label{fig:4-rewrite-to-nzn-calls} Rewriting rules for partial evaluation
of \microzinc\ calls to \nanozinc.}
@ -343,45 +342,44 @@ directly, such as arithmetic and Boolean operations on fixed values. The rule
simply returns the result of evaluating the built-in function on the evaluated
parameter values.
The (CallPrimitive) rule applies to non-built-in functions that are defined
without a function body. These are considered primitives, and as such simply
need to be transferred into the \nanozinc\ program. The rule evaluates all
actual parameters, and creates a new variable \(x\) to store the result, which is
simply the function applied to the evaluated parameters.
The (CallPredicate) rule applies to calls to solver predicates. These are
considered primitives, and as such simply need to be transferred into the
\nanozinc\ program. The rule evaluates all actual parameters, and creates a new
variable \(x\) to store the result, which is simply the function applied to the
evaluated parameters.
\begin{figure*}
\centering
\begin{prooftree}
\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env} \Rightarrow (\Ctx, \Env')}
\hypo{\Sem{\(E\)}{\Prog, \Env'} \Rightarrow \tuple{v, \Env''}}
\hypo{x \text{~fresh}}
\infer3[(LetC)]{\Sem{let \{ \(\mathbf{I}\) \} in \(E\)}{\Prog, \Env} \Rightarrow
\tuple{x, \{ x \mapsto v \sep \Ctx \} \cup \Env''}}
\hypo{\Sem{\(E\)}{\Prog, \Env'} \Rightarrow \tuple{x, \{ t : x \}\cup{}\Env''}}
\infer2[(LetC)]{\Sem{let \{ \(\mathbf{I}\) \} in \(E\)}{\Prog, \Env} \Rightarrow
\tuple{x, \{ t : x_{\texttt{~└──~}} \Ctx \} \cup \Env''}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{}
\infer1[(Item0)]{\Sem{\(\epsilon\)}{\Prog, \Env} \Rightarrow \tuple{\ptinline{true}, \Env}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env\ \cup\ \{x \mapsto\ \texttt{mkvar()} \sep\ []\}} \Rightarrow \tuple{\Ctx, \Env'}}
\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env\ \cup\ \{t : x\}} \Rightarrow \tuple{\Ctx, \Env'}}
\infer1[(ItemT)]{\Sem{\(t:x, \mathbf{I}\)}{\Prog, \Env} \Rightarrow \tuple{\Ctx, \Env'}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(E\)}{\Prog, \Env} \Rightarrow \tuple{v, \Env'}}
\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env' \cup\ \{x \mapsto\ v \sep\ [] \}} \Rightarrow \tuple{\Ctx, \Env''}}
\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env'_{[x \mapsto v]}} \Rightarrow \tuple{\Ctx, \Env''}}
\infer2[(ItemTE)]{\Sem{\(t:x = E, \mathbf{I}\)}{\Prog, \Env} \Rightarrow \tuple{\Ctx, \Env''}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(E\)}{\Prog, \Env} \Rightarrow \tuple{\left(v_{1}, \ldots, v_{n}\right), \Env'}}
\infer[no rule]1{\Sem{\(\mathbf{I}\)}{\Prog, \Env' \cup\ \{x_{1} \mapsto\ v_{1} \sep\ [] \} \cup\ \ldots\ \cup\ \{x_{n} \mapsto\ v_{n} \sep\ [] \}} \Rightarrow \tuple{\Ctx, \Env''}}
\infer1[(ItemTD)]{\Sem{\(\left(t_{1}: x_{1}, \ldots, t_{n}: x_{n}\right) = E, \mathbf{I}\)}{\Prog, \Env} \Rightarrow
\hypo{\Sem{\(E\)}{\Prog, \Env} \Rightarrow \tuple{\left(v_{\langle{}1\rangle}, \ldots, v_{\langle{}k\rangle}\right), \Env'}}
\infer[no rule]1{\Sem{\(\mathbf{I}\)}{\Prog, \Env'_{[x_{i} \mapsto v_{\langle{}i\rangle}, \forall{} 1 \leq{} i \leq{} k]}} \Rightarrow\ \tuple{\Ctx, \Env''}}
\infer1[(ItemTD)]{\Sem{\(\left(t_{1}: x_{1}, \ldots, t_{k}: x_{k}\right) = E, \mathbf{I}\)}{\Prog, \Env} \Rightarrow\
\tuple{\Ctx, \Env''}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(C\)}{\Prog, \Env} \Rightarrow \tuple{v, \Env'}}
\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env'} \Rightarrow \tuple{\Ctx, \Env''}}
@ -405,56 +403,53 @@ is the base case for a list of let items.
\begin{figure*}
\centering
% \begin{prooftree}
% \hypo{x \in \langle \text{ident} \rangle}
% \hypo{v \in \langle \text{literal} \rangle}
% \infer2[(IdC)]{\Sem{\(x\)}{\Prog, \{x \mapsto\ v \sep\ []\} \cup\ \Env} \Rightarrow \tuple{v, \{x \mapsto v \sep [] \} \cup \Env}}
% \end{prooftree} \\
% \bigskip
\begin{prooftree}
\hypo{x \in \syntax{<ident>}}
\hypo{v \in \syntax{<literal>}}
\infer2[(IdC)]{\Sem{\(x\)}{\Prog, \{x \mapsto\ v \sep\ []\} \cup\ \Env} \Rightarrow \tuple{v, \{x \mapsto v \sep [] \} \cup \Env}}
\end{prooftree} \\
\medskip
\begin{prooftree}
\hypo{x \in \syntax{<ident>}}
\hypo{x \in \langle \text{ident} \rangle}
\hypo{v}
\hypo{\phi \text{~otherwise}}
\infer3[(IdX)]{\Sem{\(x\)}{\Prog, \{x \mapsto\ v \sep\ \phi\ \} \cup\ \Env} \Rightarrow \tuple{x, \{x \mapsto v \sep \phi \} \cup \Env}}
\infer3[(IdX)]{\Sem{\(x\)}{\Prog, \{t: x^{\texttt{~└──~}} \phi\ \} \cup\ \Env} \Rightarrow \tuple{x, \{t: x^{\texttt{~└──~}} \phi \cup \phi}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{c \text{~constant}}
\infer1[(Const)]{\Sem{c}{\Prog, \Env} \Rightarrow \tuple{c, \Env}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(E_{i}\)}{\Prog,\Env^{i-1}} \Rightarrow \tuple{v_{i}, \Env^{i}}, \forall{} 1 \leq{} i \leq{} k}
\infer1[(Tuple)]{\Sem{\(\left(E_{1}, \ldots, E_{k}\right)\)}{\Prog, \Env^{0}} \Rightarrow \tuple{x, \{x \mapsto \left(v_{1}, \ldots, v_{n}\right) \sep [] \} \cup \Env^{k}}}
\infer1[(Tuple)]{\Sem{\(\left(E_{1}, \ldots, E_{k}\right)\)}{\Prog, \Env^{0}} \Rightarrow \tuple{x, \Env^{k}_{[x_{\langle{}i\rangle} \mapsto v_{i}, \forall{} 1 \leq{} i \leq{} k]}}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(C\)}{\Prog, \Env} \Rightarrow \tuple{\ptinline{true}, \_}}
\infer1[(If\(_T\))]{\Sem{if \(C\) then \(E_t\) else \(E_e\) endif}{\Prog, \Env} \Rightarrow \Sem{\(E_t\)}{\Prog, \Env}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(C\)}{\Prog, \Env} \Rightarrow \tuple{\ptinline{false}, \_}}
\infer1[(If\(_F\))]{\Sem{if \(C\) then \(E_t\) else \(E_e\) endif}{\Prog, \Env} \Rightarrow \Sem{\(E_e\)}{\Prog, \Env}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{G}{\Prog,\Env} \Rightarrow \tuple{ \ptinline{true}, \_}}
\hypo{\Sem{\(E\)}{\Prog, \Env} \Rightarrow \tuple {v, \Env'}}
\infer2[(WhereT)]{\Sem{\([E ~|~ \mbox{where~} G]\)}{\Prog, \Env} \Rightarrow \tuple{[v], \Env'}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{G}{\Prog,\Env} \Rightarrow \tuple{\ptinline{false}, \_}}
\infer1[(WhereF)]{\Sem{\([E ~|~ \mbox{where~} G]\)}{\Prog, \Env} \Rightarrow \tuple{[], \Env}}
\end{prooftree} \\
\medskip
\bigskip
\begin{prooftree}
\hypo{\Sem{\(\mathit{PE}\)}{\Prog,\Env} \Rightarrow \tuple{S, \_}}
\infer[no rule]1{\Sem{\([E ~|~ \mathit{GE} \mbox{~where~} G]\)}{\Prog, \Env\ \cup\ \{ x\mapsto\
v \sep\ []\}}}
\infer[no rule]1{\hspace{8em} \Rightarrow \tuple{A_v, \Env \cup \{x \mapsto v \sep []\} \cup \Env_v}, \forall
v \in S }
\infer[no rule]1{\Sem{\([E ~|~ \mathit{GE} \mbox{~where~} G]\)}{\Prog, \Env_{[x \mapsto v]}} \Rightarrow \tuple{A_v, \Env_{[x \mapsto v]} \cup \Env_v}, \forall{} v \in S }
\infer1[(ListG)]{\Sem{\([E~|~ x \mbox{~in~} \mathit{PE}, \mathit{GE} \mbox{~where~} G]\)}{\Prog, \Env} \Rightarrow
\tuple{\mbox{concat} [ A_v ~|~ v \in S ], \Env \cup \bigcup_{v \in S} \Env_v}}
\end{prooftree}