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doc/build/html/_sources/references/gm-state-transition-rules.rst.txt
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347
doc/build/html/_sources/references/gm-state-transition-rules.rst.txt
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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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1. 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
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& d
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& h
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& m
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\begin{bmatrix}
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f : a
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\end{bmatrix}
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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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2. 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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3. 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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4. 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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5. 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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6. If a number is on top of the stack, :code:`Unwind` leaves the machine in a
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halt state
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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{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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& d
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& h
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& m
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}
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7. 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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}
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{ \mathtt{Unwind} : \nillist
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& f : 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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8. 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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.. 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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}
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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
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& m
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}
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9. Pop the stack, and update the nth node to point to the popped address
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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
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& h
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\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
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}
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{ i
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& f : a_1 : \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_n : \mathtt{NInd} \; e
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\end{bmatrix}
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& m
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}
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10. 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
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& h
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& m
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}
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{ 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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11. 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
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& d
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& h
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\begin{bmatrix}
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a : \mathtt{NInd} \; a'
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\end{bmatrix}
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& m
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}
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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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& m
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}
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12. Allocate uninitialised heap space
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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
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& m
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}
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{ i
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& a_1 : \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_1 : \mathtt{NUninitialised} \\
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\vdots \\
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a_n : \mathtt{NUninitialised} \\
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\end{bmatrix}
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& m
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}
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13. When unwinding, if the top of the stack is in WHNF, pop the dump
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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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14. Evaluate the top of the stack to WHNF
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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
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& h
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& m
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}
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{ i'
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& a : \nillist
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& \langle i, s \rangle
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& h
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& m
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}
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***************
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Extension Rules
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***************
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1. A sneaky trick to enable sharing of :code:`NNum` nodes. We note that the
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global environment is a mapping of :code:`Name` objects (i.e. identifiers) to
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heap addresses. Strings of digits are not considered valid identifiers! We
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abuse this by modifying Core Rule 2 to update the global environment with the
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new node's address. Consider how this rule might impact garbage collection
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(remember that the environment is intended for *globals*).
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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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& 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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& 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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\begin{bmatrix}
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n' : a
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\end{bmatrix}
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\\
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\SetCell[c=5]{c}
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\text{where $n'$ is the base-10 string rep. of $n$}
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}
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2. In order for Extension Rule 1. to be effective, we are also required to take
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action when a number already exists in the environment:
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.. math::
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\transrule
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{ \mathtt{PushInt} \; n : i
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& s
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& h
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& m
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\begin{bmatrix}
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n' : a
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\end{bmatrix}
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}
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{ i
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& a : s
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& h
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& m
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\\
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\SetCell[c=5]{c}
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\text{where $n'$ is the base-10 string rep. of $n$}
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}
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