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doc/src/references/gm-state-transitions.rst
Normal file
425
doc/src/references/gm-state-transitions.rst
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================================
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G-Machine State Transition Rules
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================================
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*********************
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Core Transition Rules
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*********************
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#. Lookup a global by name and push its value onto the stack
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.. math::
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\gmrule
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{ \mathtt{PushGlobal} \; f : i
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& s
|
||||
& d
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& h
|
||||
& m
|
||||
\begin{bmatrix}
|
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f : a
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||||
\end{bmatrix}
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}
|
||||
{ i
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||||
& a : s
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& d
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& h
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& m
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}
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#. Allocate an int node on the heap, and push the address of the newly created
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node onto the stack
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.. math::
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\gmrule
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{ \mathtt{PushInt} \; n : i
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& s
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& d
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& h
|
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& m
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}
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{ i
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& a : s
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& d
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& h
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\begin{bmatrix}
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a : \mathtt{NNum} \; n
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\end{bmatrix}
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& m
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}
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#. Allocate an application node on the heap, applying the top of the stack to
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the address directly below it. The address of the application node is pushed
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onto the stack.
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.. math::
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\gmrule
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{ \mathtt{MkAp} : i
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& f : x : s
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& d
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& h
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& m
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}
|
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{ i
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& a : s
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& d
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& h
|
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\begin{bmatrix}
|
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a : \mathtt{NAp} \; f \; x
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\end{bmatrix}
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& m
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}
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#. Push a function's argument onto the stack
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.. math::
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\gmrule
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{ \mathtt{Push} \; n : i
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& a_0 : \ldots : a_n : s
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& d
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& h
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& m
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}
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{ i
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& a_n : a_0 : \ldots : a_n : s
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& d
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& h
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& m
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}
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#. Tidy up the stack after instantiating a supercombinator
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.. math::
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\gmrule
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{ \mathtt{Slide} \; n : i
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& a_0 : \ldots : a_n : s
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& d
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& h
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& m
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}
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{ i
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& a_0 : s
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& d
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& h
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& m
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}
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#. If the top of the stack is in WHNF (currently this just means a number) is on
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top of the stack, :code:`Unwind` considers evaluation complete. In the case
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where the dump is **not** empty, the instruction queue and stack is restored
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from the top.
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.. math::
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\gmrule
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{ \mathtt{Unwind} : \nillist
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& a : s
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& \langle i', s' \rangle : d
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& h
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\begin{bmatrix}
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a : \mathtt{NNum} \; n
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\end{bmatrix}
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& m
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}
|
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{ i'
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& a : s'
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& d
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& h
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& m
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}
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#. Bulding on the previous rule, in the case where the dump **is** empty, leave
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the machine in a halt state (i.e. with an empty instruction queue).
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.. math::
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\gmrule
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{ \mathtt{Unwind} : \nillist
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& a : s
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& \nillist
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& h
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\begin{bmatrix}
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a : \mathtt{NNum} \; n
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\end{bmatrix}
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& m
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}
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{ \nillist
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& a : s
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& \nillist
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& h
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& m
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}
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|
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#. Again, building on the previous rules, this rule makes the machine consider
|
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unapplied supercombinators to be in WHNF
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.. math::
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\gmrule
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{ \mathtt{Unwind} : \nillist
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& a_0 : \ldots : a_n : \nillist
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& \langle i, s \rangle : d
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& h
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\begin{bmatrix}
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a_0 : \mathtt{NGlobal} \; k \; c
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\end{bmatrix}
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& m
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}
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{ i
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& a_n : s
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& d
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& h
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& m \\
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\SetCell[c=2]{c}
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\text{when $n < k$}
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}
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#. If an application is on top of the stack, :code:`Unwind` continues unwinding
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.. math::
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\gmrule
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{ \mathtt{Unwind} : \nillist
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& a : s
|
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& d
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& h
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\begin{bmatrix}
|
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a : \mathtt{NAp} \; f \; x
|
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\end{bmatrix}
|
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& m
|
||||
}
|
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{ \mathtt{Unwind} : \nillist
|
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& f : a : s
|
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& d
|
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& h
|
||||
& m
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||||
}
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|
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#. When a supercombinator is on top of the stack (and the correct number of
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arguments have been provided), :code:`Unwind` sets up the stack and jumps to
|
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the supercombinator's code (:math:`\beta`-reduction)
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|
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.. math::
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\gmrule
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{ \mathtt{Unwind} : \nillist
|
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& a_0 : \ldots : a_n : s
|
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& d
|
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& h
|
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\begin{bmatrix}
|
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a_0 : \mathtt{NGlobal} \; n \; c \\
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a_1 : \mathtt{NAp} \; a_0 \; e_1 \\
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\vdots \\
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a_n : \mathtt{NAp} \; a_{n-1} \; e_n \\
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\end{bmatrix}
|
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& m
|
||||
}
|
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{ c
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& e_1 : \ldots : e_n : a_n : s
|
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& d
|
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& h
|
||||
& m
|
||||
}
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|
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#. Pop the stack, and update the nth node to point to the popped address
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|
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.. math::
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\gmrule
|
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{ \mathtt{Update} \; n : i
|
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& e : f : a_1 : \ldots : a_n : s
|
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& d
|
||||
& h
|
||||
\begin{bmatrix}
|
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a_1 : \mathtt{NAp} \; f \; e \\
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\vdots \\
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a_n : \mathtt{NAp} \; a_{n-1} \; e_n
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\end{bmatrix}
|
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& m
|
||||
}
|
||||
{ i
|
||||
& f : a_1 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_n : \mathtt{NInd} \; e
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\end{bmatrix}
|
||||
& m
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||||
}
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#. Pop the stack.
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.. math::
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\gmrule
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{ \mathtt{Pop} \; n : i
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& a_1 : \ldots : a_n : s
|
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& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
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||||
}
|
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|
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#. Follow indirections while unwinding
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.. math::
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\gmrule
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{ \mathtt{Unwind} : \nillist
|
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& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NInd} \; a'
|
||||
\end{bmatrix}
|
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& m
|
||||
}
|
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{ \mathtt{Unwind} : \nillist
|
||||
& a' : s
|
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& d
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||||
& h
|
||||
& m
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}
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#. Allocate uninitialised heap space
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|
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.. math::
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\gmrule
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{ \mathtt{Alloc} \; n : i
|
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& s
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& d
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& h
|
||||
& m
|
||||
}
|
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{ i
|
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& a_1 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
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a_1 : \mathtt{NUninitialised} \\
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||||
\vdots \\
|
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a_n : \mathtt{NUninitialised} \\
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
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|
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#. Evaluate the top of the stack to WHNF
|
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|
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.. math::
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\gmrule
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{ \mathtt{Eval} : i
|
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& a : s
|
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& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a : \nillist
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& \langle i, s \rangle : d
|
||||
& h
|
||||
& m
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}
|
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|
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#. Reduce a primitive binary operator :math:`*`.
|
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.. math::
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\gmrule
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{ * : i
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& a_1 : a_2 : s
|
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& d
|
||||
& h
|
||||
\begin{bmatrix}
|
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a_1 : x \\
|
||||
a_2 : y
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ i
|
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& a' : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
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a' : (x * y)
|
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\end{bmatrix}
|
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& m
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||||
}
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#. Reduce a primitive unary operator :math:`\neg`.
|
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|
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.. math::
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\gmrule
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{ \neg : i
|
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& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
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a : x
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ i
|
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& a' : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
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a' : (\neg x)
|
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\end{bmatrix}
|
||||
& m
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}
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***************
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Extension Rules
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***************
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#. A sneaky trick to enable sharing of :code:`NNum` nodes. We note that the
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global environment is a mapping of plain old strings to heap addresses.
|
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Strings of digits are not considered valid identifiers, so putting them on
|
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the global environment will never conflict with a supercombinator! We abuse
|
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this by modifying Core Rule 2 to update the global environment with the new
|
||||
node's address. Consider how this rule might impact garbage collection
|
||||
(remember that the environment is intended for *globals*).
|
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|
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.. math::
|
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\gmrule
|
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{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
\begin{bmatrix}
|
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n' : a
|
||||
\end{bmatrix}
|
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\\
|
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\SetCell[c=5]{c}
|
||||
\text{where $n'$ is the base-10 string rep. of $n$}
|
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}
|
||||
|
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#. In order for the previous rule to be effective, we are also required to take
|
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action when a number already exists in the environment:
|
||||
|
||||
.. math::
|
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\gmrule
|
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{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
n' : a
|
||||
\end{bmatrix}
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\\
|
||||
\SetCell[c=5]{c}
|
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\text{where $n'$ is the base-10 string rep. of $n$}
|
||||
}
|
||||
|
||||
334
doc/src/references/ti-state-transitions.rst
Normal file
334
doc/src/references/ti-state-transitions.rst
Normal file
@@ -0,0 +1,334 @@
|
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============================================
|
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Template Instantiator State Transition Rules
|
||||
============================================
|
||||
|
||||
Evaluation is complete when a single :code:`NNum` remains on the stack and the
|
||||
dump is empty.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ a : \nillist
|
||||
& \nillist
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ \mathtt{HALT}
|
||||
}
|
||||
|
||||
Dereference an indirection passed as an argument to a function.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{a : s & d & h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NAp} \; a_1 \; a_2 \\
|
||||
a_2 : \mathtt{NInd} \; a_3
|
||||
\end{bmatrix} & g}
|
||||
{a : s & d & h[a : \mathtt{NAp} \; a_1 \; a_3] & g}
|
||||
|
||||
Dereference an indirection on top of the stack.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{p : s & d & h
|
||||
\begin{bmatrix}
|
||||
p : \mathtt{NInd} \; a
|
||||
\end{bmatrix} & g}
|
||||
{a : s & d & h & g}
|
||||
|
||||
Perform a unary operation :math:`o(n)` with internal :code:`Prim` constructor
|
||||
:code:`O` on an argument in normal form.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ f : a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
f : \mathtt{NPrim} \; \mathtt{O} \\
|
||||
a : \mathtt{NAp} \; f \; x \\
|
||||
x : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; (o(n))
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
|
||||
Evaluate the argument of a unary operation with internal :code:`Prim`
|
||||
constructor :code:`O`.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ f : a : \nillist
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
f : \mathtt{NPrim} \; \mathtt{O} \\
|
||||
a : \mathtt{NAp} \; f \; x
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ x : \nillist
|
||||
& (f : a : \nillist) : d
|
||||
& h
|
||||
& g
|
||||
}
|
||||
|
||||
Restore the stack when a sub-computation has completed.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ a : \nillist
|
||||
& s : d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ s
|
||||
& d
|
||||
& h
|
||||
& g
|
||||
}
|
||||
|
||||
Reduce a supercombinator and update the root with the :math:`\beta`-reduced form
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ a_0 : a_1 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_0 : \mathtt{NSupercomb} \; [x_1,\ldots,x_n] \; e
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ a_n : s
|
||||
& d
|
||||
& h'
|
||||
& g
|
||||
\\
|
||||
& \SetCell[c=3]{c}
|
||||
\text{where } h' = \mathtt{instantiateU} \; e \; a_n \; h \; g
|
||||
}
|
||||
|
||||
Perform a binary operation :math:`o(x,y)` associated with internal :code:`Prim`
|
||||
constructor :code:`O` on two :code:`NNum` s both in normal form.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ f : a_1 : a_2 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
f : \mathtt{NPrim} \; \mathtt{O} \\
|
||||
a_1 : \mathtt{NAp} \; f \; (\mathtt{NNum} \; x) \\
|
||||
a_2 : \mathtt{NAp} \; a_1 \; (\mathtt{NNum} \; y)
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ a_2 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_2 : \mathtt{NNum} \; (o(x,y))
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
|
||||
In a conditional primitive, perform the reduction if the condition has been
|
||||
evaluated as True (:code:`NData 1 []`).
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ f : a_1 : a_2 : a_3 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
f : \mathtt{NPrim} \; \mathtt{IfP} \\
|
||||
c : \mathtt{NPrim} \; (\mathtt{NData} \; 1 \; \nillist) \\
|
||||
a_1 : \mathtt{NAp} \; f \; c \\
|
||||
a_2 : \mathtt{NAp} \; a_1 \; x \\
|
||||
a_3 : \mathtt{NAp} \; a_2 \; y
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ x : s
|
||||
& d
|
||||
& h
|
||||
& g
|
||||
}
|
||||
|
||||
In a conditional primitive, perform the reduction if the condition has been
|
||||
evaluated as False (:code:`NData 0 []`).
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ f : a_1 : a_2 : a_3 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
f : \mathtt{NPrim} \; \mathtt{IfP} \\
|
||||
c : \mathtt{NPrim} \; (\mathtt{NData} \; 0 \; \nillist) \\
|
||||
a_1 : \mathtt{NAp} \; f \; c \\
|
||||
a_2 : \mathtt{NAp} \; a_1 \; x \\
|
||||
a_3 : \mathtt{NAp} \; a_2 \; y
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ y : s
|
||||
& d
|
||||
& h
|
||||
& g
|
||||
}
|
||||
|
||||
|
||||
In a conditional primitive, evaluate the condition.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ f : a_1 : \nillist
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
f : \mathtt{NPrim} \; \mathtt{IfP} \\
|
||||
a_1 : \mathtt{NAp} \; f \; x
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ x : \nillist
|
||||
& (f : a_1 : \nillist) : d
|
||||
& h
|
||||
& g
|
||||
}
|
||||
|
||||
Construct :code:`NData` out of a constructor and its arguments
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ c : a_1 : \ldots : a_n : \nillist
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
c : \mathtt{NPrim} \; (\mathtt{ConP} \; t \; n) \\
|
||||
a_1 : \mathtt{NAp} \; c \; x_1 \\
|
||||
\vdots \\
|
||||
a_n : \mathtt{NAp} \; a_{n-1} \; x_n
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ a_n : \nillist
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_n : \mathtt{NData} \; t \; [x_1, \ldots, x_n]
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
|
||||
Pairs
|
||||
-----
|
||||
|
||||
Evaluate the first argument if necessary
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ c : a_1 : a_2 : \nillist
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
c : \mathtt{NPrim} \; \mathtt{CasePairP} \\
|
||||
p : \mathtt{NAp} \; \_ \: \_ \\
|
||||
a_1 : \mathtt{NAp} \; c \; p \\
|
||||
a_2 : \mathtt{NAp} \; a_2 \; f
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ p : \nillist
|
||||
& (a_1 : a_2 : \nillist) : d
|
||||
& h
|
||||
& g
|
||||
}
|
||||
|
||||
Perform the reduction if the first argument is in normal form
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ c : a_1 : a_2 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
c : \mathtt{NPrim} \; \mathtt{CasePairP} \\
|
||||
p : \mathtt{NData} \; 0 \; [x,y] \\
|
||||
a_1 : \mathtt{NAp} \; c \; p \\
|
||||
a_2 : \mathtt{NAp} \; a_1 \; f
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ a_1 : a_2 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_1 : \mathtt{NAp} \; f \; x \\
|
||||
a_2 : \mathtt{NAp} \; a_1 \; y
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
|
||||
Lists
|
||||
-----
|
||||
|
||||
Evaluate the scrutinee
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ c : a_1 : a_2 : a_3 : \nillist
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
c : \mathtt{NPrim} \; \mathtt{CaseListP} \\
|
||||
a_1 : \mathtt{NAp} \; c \; x
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ x
|
||||
& (a_1 : a_2 : a_3) : \nillist
|
||||
& h
|
||||
& g
|
||||
}
|
||||
|
||||
If the scrutinee is :code:`Nil`, perform the appropriate reduction.
|
||||
|
||||
.. math::
|
||||
\transrule
|
||||
{ c : a_1 : a_2 : a_3 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
c : \mathtt{NPrim} \; \mathtt{CaseListP} \\
|
||||
p : \mathtt{NData} \; 1 \; \nillist \\
|
||||
a_1 : \mathtt{NAp} \; c \; p \\
|
||||
a_2 : \mathtt{NAp} \; p \; f_\text{nil} \\
|
||||
a_3 : \mathtt{NAp} \; a_2 \; f_\text{cons}
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
{ a_3 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_3 : \mathtt{NAp} \; f_\text{nil}
|
||||
\end{bmatrix}
|
||||
& g
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user