Update some of the docs.
This commit is contained in:
Simon Forman
+103
-106
@@ -1,8 +1,8 @@
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.. code:: ipython2
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.. code:: ipython3
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from notebook_preamble import D, DefinitionWrapper, J, V, define
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On “Two Exercises Found in a Book on Algorithmics”
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On "Two Exercises Found in a Book on Algorithmics"
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==================================================
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Bird & Meertens
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@@ -13,27 +13,25 @@ here <https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.694.2614>`__
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Define ``scan`` in terms of a reduction.
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----------------------------------------
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Problem I. The reduction operator ``/`` of APL takes some binary
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operator ``⨁`` on its left and a vector ``x`` of values on its right.
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The meaning of ``⨁/x`` for ``x = [a b ... z]`` is the value
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``a⨁b⨁...⨁z``. For this to be well-defined in the absence of
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brackets, the operation ``⨁`` has to be associative. Now there is
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another operator ``\`` of APL called ``scan``. Its effect is closely
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related to reduction in that we have:
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Problem I. The reduction operator ``/`` of APL takes some binary
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operator ``⨁`` on its left and a vector ``x`` of values on its
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right. The meaning of ``⨁/x`` for ``x = [a b ... z]`` is the value
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``a⨁b⨁...⨁z``. For this to be well-defined in the absence of
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brackets, the operation ``⨁`` has to be associative. Now there is
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another operator ``\`` of APL called ``scan``. Its effect is closely
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related to reduction in that we have:
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::
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⨁\x = [a a⨁b a⨁b⨁c ... a⨁b⨁...⨁z]
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⨁\x = [a a⨁b a⨁b⨁c ... a⨁b⨁...⨁z]
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..
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The problem is to find some definition of ``scan`` as a reduction. In
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other words, we have to find some function ``f`` and an operator
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``⨂`` so that
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The problem is to find some definition of ``scan`` as a reduction.
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In other words, we have to find some function ``f`` and an operator
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``⨂`` so that
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::
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⨁\x = f(a)⨂f(b)⨂...⨂f(z)
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⨁\x = f(a)⨂f(b)⨂...⨂f(z)
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Designing the Recursive Function
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--------------------------------
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@@ -52,48 +50,48 @@ instead of two (the b is instead the duplicate of a.)
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::
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H3 == [P] [pop c] [[G] dupdip] [dip F] genrec
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H3 == [P] [pop c] [[G] dupdip] [dip F] genrec
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... a [G] dupdip [H3] dip F
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... a G a [H3] dip F
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... a′ a [H3] dip F
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... a′ H3 a F
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... a′ [G] dupdip [H3] dip F a F
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... a′ G a′ [H3] dip F a F
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... a″ a′ [H3] dip F a F
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... a″ H3 a′ F a F
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... a″ [G] dupdip [H3] dip F a′ F a F
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... a″ G a″ [H3] dip F a′ F a F
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... a‴ a″ [H3] dip F a′ F a F
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... a‴ H3 a″ F a′ F a F
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... a‴ pop c a″ F a′ F a F
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... c a″ F a′ F a F
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... d a′ F a F
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... d′ a F
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... d″
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... a [G] dupdip [H3] dip F
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... a G a [H3] dip F
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... a′ a [H3] dip F
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... a′ H3 a F
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... a′ [G] dupdip [H3] dip F a F
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... a′ G a′ [H3] dip F a F
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... a″ a′ [H3] dip F a F
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... a″ H3 a′ F a F
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... a″ [G] dupdip [H3] dip F a′ F a F
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... a″ G a″ [H3] dip F a′ F a F
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... a‴ a″ [H3] dip F a′ F a F
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... a‴ H3 a″ F a′ F a F
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... a‴ pop c a″ F a′ F a F
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... c a″ F a′ F a F
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... d a′ F a F
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... d′ a F
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... d″
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Initial Definition
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~~~~~~~~~~~~~~~~~~
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We’re building a list of values so this is an “anamorphism”. (An
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We're building a list of values so this is an "anamorphism". (An
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anamorphism uses ``[]`` for ``c`` and ``swons`` for ``F``.)
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::
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scan == [P] [pop []] [[G] dupdip] [dip swons] genrec
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scan == [P] [pop []] [[G] dupdip] [dip swons] genrec
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Convert to ``ifte``:
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::
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scan == [P] [pop []] [[G] dupdip [scan] dip swons] ifte
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scan == [P] [pop []] [[G] dupdip [scan] dip swons] ifte
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On the recursive branch ``[G] dupdip`` doesn’t cut it:
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On the recursive branch ``[G] dupdip`` doesn't cut it:
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::
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[1 2 3] [G] dupdip [scan] dip swons
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[1 2 3] G [1 2 3] [scan] dip swons
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[1 2 3] [G] dupdip [scan] dip swons
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[1 2 3] G [1 2 3] [scan] dip swons
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Use ``first``
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~~~~~~~~~~~~~
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@@ -103,11 +101,11 @@ use ``first``.
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::
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scan == [P] [pop []] [[G] dupdip first] [dip swons] genrec
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scan == [P] [pop []] [[G] dupdip first] [dip swons] genrec
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[1 2 3] [G] dupdip first [scan] dip swons
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[1 2 3] G [1 2 3] first [scan] dip swons
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[1 2 3] G 1 [scan] dip swons
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[1 2 3] [G] dupdip first [scan] dip swons
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[1 2 3] G [1 2 3] first [scan] dip swons
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[1 2 3] G 1 [scan] dip swons
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``G`` applies ``⨁``
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~~~~~~~~~~~~~~~~~~~
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@@ -117,10 +115,10 @@ in the list.
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::
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[1 2 3] G
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[1 2 3] [⨁] infra
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[1 2 3] [+] infra
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[3 3]
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[1 2 3] G
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[1 2 3] [⨁] infra
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[1 2 3] [+] infra
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[3 3]
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Predicate ``P``
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~~~~~~~~~~~~~~~
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@@ -130,21 +128,21 @@ less that two items in them:
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::
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P == size 1 <=
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P == size 1 <=
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Let’s see what we’ve got so far:
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Let's see what we've got so far:
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::
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scan == [P ] [pop []] [[G] dupdip first] [dip swons] genrec
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scan == [size 1 <=] [pop []] [[[F] infra] dupdip first] [dip swons] genrec
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scan == [P ] [pop []] [[G] dupdip first] [dip swons] genrec
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scan == [size 1 <=] [pop []] [[[F] infra] dupdip first] [dip swons] genrec
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Handling the Last Term
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~~~~~~~~~~~~~~~~~~~~~~
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This works to a point, but it throws away the last term:
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.. code:: ipython2
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.. code:: ipython3
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J('[1 2 3] [size 1 <=] [pop []] [[[+] infra] dupdip first] [dip swons] genrec')
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@@ -154,9 +152,9 @@ This works to a point, but it throws away the last term:
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[1 3]
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Hmm… Let’s take out the ``pop`` for a sec…
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Hmm... Let's take out the ``pop`` for a sec...
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.. code:: ipython2
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.. code:: ipython3
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J('[1 2 3] [size 1 <=] [[]] [[[+] infra] dupdip first] [dip swons] genrec')
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@@ -167,11 +165,11 @@ Hmm… Let’s take out the ``pop`` for a sec…
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That leaves the last item in our list, then it puts an empty list on the
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stack and ``swons``\ ’s the new terms onto that. If we leave out that
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empty list, they will be ``swons``\ ’d onto that list that already has
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the last item.
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stack and ``swons``'s the new terms onto that. If we leave out that
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empty list, they will be ``swons``'d onto that list that already has the
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last item.
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.. code:: ipython2
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.. code:: ipython3
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J('[1 2 3] [size 1 <=] [] [[[+] infra] dupdip first] [dip swons] genrec')
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@@ -188,28 +186,28 @@ So we have:
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::
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[⨁] scan == [size 1 <=] [] [[[⨁] infra] dupdip first] [dip swons] genrec
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[⨁] scan == [size 1 <=] [] [[[⨁] infra] dupdip first] [dip swons] genrec
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Trivially:
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::
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== [size 1 <=] [] [[[⨁] infra] dupdip first] [dip swons] genrec
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== [[[⨁] infra] dupdip first] [size 1 <=] [] roll< [dip swons] genrec
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== [[⨁] infra] [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec
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== [⨁] [infra] cons [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec
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== [size 1 <=] [] [[[⨁] infra] dupdip first] [dip swons] genrec
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== [[[⨁] infra] dupdip first] [size 1 <=] [] roll< [dip swons] genrec
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== [[⨁] infra] [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec
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== [⨁] [infra] cons [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec
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And so:
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::
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scan == [infra] cons [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec
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scan == [infra] cons [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec
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.. code:: ipython2
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.. code:: ipython3
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define('scan == [infra] cons [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec')
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define('scan [infra] cons [dupdip first] cons [size 1 <=] [] roll< [dip swons] genrec')
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.. code:: ipython2
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.. code:: ipython3
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J('[1 2 3 4] [+] scan')
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@@ -219,7 +217,7 @@ And so:
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[1 3 6 10]
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.. code:: ipython2
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.. code:: ipython3
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J('[1 2 3 4] [*] scan')
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@@ -229,7 +227,7 @@ And so:
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[1 2 6 24]
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.. code:: ipython2
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.. code:: ipython3
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J('[1 2 3 4 5 6 7] [neg +] scan')
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@@ -242,24 +240,24 @@ And so:
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Problem 2.
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----------
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Define a line to be a sequence of characters not containing the
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newline character. It is easy to define a function ``Unlines`` that
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converts a non-empty sequence of lines into a sequence of characters
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by inserting newline characters between every two lines.
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Define a line to be a sequence of characters not containing the
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newline character. It is easy to define a function ``Unlines`` that
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converts a non-empty sequence of lines into a sequence of characters
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by inserting newline characters between every two lines.
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Since ``Unlines`` is injective, the function ``Lines``, which
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converts a sequence of characters into a sequence of lines by
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splitting on newline characters, can be specified as the inverse of
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``Unlines``.
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Since ``Unlines`` is injective, the function ``Lines``, which
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converts a sequence of characters into a sequence of lines by
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splitting on newline characters, can be specified as the inverse of
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``Unlines``.
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The problem, just as in Problem 1. is to find a definition by
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reduction of the function ``Lines``.
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The problem, just as in Problem 1. is to find a definition by
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reduction of the function ``Lines``.
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::
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Unlines = uncons ['\n' swap + +] step
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Unlines = uncons ['\n' swap + +] step
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.. code:: ipython2
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.. code:: ipython3
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J('["hello" "world"] uncons ["\n" swap + +] step')
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@@ -269,41 +267,41 @@ Problem 2.
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'hello\nworld'
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Again ignoring the actual task let’s just derive ``Lines``:
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Again ignoring the actual task let's just derive ``Lines``:
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::
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"abc\nefg\nhij" Lines
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---------------------------
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["abc" "efg" "hij"]
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"abc\nefg\nhij" Lines
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---------------------------
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["abc" "efg" "hij"]
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Instead of ``P == [size 1 <=]`` we want ``["\n" in]``, and for the
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base-case of a string with no newlines in it we want to use ``unit``:
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::
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Lines == ["\n" in] [unit] [R0] [dip swons] genrec
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Lines == ["\n" in] [unit] [R0 [Lines] dip swons] ifte
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Lines == ["\n" in] [unit] [R0] [dip swons] genrec
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Lines == ["\n" in] [unit] [R0 [Lines] dip swons] ifte
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Derive ``R0``:
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::
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"a \n b" R0 [Lines] dip swons
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"a \n b" split-at-newline swap [Lines] dip swons
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"a " " b" swap [Lines] dip swons
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" b" "a " [Lines] dip swons
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" b" Lines "a " swons
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[" b"] "a " swons
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["a " " b"]
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"a \n b" R0 [Lines] dip swons
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"a \n b" split-at-newline swap [Lines] dip swons
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"a " " b" swap [Lines] dip swons
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" b" "a " [Lines] dip swons
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" b" Lines "a " swons
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[" b"] "a " swons
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["a " " b"]
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So:
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::
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R0 == split-at-newline swap
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R0 == split-at-newline swap
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Lines == ["\n" in] [unit] [split-at-newline swap] [dip swons] genrec
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Lines == ["\n" in] [unit] [split-at-newline swap] [dip swons] genrec
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Missing the Point?
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------------------
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@@ -313,27 +311,26 @@ properties are discussed. Am I missing the point?
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::
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0 [a b c d] [F] step == 0 [a b] [F] step 0 [c d] [F] step concat
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0 [a b c d] [F] step == 0 [a b] [F] step 0 [c d] [F] step concat
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For associative function ``F`` and a “unit” element for that function,
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For associative function ``F`` and a "unit" element for that function,
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here represented by ``0``.
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For functions that don’t have a “unit” we can fake it (the example is
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For functions that don't have a "unit" we can fake it (the example is
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given of infinity for the ``min(a, b)`` function.) We can also use:
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::
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safe_step == [size 1 <=] [] [uncons [F] step] ifte
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safe_step == [size 1 <=] [] [uncons [F] step] ifte
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Or:
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::
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safe_step == [pop size 1 <=] [pop] [[uncons] dip step] ifte
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safe_step == [pop size 1 <=] [pop] [[uncons] dip step] ifte
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[a b c] [F] safe_step
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---------------------------
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a [b c] [F] step
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[a b c] [F] safe_step
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---------------------------
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a [b c] [F] step
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To limit ``F`` to working on pairs of terms from its domain.
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Reference in New Issue
Block a user