prim arith hooray
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crumbtoo
@@ -6,296 +6,345 @@ G-Machine State Transition Rules
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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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#. 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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.. 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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#. 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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.. 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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#. 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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.. 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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#. 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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.. 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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#. 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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.. 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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#. 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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& 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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.. 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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7. If an application is on top of the stack, :code:`Unwind` continues unwinding
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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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& 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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.. 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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8. When a supercombinator is on top of the stack (and the correct number of
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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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}
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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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#. 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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.. 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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#. 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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.. 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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#. 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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.. 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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#. 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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.. 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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#. 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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.. 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 (currently this just
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means a number), pop the dump
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#. Evaluate the top of the stack to WHNF
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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}
|
||||
& m
|
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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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.. 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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{ \mathtt{Unwind} : \nillist
|
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& a : \nillist
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& \langle i, s \rangle : d
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& h
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& m
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}
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|
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14. Evaluate the top of the stack to WHNF
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#. Reduce a primitive binary operator :math:`*`.
|
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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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{ \mathtt{Unwind} : \nillist
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& a : \nillist
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& \langle i, s \rangle : d
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& h
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& m
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}
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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
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& h
|
||||
\begin{bmatrix}
|
||||
a_1 : x \\
|
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a_2 : y
|
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\end{bmatrix}
|
||||
& m
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||||
}
|
||||
{ i
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& a' : s
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& d
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||||
& h
|
||||
\begin{bmatrix}
|
||||
a' : (x * y)
|
||||
\end{bmatrix}
|
||||
& m
|
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}
|
||||
|
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#. Reduce a primitive unary operator :math:`\neg`.
|
||||
|
||||
.. math::
|
||||
\gmrule
|
||||
{ \neg : i
|
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& a : s
|
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& d
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||||
& h
|
||||
\begin{bmatrix}
|
||||
a : x
|
||||
\end{bmatrix}
|
||||
& m
|
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}
|
||||
{ i
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& a' : s
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& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a' : (\neg x)
|
||||
\end{bmatrix}
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||||
& m
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}
|
||||
|
||||
***************
|
||||
Extension Rules
|
||||
***************
|
||||
|
||||
1. A sneaky trick to enable sharing of :code:`NNum` nodes. We note that the
|
||||
#. A sneaky trick to enable sharing of :code:`NNum` nodes. We note that the
|
||||
global environment is a mapping of plain old strings to heap addresses.
|
||||
Strings of digits are not considered valid identifiers, so putting them on
|
||||
the global environment will never conflict with a supercombinator! We abuse
|
||||
@@ -303,51 +352,51 @@ Extension Rules
|
||||
node's address. Consider how this rule might impact garbage collection
|
||||
(remember that the environment is intended for *globals*).
|
||||
|
||||
.. math::
|
||||
\gmrule
|
||||
{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
n' : a
|
||||
\end{bmatrix}
|
||||
\\
|
||||
\SetCell[c=5]{c}
|
||||
\text{where $n'$ is the base-10 string rep. of $n$}
|
||||
}
|
||||
.. math::
|
||||
\gmrule
|
||||
{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
n' : a
|
||||
\end{bmatrix}
|
||||
\\
|
||||
\SetCell[c=5]{c}
|
||||
\text{where $n'$ is the base-10 string rep. of $n$}
|
||||
}
|
||||
|
||||
2. In order for Extension Rule 1. to be effective, we are also required to take
|
||||
#. In order for the previous rule to be effective, we are also required to take
|
||||
action when a number already exists in the environment:
|
||||
|
||||
.. math::
|
||||
\gmrule
|
||||
{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
n' : a
|
||||
\end{bmatrix}
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\\
|
||||
\SetCell[c=5]{c}
|
||||
\text{where $n'$ is the base-10 string rep. of $n$}
|
||||
}
|
||||
.. math::
|
||||
\gmrule
|
||||
{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
n' : a
|
||||
\end{bmatrix}
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\\
|
||||
\SetCell[c=5]{c}
|
||||
\text{where $n'$ is the base-10 string rep. of $n$}
|
||||
}
|
||||
|
||||
|
||||
Reference in New Issue
Block a user