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tryout.dtx
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tryout.dtx
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%
% \iffalse
%<*driver>
\documentclass{mtmtcl}
\newcommand{\PROP}{PROP}
\newcommand{\PROPs}{\PROP s}
\begin{document}
\DocInput{tryout.dtx}
\end{document}
%</driver>
% \fi
%
% \title{Some code fragments for trying out \mtl\ features}
% \author{Lars Hellstr\"om}
% \maketitle
%
% \begin{abstract}
% Since the term `test' has rather formal interpretation
% (\textsf{tcltest}, \texttt{.test} file) within the \Tcl\
% community, these less organised tests are instead called tryouts.
% Basically they aim to achieve something with some feature, but
% may in the process make use of lots of other features, so they're
% not very isolated.
% \end{abstract}
%
%
% \section{Prefix}
%
% The following loads the code in the basic files, using the
% \textsf{docstrip} package. This is to facilitate |source|ing this
% code into an interpreter.
% \begin{tcl}
%<*loadbase>
if {[info script] ne ""} then {
set dir [file dirname [info script]]
} elseif {[info exists env(MTMTCL_SOURCE_DIR)]} then {
set dir $env(MTMTCL_SOURCE_DIR)
} else {
set dir ~/Projekt/Tcl-projekt/Algebra
}
if {[catch {
package require APIutil
}]} then {
source [file join [file dirname $dir] API apiutil.tcl]
}
package require docstrip
docstrip::sourcefrom [file join $dir export.dtx] {openmath pkg}
docstrip::sourcefrom [file join $dir support.dtx] {pkg}
docstrip::sourcefrom [file join $dir groups.dtx] {pkg}
docstrip::sourcefrom [file join $dir rings.dtx] {integers SemigroupAlgebra}
%</loadbase>
% \end{tcl}
%
%
% \section{Exporting a polynomial}
%
% The idea for the following is to construct the polynomial
% $\prod_{i=1}^4 (x -\nobreak i)$ and export it.
%
% The first order of business is to construct a polynomial ring
% $\mathbb{Z}[x]$. It can be done as the semigroup algabra over $\Z$
% of the abelian monoid generated by the finite set $\{x\}$, so we
% end up with
% \begin{tcl}
%<*univpoly>
interp alias {} Zx {} ::mtmtcl::rings::semigroup_algebra {
::mtmtcl::rings::integers::all
} {
::mtmtcl::groups::power_product_monoid_1.3 {
::mtmtcl::sets::variable_set {x} {}
}
}
% \end{tcl}
% For the future, it might actually be easier to keep that as a
% command prefix in a variable, but for now an alias is fine.
% It is also likely that this construction should be provided by a
% procedure, so that users won't have to concern themselves with the
% syntax of the component structures (much).
%
% Now for the actual polynomial. Since it is constructed as a
% product, it is naturally computed in a loop where each iteration
% computes one factor and multiplies it into the product. For the
% purpose of computing the factor, it is however convenient to have
% precomputed the |Zx| representations of $1$ and $x$.
% \begin{tcl}
set unit [Zx 1]
set x [Zx basiselement [Zx basis named x]]
set prod $unit
foreach i {1 2 3 4} {
set prod [Zx * $prod [
Zx - $x [Zx . [Zx scalar 1 $i] $unit]
]]
}
% \end{tcl}
% Even then the computation is rather roundabout, so one should
% probably see to that |semigroup_algebra| implements an |integer.|
% method.
%
% But with that taken care of, we get to the main point of business:
% exporting this polynomial.
% \begin{tcl}
set raw [Zx export $prod {}]
set typo [mtmtcl::present::typography $raw]
% \end{tcl}
% As far as exercises of edge cases go, this product is rather good.
% It shows both a naked power term ($x^4$), product-and-power terms
% ($-10x^3$ and $35x^2$), term without power ($-50x$), and naked
% integer term ($24$).
%
% If we're targeting OpenMath, then the next step would be
% \begin{tcl}
set strict [mtmtcl::openmath::stricten\
{*}[mtmtcl::openmath::oma $typo *] 1]
% \end{tcl}
% In order to generate XML, it is convenient to use the \textsf{tdom}
% package so that we get some indentation.
% \begin{tcl}
%<*tdom>
package require tdom
dom createDocumentNode obj
$obj appendFromList [list OMOBJ {} [list $strict]]
puts [$obj asXML]
unset obj
%</tdom>
% \end{tcl}
%
% If we're targetting \LaTeX, there is a command which does all the
% formatting.
% \begin{tcl}
set code [mtmtcl::latex::convertto $raw]
puts $code
% \end{tcl}
%
%
%
% \begin{tcl}
%</univpoly>
% \end{tcl}
%
%
% \section{Exporting to \LaTeX}
%
% The following tests the generation of \LaTeX\ code for formulae.
% \begin{tcl}
%<*LaTeX.export>
% \end{tcl}
% Since the last steps were completed first, that is where this will
% start.
%
% The first formula is \(f\bigl( g(a+b) - g(a,b)^2 \bigr)\).
% \begin{tcl}
set tree {mrow {} {
{{#text} f}
{mfenced {} {
{mrow {} {
{mrow {} {
{#text g}
{mfenced {} { {mrow {} {
{#text a} {#text +} {#text b}
}} }}
}}
{#text -}
{mrow {} {
{mrow {} {
{#text g}
{mfenced {expand 1} {
{#text a} {#text b}
}}
}}
{#text ^}
{maybewrap {} { {#text 2} }}
}}
}}
}}
}}
set temp [mtmtcl::latex::metax:: {*}$tree]
set tree [lindex $temp 0]
set code [mtmtcl::latex::fold $tree 20 1]
% \end{tcl}
%
%
% \begin{tcl}
%</LaTeX.export>
% \end{tcl}
%
%
% \section{PROP-expression comparisons}
%
% The following is an attempt to find coefficients in a
% representation of the free bialgebra PROP under which the matrix
% order would order the axioms as I want them.
%
% \begin{tcl}
%<*PROPcompare>
%<*loadbase>
docstrip::sourcefrom [file join $dir props.dtx] {pkg}
%</loadbase>
% \end{tcl}
%
% \subsection{First attempt}
%
% In this approach, the multiplication and comultiplication
% representatatives are upper triangular with $1$ in the upper left
% corner.
%
% \begin{tcl}
interp alias {} Coeff_ring {} ::mtmtcl::rings::semigroup_algebra {
::mtmtcl::rings::integers::all
} {
::mtmtcl::groups::power_product_monoid_1.3 {
::mtmtcl::sets::variable_set {A B C D E F a b c d e f} {}
}
}
interp alias {} PROP {} mtmtcl::matprop::homfdpower 2 Coeff_ring
% \end{tcl}
%
% \begin{tcl}
set epsilon [list [
list [Coeff_ring basiselement [Coeff_ring basis named E]]\
[Coeff_ring basiselement [Coeff_ring basis named F]]
]]
set code [mtmtcl::latex::convertto [PROP export $epsilon {}]]
% \end{tcl}
%
%
%
% \begin{tcl}
set Delta [
list [
list [Coeff_ring 1] [Coeff_ring named A]
] [
list [Coeff_ring 0] [Coeff_ring named B]
] [
list [Coeff_ring 0] [Coeff_ring named C]
] [
list [Coeff_ring 0] [Coeff_ring named D]
]
]
set code [mtmtcl::latex::convertto [PROP export $Delta {}]]
% \end{tcl}
%
% \begin{tcl}
set DeltaDelta [PROP tensor $Delta $Delta]
set code [mtmtcl::latex::convertto [PROP export $DeltaDelta {}]]
% \end{tcl}
%
% \begin{tcl}
set mult [
list [
list [Coeff_ring 1] [Coeff_ring named a] [Coeff_ring named b]\
[Coeff_ring named c]
] [
list [Coeff_ring 0] [Coeff_ring 0] [Coeff_ring 0]\
[Coeff_ring named d]
]
]
set code [mtmtcl::latex::convertto [PROP export $mult {}]]
% \end{tcl}
%
% \begin{tcl}
set Delta_mult [PROP * $Delta $mult]
set reduced_Delta_mult [PROP * [
PROP tensor $mult $mult
] [
PROP perm. {0 2 1 3} $DeltaDelta
]]
set diff [PROP - $Delta_mult $reduced_Delta_mult]
set code [mtmtcl::latex::convertto [PROP export $diff {}]]
% \end{tcl}
%
% \begin{equation} \label{Eq:Delta_mult}
% \hss
% \begin{pmatrix} 0 & a - A - B a - C a - D a^2 &
% b - A - B b - C b - D b^2 &
% \parbox{0.4\linewidth}{\(c + A d - A^2 - A B a - A B b -
% B^2 c - A C a - A D a^2 - 2 B C a b - B D a c - A C b -
% A D b^2 - B D b c - C^2 c - C D a c - C D b c - D^2 c^2\)}
% \\ 0 & 0 & 0 & B d - B^2 d - B D a d - B D b d - D^2 c d \\
% 0 & 0 & 0 & C d - C^2 d - C D a d - C D b d - D^2 c d \\
% 0 & 0 & 0 & D d - D^2 d^2 \end{pmatrix} > 0
% \end{equation}
%
% \begin{tcl}
set counit_Delta1 [PROP * [PROP tensorpad $epsilon 1] $Delta]
set counit_Delta2 [PROP *\
[PROP tensor [PROP permutation 0] $epsilon] $Delta]
set diff1 [PROP - $counit_Delta1 [PROP permutation 0]]
set diff2 [PROP - $counit_Delta2 [PROP permutation 0]]
set code "
[mtmtcl::latex::convertto [PROP export $diff1 {}]],
[mtmtcl::latex::convertto [PROP export $diff2 {}]]"
% \end{tcl}
% \begin{equation} \label{Eq:epsilon_Delta}
% \begin{pmatrix} E - 1 & A E + B F \\ 0 & C E + D F - 1
% \end{pmatrix} ,
% \begin{pmatrix} E - 1 & A E + C F \\ 0 & B E + D F - 1
% \end{pmatrix}
% > 0
% \end{equation}
%
% \begin{tcl}
set counit_counit [PROP tensor $epsilon $epsilon]
set counit_mult [PROP * $epsilon $mult]
set diff [PROP - $counit_mult $counit_counit]
set code [mtmtcl::latex::convertto [PROP export $diff {}]]
% \end{tcl}
% \begin{equation} \label{Eq:epsilon_mult}
% \begin{pmatrix}
% E - E^2 & E a - E F & E b - E F & E c + F d - F^2
% \end{pmatrix} > 0
% \end{equation}
%
% Conclusion: \(E = 1\). At least one variable less! Is it the same
% for $e$?
%
% \begin{tcl}
set unit [list [
list [Coeff_ring named e]
] [
list [Coeff_ring named f]
]]
set code [mtmtcl::latex::convertto [PROP export $unit {}]]
% \end{tcl}
%
% \begin{tcl}
set mult_unit1 [PROP * $mult [PROP tensorpad $unit 1]]
set mult_unit2 [PROP * $mult [PROP tensor [PROP permutation 0] $unit]]
set diff1 [PROP - $mult_unit1 [PROP permutation 0]]
set diff2 [PROP - $mult_unit2 [PROP permutation 0]]
set code "
[mtmtcl::latex::convertto [PROP export $diff1 {}]],
[mtmtcl::latex::convertto [PROP export $diff2 {}]]"
% \end{tcl}
% \begin{equation} \label{Eq:mult_unit}
% \begin{pmatrix} e + a f - 1 & b e + c f \\ 0 & d f - 1 \end{pmatrix} ,
% \begin{pmatrix} e + b f - 1 & a e + c f \\ 0 & d f - 1 \end{pmatrix}
% > 0
% \end{equation}
% It appears not, but what is then the conclusion?
%
% \begin{tcl}
set unit_unit [PROP tensor $unit $unit]
set Delta_unit [PROP * $Delta $unit]
set diff [PROP - $Delta_unit $unit_unit]
set code [mtmtcl::latex::convertto [PROP export $diff {}]]
% \end{tcl}
% \begin{equation} \label{Eq:Delta_unit}
% \begin{pmatrix}
% e + A f - e^2 \\ B f - e f \\ C f - e f \\ D f - f^2
% \end{pmatrix} > 0
% \end{equation}
%
% Easy conclusions:
% \begin{enumerate}
% \item
% By \eqref{Eq:epsilon_Delta} and \eqref{Eq:epsilon_mult},
% \(E=1\). Hence
% \begin{align*}
% a \geqslant{}& F &
% b \geqslant{}& F \\
% c \geqslant{}& F^2 - Fd \\
% DF \geqslant{}& 1-C &
% DF \geqslant{}& 1-B
% \end{align*}
% \item
% By \eqref{Eq:Delta_mult}, \(Dd \leqslant 1\).
% \item
% By \eqref{Eq:mult_unit}, \(df \geqslant 1\).
% \item
% By \eqref{Eq:Delta_unit}, \(Af \geqslant e^2 - e\).
% Also, \(B,C \geqslant e\) and \(D \geqslant f\).
% \end{enumerate}
% Unfortunately, \(C d - C^2 d - C D a d - C D b d - D^2 c d =
% (C - C^2 - C D a - C D b - D^2 c) d \geqslant 0\) by
% \eqref{Eq:Delta_mult} but
% \begin{multline*}
% C - C^2 - C D a - C D b =
% C ( 1 - C - D a - D b) \leqslant
% C ( DF - D a - D b ) =
% CD ( F - a - b) < 0
% \end{multline*}
% so there is a contradiction.
%
% The best way to overcome this appears to be to let go of the
% restriction that the upper left corner element in $\mathit{mult}$ and
% $\Delta$ is $1$. It may seem that rescaling shouldn't buy us
% anything if we anyway control all coefficients, but that isn't
% true; the $1$s in the identity map (and other permutations) and the
% $1$ of the nullary tensor product are unavoidable. Letting the
% upper left corner drop below $1$ may well hold the key to making
% $\Delta\otimes\Delta$ less than $\Delta$.
%
%
% \subsection{Second attempt}
%
% So, we add two variables $G$ and $g$ for what used to be $1$. We
% also list the variables in a different order to make $D$ and $d$
% show up next to each other.
% \begin{tcl}
interp alias {} Coeff_ring {} ::mtmtcl::rings::semigroup_algebra {
::mtmtcl::rings::integers::all
} {
::mtmtcl::groups::power_product_monoid_1.3 {
::mtmtcl::sets::variable_set {A a B b C c D d E e F f G g} {}
}
}
interp alias {} PROP {} mtmtcl::matprop::homfdpower 2 Coeff_ring
% \end{tcl}
% Another difference is to use the |fuse| method when constructing
% terms to compare.
%
% \begin{proc}{fusediff}
% This procedure takes two arguments, which have the form of |fuse|
% argument sequences except that the ``factors'' are indices into
% the |PROPgen| array of the actual factors. These are then
% |fuse|d, and the second is subtracted from the first, after which
% the difference is returned.
% \begin{tcl}
proc fusediff {term1 term2} {
variable PROPgen
set call1 [list PROP fuse {*}[lrange $term1 0 1]]
foreach {name h t} [lrange $term1 2 end] {
lappend call1 $PROPgen($name) $h $t
}
set call2 [list PROP fuse {*}[lrange $term2 0 1]]
foreach {name h t} [lrange $term2 2 end] {
lappend call2 $PROPgen($name) $h $t
}
PROP - [eval $call1] [eval $call2]
}
% \end{tcl}
% \end{proc}
%
% \begin{arrayvar}{PROPgen}
% Except for the $g$ and $G$ entries, the generator values are
% constructed as above.
% \begin{tcl}
set PROPgen(epsilon) [list [
list [Coeff_ring named E] [Coeff_ring named F]
]]
set PROPgen(Delta) [
list [
list [Coeff_ring named G] [Coeff_ring named A]
] [
list [Coeff_ring 0] [Coeff_ring named B]
] [
list [Coeff_ring 0] [Coeff_ring named C]
] [
list [Coeff_ring 0] [Coeff_ring named D]
]
]
set PROPgen(mu) [
list [
list [Coeff_ring named g] [Coeff_ring named a]\
[Coeff_ring named b] [Coeff_ring named c]
] [
list [Coeff_ring 0] [Coeff_ring 0] [Coeff_ring 0]\
[Coeff_ring named d]
]
]
set PROPgen(unit) [
list [
list [Coeff_ring named e]
] [
list [Coeff_ring named f]
]
]
% \end{tcl}
% \end{arrayvar}
%
% Now for the list of pairs to compare.
% \begin{tcl}
set diffL {}
lappend diffL [fusediff {o1 o1 mu {m1 m2} m1 unit {} m2} {e e}]
lappend diffL [fusediff {o1 o1 mu {m2 m1} m1 unit {} m2} {e e}]
lappend diffL [fusediff {
o1 o1 mu {m1 m2} m2 mu {i1 i2} {m1 i1 i2}
} {
o1 o1 mu {m1 m2} m1 mu {i1 i2} {i1 i2 m2}
}]
%
lappend diffL [fusediff {
o1 {o1 m2} Delta i1 {} epsilon m2 i1
} {e e}]
lappend diffL [fusediff {
o1 {m2 o1} Delta i1 {} epsilon m2 i1
} {e e}]
lappend diffL [fusediff {
{o1 o2 o3} {o2 o3} Delta m1 {o1 m1} Delta i1 i1
} {
{o1 o2 o3} {o1 o2} Delta m1 {m1 o3} Delta i1 i1
}]
%
lappend diffL [fusediff {
{} {} epsilon m1 m1 unit {} {}
} {"" ""}]
lappend diffL [fusediff {
{} {} epsilon m1 m1 mu {i1 i2} {i1 i2}
} {
{} {} epsilon i1 {} epsilon i2 {i1 i2}
}]
lappend diffL [fusediff {
{o1 o2} {o1 o2} Delta m1 m1 unit {} {}
} {
{o1 o2} o1 unit {} o2 unit {} {}
}]
lappend diffL [fusediff {
{o1 o2} {o1 o2} Delta m1 m1 mu {i1 i2} {i1 i2}
} {
{o1 o2} o1 mu {m0 m1} o2 mu {m2 m3}
{m0 m2} Delta i1 {m1 m3} Delta i2 {i1 i2}
}]
% \end{tcl}
%
% And finally, made code of this list.
% \begin{tcl}
set codeL {}
foreach item $diffL {
lappend codeL [mtmtcl::latex::convertto [PROP export $item {}]]
}
set code "\\begin{gather}\n [
join $codeL "\n \\\\\n "
]\n\\end{gather}"
% \end{tcl}
% And the result is:
% \begin{gather}
% \begin{pmatrix} e g + a f - 1 & b e + c f \\ 0 & d f - 1
% \end{pmatrix}
% \\
% \begin{pmatrix} e g + b f - 1 & a e + c f \\ 0 & d f - 1
% \end{pmatrix}
% \\
% \begin{pmatrix} 0 & 0 & a g - b g & a^2 - c g - a d & 0 & 0 &
% c g + b d - b^2 & a c - b c
% \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}
% \\
% \begin{pmatrix} E G - 1 & A E + C F \\ 0 & B E + D F - 1
% \end{pmatrix}
% \\
% \begin{pmatrix} E G - 1 & A E + B F \\ 0 & C E + D F - 1
% \end{pmatrix}
% \label{Eq2:epsilon_Delta}
% \\
% \begin{pmatrix} 0 & A C - A B \\ 0 & B G + A D - B^2 \\ 0 & 0 \\
% 0 & 0 \\ 0 & C^2 - C G - A D \\ 0 & C D - B D \\ 0 & 0 \\ 0 & 0
% \end{pmatrix}
% \\
% \begin{pmatrix} E e + F f - 1 \end{pmatrix}
% \\
% \begin{pmatrix} E g - E^2 & a E - E F & b E - E F & c E + d F - F^2
% \end{pmatrix}
% \label{Eq2:epsilon_mu}
% \\
% \begin{pmatrix} e G + A f - e^2 \\ B f - e f \\ C f - e f \\ D f - f^2
% \end{pmatrix}
% \\
% \hspace{-3cm}
% \begin{pmatrix}
% G g - G^2 g^2 &
% \parbox{0.25\linewidth}{\(
% a G - A G g^2 - a B G g - a C G g - a^2 D G
% \)}&
% \parbox{0.25\linewidth}{\(
% b G - A G g^2 - b C G g - B b G g - b^2 D G
% \)}&
% \parbox{0.4\linewidth}{\(
% c G + A d - A^2 g^2 - A b C g - A a B g - 2 a B b C - A B b g -
% A b^2 D - B^2 c g - B b c D - A a C g - C^2 c g - A a^2 D -
% a C c D - b C c D - a B c D - c^2 D^2
% \)\vadjust{\kern4pt}}
% \\ 0 & 0 & 0 & B d - B^2 d g - B b D d - a B D d - c D^2 d \\
% 0 & 0 & 0 & C d - C^2 d g - a C D d - b C D d - c D^2 d \\
% 0 & 0 & 0 & D d - D^2 d^2 \end{pmatrix}
% \label{Eq2:Delta_mu}
% \end{gather}
% Unfortunately, this still runs into the same problem\Ldash which
% amazingly depends on only three PROP inequalities: \(\mathrm{id}
% \otimes \varepsilon \circ \Delta \geqslant \mathrm{id}\)
% \eqref{Eq2:epsilon_Delta} for \(EG \geqslant 1\) and \(1-CE
% \leqslant DF\), \(\varepsilon \circ \mu \geqslant \varepsilon
% \otimes \varepsilon\) \eqref{Eq2:epsilon_mu} for \(E (g - E)
% \geqslant 0\), \(E(a-F) \geqslant 0\), and \(E(b-F) \geqslant 0\),
% and finally \(\Delta \circ \mu \geqslant \mu\otimes\mu \circ (1\,2)
% \circ \Delta\otimes\Delta\) \eqref{Eq2:Delta_mu} for \(G g \geqslant
% G^2 g^2\) and \(C d - C^2 d g - a C D d - b C D d - c D^2 d
% \geqslant 0\).
%
% First, \(1 \geqslant Gg \geqslant GE \geqslant 1\) so \(E=g\).
% Second, \(-a \leqslant -F\) and \(-b \leqslant -F\). Finally,
% \(C - C^2 g - a C D - b C D - c D^2 \geqslant 0\) so \(cD^2
% \leqslant C (1 - Cg - aD - bD) \leqslant C (1 - CE - FD - FD)
% \leqslant -CFD\). This contradicts the positivity of these
% variables, which is needed for preserving strict compatibility of
% the matrix order with matrix multiplication. A similar
% contradiction can be constructed with the dual axiom inequalities.
%
%
% \subsection{Third attempt}
%
% At this point, all hope might seem to be lost; although the
% possibilities have not been exhausted, the complexity of the
% inequality systems that need to be solved (and the computational
% complexity of computing the value of realistic networks) are
% overwhelming. However, late night inspiration struck and suggested
% a third PROP\Ldash quickly dubbed `the biaffine PROP'\Dash might be
% of use in this situation. The definition of this PROP is a
% restatement of the old path-counting order for networks, and it
% combines the complexity of the trivial matrix PROP with the
% $\hom$-PROP's capability to sensibly deal with nullary and
% nullcoary elements.
%
% \begin{tcl}
interp alias {} PROP {}\
mtmtcl::matprop::biaffine ::mtmtcl::rings::integers::all
if {[catch {
lrepeat 0 0
}]} then {
rename lrepeat _lrepeat
proc lrepeat {num args} {
if {$num>0 && [llength $args]} then {_lrepeat $num {*}$args}
}
}
% \end{tcl}
% As in the previous attempt, the |fuse| method it used for constructing
% terms to compare. The defintion of |fusediff| need not be repeated,
% as it is the same as above.
%
% \begin{arrayvar}{PROPgen}
% The array of generator values is however quite different.
% \begin{tcl}
set PROPgen(epsilon) [
PROP fromparts 0 {} 1 {}
]
set PROPgen(Delta) [
PROP fromparts 0 {0 0} 1 {1 1}
]
% \end{tcl}
% The first |1| in that definition represents the great innovation
% in this implementation round: a coproduct counts as being the
% endpoint for one path, as well as letting two paths through. This
% turns out to slant a surprising number of inequalities the right
% way.
% \begin{tcl}
set PROPgen(m) [
PROP fromparts 0 1 {0 0} {{1 1}}
]
set PROPgen(unit) [
PROP fromparts 0 1 {} {}
]
set PROPgen(S) [
PROP fromparts 0 0 0 2
]
% \end{tcl}
% Giving the antipode a weight greater than $1$ is another
% innovation. When combined with the end-point in a coproduct or
% begin-point in a product, it actually forces the antipode past
% the product or coproduct, even though this increases the total
% number of antipodes.
% \begin{tcl}
set PROPgen(twist) [PROP permutation {1 0}]
% \end{tcl}
% It does not current seem like twists can be penalised at this
% level, so they are just treated as simple permutations.
% \end{arrayvar}
%
% Now for the list of pairs to compare.
% \begin{tcl}
set diffL {}
lappend diffL [fusediff {o1 o1 m {m1 m2} m1 unit {} m2} {e e}]
lappend diffL [fusediff {o1 o1 m {m2 m1} m1 unit {} m2} {e e}]
lappend diffL [fusediff {
o1 o1 m {m1 m2} m2 m {i1 i2} {m1 i1 i2}
} {
o1 o1 m {m1 m2} m1 m {i1 i2} {i1 i2 m2}
}]
%
lappend diffL [fusediff {
o1 {o1 m2} Delta i1 {} epsilon m2 i1
} {e e}]
lappend diffL [fusediff {
o1 {m2 o1} Delta i1 {} epsilon m2 i1
} {e e}]
lappend diffL [fusediff {
{o1 o2 o3} {o2 o3} Delta m1 {o1 m1} Delta i1 i1
} {
{o1 o2 o3} {o1 o2} Delta m1 {m1 o3} Delta i1 i1
}]
%
lappend diffL [fusediff {
{} {} epsilon m1 m1 unit {} {}
} {"" ""}]
lappend diffL [fusediff {
{} {} epsilon m1 m1 m {i1 i2} {i1 i2}
} {
{} {} epsilon i1 {} epsilon i2 {i1 i2}
}]
lappend diffL [fusediff {
{o1 o2} {o1 o2} Delta m1 m1 unit {} {}
} {
{o1 o2} o1 unit {} o2 unit {} {}
}]
lappend diffL [fusediff {
{o1 o2} {o1 o2} Delta m1 m1 m {i1 i2} {i1 i2}
} {
{o1 o2} o1 m {m0 m1} o2 m {m2 m3}
{m0 m2} Delta i1 {m1 m3} Delta i2 {i1 i2}
}]
%
lappend diffL [fusediff {
o o m {1 2} 2 S 3 {1 3} Delta i i
} {
o {} epsilon i o unit {} i
}]
lappend diffL [fusediff {
o o m {1 2} 1 S 3 {3 2} Delta i i
} {
o {} epsilon i o unit {} i
}]
% \end{tcl}
% All of these come out non-negative, and all but the
% (co)associativity rules come out with at least one positive
% element, which is the best that could be hoped for. Hence this
% order is a success as far as the axioms go. Time to test it on the
% total list of derived rules and equalities!
%
% \begin{proc}{rulediff}
% This procedure has the call syntax
% \begin{displaysyntax}
% rulediff \word{map} \word{rule}
% \end{displaysyntax}
% where \word{rule} is a \textsf{cmplutil1} rule and \word{map} is
% a dictionary mapping vertex annotations to |PROP| elements. It
% |PROP fuse|s both sides of the \word{rule} and returns their
% difference (left hand side minus right hand side).
% \begin{tcl}
proc rulediff {map rule} {
set sides {}
foreach NW [list [lindex $rule 0 0] [lindex $rule 2 0]] {
set call [list PROP fuse]
lappend call [lindex $NW 0 0 2]
foreach v [lrange [lindex $NW 0] 2 end] {
lappend call [lindex $v 1] [dict get $map [lindex $v 0]]\
[lindex $v 2]
}
lappend call [lindex $NW 0 1 1]
lappend sides [eval $call]
}
PROP - {*}$sides
}
% \end{tcl}
% \end{proc}
%
% \begin{proc}{easysgn}
% This procedure is a hacky test of whether a difference is
% positive, zero, negative, or other (implying incomparability),
% where the ordering is simply element-wise. This is quicker to
% do with regexps than by properly looping over the elements.
% \begin{tcl}
proc easysgn {element} {
lindex {0 - + incomparable} [
expr {([string first - $element]>=0) +\
2*[regexp {(^|\{| )[1-9]} $element]}
]
}
% \end{tcl}
% \end{proc}
%
% At this point, one should e.g.~|source Hopf+twist2.dump-tcl;\
% llength $rulesL|. The |llength| prevents dumping a megabyte-size
% list in the console.
%
% \begin{tcl}
array unset Histogram
set mainMap [array get PROPgen]
foreach list {rulesL equalitiesL} {
set n -1; foreach rule [set $list] {incr n
lappend Histogram($list,[easysgn [rulediff $mainMap $rule]]) $n
}
}
% \end{tcl}
% Results are:
% \begin{itemize}
% \item
% $457$ rules were resolved as positive, $214$ are still neutral,
% none negative or incomparable.
% \item
% All $89$ equalities remained neutral.
% \end{itemize}
% That nothing incomparable is encountered is very promising!
%
% The next step is to impose a distinction also between left- and
% right-leaning combinations of the product and coproduct
% respectively. The following |leftMap| penalises right-branching by
% a factor $2$, but does as little as possible otherwise.
% \begin{tcl}
dict set leftMap epsilon [dict get $mainMap epsilon]
dict set leftMap unit [dict get $mainMap unit]
dict set leftMap Delta [
PROP fromparts 0 {0 0} 0 {1 2}
]
dict set leftMap m [
PROP fromparts 0 0 {0 0} {{1 2}}
]
dict set leftMap S [
PROP fromparts 0 0 0 1
]
dict set leftMap twist [PROP permutation {1 0}]
% \end{tcl}
%
% \begin{tcl}
foreach list {rulesL equalitiesL} {
set entry "Histogram($list,0)"
foreach n [set $entry] {
lappend Histogram_leftMap($entry,[easysgn [
rulediff $leftMap [lindex [set $list] $n]
]]) $n
}
}
% \end{tcl}
% Produces some incomparabilities, and at least the first few are in
% the open part. What if one only distinguishes left from right in
% the multiplication?
% \begin{tcl}
set leftMap2 $leftMap
dict set leftMap2 Delta [
PROP fromparts 0 {0 0} 0 {1 1}
]
foreach list {rulesL equalitiesL} {
set entry "Histogram($list,0)"
foreach n [set $entry] {
lappend Histogram_leftMap2($entry,[easysgn [
rulediff $leftMap2 [lindex [set $list] $n]
]]) $n
}
}
% \end{tcl}
%
% Hmm\dots\ it might be better to order by several maps in sequence.
%
% \begin{proc}{direct_rule}
% This procedure tries to direct a rule, by applying maps from a
% list until one is found which throws it off the neutral. The call
% syntax is
% \begin{displaysyntax}
% |direct_rule| \word{map-list} \word{rule}
% \end{displaysyntax}
% and the return value is a list
% \begin{displaysyntax}
% \word{index} \word{sign}
% \end{displaysyntax}
% where \word{index} is the index in the \word{map-list} of the
% last map tried, and \word{sign} is the |easysgn| of the
% corresponding |rulediff|.
% \begin{tcl}
proc direct_rule {mapL rule} {
set n -1; foreach map $mapL {incr n
set sgn [easysgn [rulediff $map $rule]]
if {$sgn != "0"} then {break}
}
return [list $n $sgn]
}
% \end{tcl}
% \end{proc}
%
% \begin{proc}{direct_rulelist}
% This procedure runs |direct_rule| on all rules in a list, and
% appends the rule indices to different entries in a histogram
% array depending on the return value from the |direct_rule| call.
% The syntax is
% \begin{displaysyntax}
% |direct_rulelist| \word{array-name} \word{prefix}
% \word{rule list} \word{map}\regstar
% \end{displaysyntax}
% where \word{prefix} is prepended to each rule index before it is
% appended to some array entry.
% \begin{tcl}
proc direct_rulelist {arrname prefix ruleL args} {
upvar 1 $arrname Hist
set n -1; foreach rule $ruleL {incr n
lappend Hist([join [direct_rule $args $rule] ,]) $prefix$n
}
}
% \end{tcl}
% \end{proc}
%
% \begin{tcl}
set vertexCountMap [dict create unit [
PROP fromparts 1 1 {} {}
] epsilon [
PROP fromparts 1 {} 1 {}
] Delta [
PROP fromparts 1 {0 0} 1 {{1} {1}}
] m [
PROP fromparts 1 {1} {0 0} {{1 1}}
] S [
PROP fromparts 1 0 0 2
] twist [
PROP fromparts 1 {0 0} {0 0} {{0 1} {1 0}}
]]
% \end{tcl}
%
% \begin{tcl}
set slashMap [dict create unit [
PROP fromparts 0 1 {} {}
] epsilon [
PROP fromparts 0 {} 1 {}
] Delta [
PROP fromparts 0 {0 0} 1 {{1} {1}}
] m [
PROP fromparts 0 {1} {0 0} {{1 1}}
] S [
PROP fromparts 0 0 0 2
] twist [
PROP fromparts 0 {0 0} {0 0} {{0 2} {1 0}}
]]
% \end{tcl}
%
% Roughly the |m| side of what was used to derive the
% \texttt{Hopf+twist2} data.
% \begin{tcl}
set mslashmMap [dict create unit [
PROP fromparts 0 0 {} {}
] epsilon [
PROP fromparts 0 {} 0 {}
] Delta [
PROP fromparts 0 {0 0} 0 {{0} {0}}
] m [
PROP fromparts 0 {1} {0 0} {{0 1}}
] S [
PROP fromparts 0 0 0 0
] twist [
PROP fromparts 0 {0 0} {0 0} {{0 1} {0 0}}
]]
% \end{tcl}
%
% Impose left-associativity using an end-entry.
% \begin{tcl}
set mleft2Map $mainMap
dict set mleft2Map m [
PROP fromparts 0 {1} {0 1} {{1 1}}
]
% \end{tcl}
%
% Slashify twist with end- and begin-points.
% \begin{tcl}
set slashMap2 $mainMap
dict set slashMap2 twist [
PROP fromparts 0 {0 1} {0 1} {{0 1} {1 0}}
]
% \end{tcl}
% Not too bad, but causes some incompatibilities.
%
% Symmetrically put endpoints at all corners of a twist.
% \begin{tcl}
set twistMap4 $mainMap
dict set twistMap4 twist [
PROP fromparts 0 {1 1} {1 1} {{0 1} {1 0}}
]
% \end{tcl}
% Causes \emph{lots} of incompatibilities.
%
% Connect every twist input with every twist output.
% \begin{tcl}
set zhaMap $mainMap
dict set zhaMap twist [
PROP fromparts 0 {0 0} {0 0} {{1 1} {1 1}}
]
% \end{tcl}
% Doesn't resolve a single rule, but makes almost every one that
% wasn't already resolved incompatible.
%
% Penalise passage through a twist like passage through an antipode.
% \begin{tcl}
set twist2Map $mainMap
dict set twist2Map twist [
PROP fromparts 0 {0 0} {0 0} {{0 2} {2 0}}
]
% \end{tcl}
% Doesn't resolve a single rule, but makes almost every one that
% wasn't already resolved incompatible. This is disappointing, since
% it is very natural in the context of a super-Hopf algebra (a
% projection onto the odd part should be larger than the identity).
%
% In conclusion, it is easy to invent ways of directing rules, but
% apparently hard to find anything that doesn't cause
% incomparabilities. In order to advance on that front, one might
% return to the previous approach of indeterminate coefficients and
% see how things work out for selected rules.
%
% \subsubsection{With indeterminate experience}
%
% The experiments below with indeterminates for matrix elements
% suggest that it is hard to favour the left-leaning sides in the
% associativity and coassociativity rules; very often the asymmetry
% that should have this effect would leave other rules with more than
% one leading term, so one is forced back to the symmteric situation
% for multiplication and comultiplication. Interestingly enough the
% same does not seem to hold for the |twist|, as its right hand side
% can be penalised with few or no problems. There are some
% indications that \(T_{2,2} \geqslant T_{0,2},T_{2,0}\), but that's