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<div class="section" id="sat-solvers">
<h1>SAT Solvers<a class="headerlink" href="#sat-solvers" title="Permalink to this headline">¶</a></h1>
<div class="section" id="overview">
<h2>Overview<a class="headerlink" href="#overview" title="Permalink to this headline">¶</a></h2>
<p>A <a class="reference external" href="https://en.wikipedia.org/wiki/Boolean_satisfiability_problem">SAT Solver</a>
is tool for finding solutions to problems that can be specified using
<a class="reference external" href="https://en.wikipedia.org/wiki/Propositional_calculus">propositional logic formulas</a>.</p>
<p>Here’s an <a class="reference external" href="http://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130A_examples.htm">example from Delmar E. Searls</a>.</p>
<p>Start with a problem statement, “If the tire is flat then I will have to remove it and take it to the gas
station.”</p>
<p>Translate each basic statement to a variable so that <em>P</em> means “the tire
is flat”, <em>Q</em> means “I have to remove the tire”, and <em>R</em> means “I have
to take the tire to the gas station.</p>
<p>Next, write the statement in symbolic form: P → (Q ∧ R) and make
a truth table that shows when the statements is satisfied:</p>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="12%" />
<col width="12%" />
<col width="12%" />
<col width="65%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">P</th>
<th class="head">Q</th>
<th class="head">R</th>
<th class="head">P → (Q ∧ R)</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td>T</td>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr class="row-odd"><td>T</td>
<td>T</td>
<td>F</td>
<td>F</td>
</tr>
<tr class="row-even"><td>T</td>
<td>F</td>
<td>T</td>
<td>F</td>
</tr>
<tr class="row-odd"><td>T</td>
<td>F</td>
<td>F</td>
<td>F</td>
</tr>
<tr class="row-even"><td>F</td>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr class="row-odd"><td>F</td>
<td>T</td>
<td>F</td>
<td>T</td>
</tr>
<tr class="row-even"><td>F</td>
<td>F</td>
<td>T</td>
<td>T</td>
</tr>
<tr class="row-odd"><td>F</td>
<td>F</td>
<td>F</td>
<td>T</td>
</tr>
</tbody>
</table>
</div></blockquote>
<p>Since there is at least one row that evaluates to <em>True</em>,
we say the expression is <em>satisfiable</em>.</p>
<p>What a SAT solver does is determine whether an expression
is <em>satisfiable</em> and will generate one, some, or all of
the rows that evaluate to true.</p>
</div>
<div class="section" id="why-do-we-care">
<h2>Why Do We Care?<a class="headerlink" href="#why-do-we-care" title="Permalink to this headline">¶</a></h2>
<p>There are many otherwise difficult problems that can be
expressed in propositional logic and computers have gotten
good at solving them.</p>
<p>While the above example was simple, the problem grows
exponentially with the number of variables. The amazing
fact of the early 21st century is that a number of
heuristics have been found that make the problem
tractable for many practical examples, even ones with
tens of thousands of variables.</p>
<p>One example is finding a layout to wire a printed
circuit board in a way that doesn’t generate electrical
interference.</p>
</div>
<div class="section" id="tools">
<h2>Tools<a class="headerlink" href="#tools" title="Permalink to this headline">¶</a></h2>
<p>A tool that is available to Pythonistas is
<a class="reference external" href="https://pypi.org/project/pycosat/">pycosat</a> which is
itself a front-end for a <a class="reference external" href="http://fmv.jku.at/picosat/">PicoSAT</a>
which is a popular SAT solver written by Armin Biere in pure C.</p>
<p>The first step is to rewrite the expression into
<a class="reference external" href="https://en.wikipedia.org/wiki/Conjunctive_normal_form">Conjunctive Normal Form (CNF)</a>
sometimes called a “product of sums”:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>(¬P ∨ Q) ∧ (¬P ∨ R)
</pre></div>
</div>
<p>In <em>pycosat</em>, <code class="docutils literal notranslate"><span class="pre">P</span></code> is written as <code class="docutils literal notranslate"><span class="pre">1</span></code> because it is the first
variable, and <code class="docutils literal notranslate"><span class="pre">¬P</span></code> is written as <code class="docutils literal notranslate"><span class="pre">-1</span></code> using a negative
value to indicate negation. We pass in a list of tuples with
each disjunction was a separate tuple:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">pprint</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">pycosat</span><span class="o">.</span><span class="n">itersolve</span><span class="p">([(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)])),</span> <span class="n">width</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="go">[[-1, 2, 3],</span>
<span class="go"> [-1, 2, -3],</span>
<span class="go"> [-1, -2, 3],</span>
<span class="go"> [-1, -2, -3],</span>
<span class="go"> [1, 2, 3]]</span>
</pre></div>
</div>
</div>
<div class="section" id="the-core-challenge">
<h2>The Core Challenge<a class="headerlink" href="#the-core-challenge" title="Permalink to this headline">¶</a></h2>
<p>With a powerful SAT solver at you disposal, the remaining challenge
is to express your problem in a way that <em>pycosat</em> can understand.</p>
<p>I wrote some utility functions that can help. For starters, we
want to write our expressions with symbols instead of numbers
and with <code class="docutils literal notranslate"><span class="pre">~</span></code> for negation instead of a minus sign:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">translate</span><span class="p">([[</span><span class="s1">'~P'</span><span class="p">,</span> <span class="s1">'Q'</span><span class="p">],[</span><span class="s1">'~P'</span><span class="p">,</span> <span class="s1">'R'</span><span class="p">]])[</span><span class="mi">0</span><span class="p">]</span>
<span class="go">[(-1, 2), (-1, 3)]</span>
</pre></div>
</div>
<p>Often, it is easier to express constraints in
<a class="reference external" href="https://en.wikipedia.org/wiki/Disjunctive_normal_form">Disjunctive normal form</a></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>¬P ∨ (Q ∧ R)
</pre></div>
</div>
<p>and then convert to the CNF required by <em>pycosat</em>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">from_dnf</span><span class="p">([[</span><span class="s1">'~P'</span><span class="p">],</span> <span class="p">[</span><span class="s1">'Q'</span><span class="p">,</span> <span class="s1">'R'</span><span class="p">]])</span>
<span class="go">[('~P', 'Q'), ('~P', 'R')]</span>
</pre></div>
</div>
<p>Lastly, there is function, <em>solve_all</em>, that takes care of
the round-trip between human readable and <em>pycosat</em> readable:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">cnf</span> <span class="o">=</span> <span class="n">from_dnf</span><span class="p">([[</span><span class="s1">'~P'</span><span class="p">],</span> <span class="p">[</span><span class="s1">'Q'</span><span class="p">,</span> <span class="s1">'R'</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">solve_all</span><span class="p">(</span><span class="n">cnf</span><span class="p">,</span> <span class="kc">True</span><span class="p">)</span>
<span class="go">[['~P', 'Q', 'R'],</span>
<span class="go"> ['~P', 'Q', '~R'],</span>
<span class="go"> ['~P', '~Q', 'R'],</span>
<span class="go"> ['~P', '~Q', '~R'],</span>
<span class="go"> ['P', 'Q', 'R']]</span>
</pre></div>
</div>
<p>It’s even more readable with humanistic variable names:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">cnf</span> <span class="o">=</span> <span class="n">from_dnf</span><span class="p">([[</span><span class="s1">'~FlatTire'</span><span class="p">],</span> <span class="p">[</span><span class="s1">'NeedToRemove'</span><span class="p">,</span> <span class="s1">'GotoGasStation'</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">solve_all</span><span class="p">(</span><span class="n">cnf</span><span class="p">,</span> <span class="kc">True</span><span class="p">)</span>
<span class="go">[['~FlatTire', 'NeedToRemove', 'GotoGasStation'],</span>
<span class="go"> ['~FlatTire', 'NeedToRemove', '~GotoGasStation'],</span>
<span class="go"> ['~FlatTire', '~NeedToRemove', 'GotoGasStation'],</span>
<span class="go"> ['~FlatTire', '~NeedToRemove', '~GotoGasStation'],</span>
<span class="go"> ['FlatTire', 'NeedToRemove', 'GotoGasStation']]</span>
</pre></div>
</div>
</div>
<div class="section" id="higher-level-convenience-functions">
<h2>Higher Level Convenience Functions<a class="headerlink" href="#higher-level-convenience-functions" title="Permalink to this headline">¶</a></h2>
<p>DNFs can be easy to reason about but are sometimes tedious to write-out.</p>
<p>I have some convenience functions that build a CNF directly
from constraints expressed with quantifiers:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Express that at least one of A, B, and C are true</span>
<span class="o">>>></span> <span class="n">some_of</span><span class="p">([</span><span class="s1">'A'</span><span class="p">,</span> <span class="s1">'B'</span><span class="p">,</span> <span class="s1">'C'</span><span class="p">])</span>
<span class="p">[[</span><span class="s1">'A'</span><span class="p">,</span> <span class="s1">'B'</span><span class="p">,</span> <span class="s1">'C'</span><span class="p">]]</span>
<span class="c1"># Express that at exactly one of A, B, and C is true</span>
<span class="o">>>></span> <span class="n">one_of</span><span class="p">([</span><span class="s1">'A'</span><span class="p">,</span> <span class="s1">'B'</span><span class="p">,</span> <span class="s1">'C'</span><span class="p">])</span>
<span class="p">[(</span><span class="s1">'A'</span><span class="p">,</span> <span class="s1">'B'</span><span class="p">,</span> <span class="s1">'C'</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'~A'</span><span class="p">,</span> <span class="s1">'~B'</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'~A'</span><span class="p">,</span> <span class="s1">'~C'</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'~B'</span><span class="p">,</span> <span class="s1">'~C'</span><span class="p">)]</span>
<span class="c1"># Express that no more than one of A, B, and C are true</span>
<span class="o">>>></span> <span class="n">Q</span><span class="p">([</span><span class="s1">'A'</span><span class="p">,</span> <span class="s1">'B'</span><span class="p">,</span> <span class="s1">'C'</span><span class="p">])</span> <span class="o"><=</span> <span class="mi">1</span>
<span class="p">[(</span><span class="s1">'~A'</span><span class="p">,</span> <span class="s1">'~B'</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'~A'</span><span class="p">,</span> <span class="s1">'~C'</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'~B'</span><span class="p">,</span> <span class="s1">'~C'</span><span class="p">)]</span>
</pre></div>
</div>
<p>With these tools as primitives, it is easier
to write higher level functions to express problem
constraints in conjunctive normal form.</p>
</div>
<div class="section" id="sudoku-puzzles">
<h2>Sudoku Puzzles<a class="headerlink" href="#sudoku-puzzles" title="Permalink to this headline">¶</a></h2>
<p>Given a puzzle in this form:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">8</span> <span class="mi">6</span><span class="o">|</span><span class="mi">4</span> <span class="mi">3</span><span class="o">|</span>
<span class="mi">5</span> <span class="o">|</span> <span class="o">|</span> <span class="mi">7</span>
<span class="o">|</span> <span class="mi">2</span> <span class="o">|</span>
<span class="o">---+---+---</span>
<span class="mi">32</span> <span class="o">|</span> <span class="mi">8</span> <span class="o">|</span> <span class="mi">5</span>
<span class="mi">8</span><span class="o">|</span> <span class="mi">5</span> <span class="o">|</span><span class="mi">4</span>
<span class="mi">1</span> <span class="o">|</span> <span class="mi">7</span> <span class="o">|</span> <span class="mi">93</span>
<span class="o">---+---+---</span>
<span class="o">|</span> <span class="mi">4</span> <span class="o">|</span>
<span class="mi">9</span> <span class="o">|</span> <span class="o">|</span> <span class="mi">4</span>
<span class="o">|</span><span class="mi">6</span> <span class="mi">7</span><span class="o">|</span><span class="mi">2</span> <span class="mi">8</span>
</pre></div>
</div>
<p>We enter it in Python as a single string row major order:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'8 64 3 5 7 2 '</span>
<span class="s1">'32 8 5 8 5 4 1 7 93'</span>
<span class="s1">' 4 9 4 6 72 8'</span><span class="p">)</span>
</pre></div>
</div>
<p>And expect the computer to solve it:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">876</span><span class="o">|</span><span class="mi">413</span><span class="o">|</span><span class="mi">529</span>
<span class="mi">452</span><span class="o">|</span><span class="mi">968</span><span class="o">|</span><span class="mi">371</span>
<span class="mi">931</span><span class="o">|</span><span class="mi">725</span><span class="o">|</span><span class="mi">684</span>
<span class="o">---+---+---</span>
<span class="mi">329</span><span class="o">|</span><span class="mi">184</span><span class="o">|</span><span class="mi">756</span>
<span class="mi">768</span><span class="o">|</span><span class="mi">359</span><span class="o">|</span><span class="mi">412</span>
<span class="mi">145</span><span class="o">|</span><span class="mi">276</span><span class="o">|</span><span class="mi">893</span>
<span class="o">---+---+---</span>
<span class="mi">283</span><span class="o">|</span><span class="mi">541</span><span class="o">|</span><span class="mi">967</span>
<span class="mi">697</span><span class="o">|</span><span class="mi">832</span><span class="o">|</span><span class="mi">145</span>
<span class="mi">514</span><span class="o">|</span><span class="mi">697</span><span class="o">|</span><span class="mi">238</span>
</pre></div>
</div>
</div>
<div class="section" id="code-for-the-sudoku-solver">
<h2>Code for the Sudoku Solver<a class="headerlink" href="#code-for-the-sudoku-solver" title="Permalink to this headline">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sat_utils</span> <span class="k">import</span> <span class="n">solve_one</span><span class="p">,</span> <span class="n">one_of</span><span class="p">,</span> <span class="n">basic_fact</span>
<span class="kn">from</span> <span class="nn">sys</span> <span class="k">import</span> <span class="n">intern</span>
<span class="kn">from</span> <span class="nn">pprint</span> <span class="k">import</span> <span class="n">pprint</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">grid</span> <span class="o">=</span> <span class="s1">'''</span><span class="se">\</span>
<span class="s1">AA AB AC BA BB BC CA CB CC</span>
<span class="s1">AD AE AF BD BE BF CD CE CF</span>
<span class="s1">AG AH AI BG BH BI CG CH CI</span>
<span class="s1">DA DB DC EA EB EC FA FB FC</span>
<span class="s1">DD DE DF ED EE EF FD FE FF</span>
<span class="s1">DG DH DI EG EH EI FG FH FI</span>
<span class="s1">GA GB GC HA HB HC IA IB IC</span>
<span class="s1">GD GE GF HD HE HF ID IE IF</span>
<span class="s1">GG GH GI HG HH HI IG IH II</span>
<span class="s1">'''</span>
<span class="n">values</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="s1">'123456789'</span><span class="p">)</span>
<span class="n">table</span> <span class="o">=</span> <span class="p">[</span><span class="n">row</span><span class="o">.</span><span class="n">split</span><span class="p">()</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">grid</span><span class="o">.</span><span class="n">splitlines</span><span class="p">()]</span>
<span class="n">points</span> <span class="o">=</span> <span class="n">grid</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
<span class="n">subsquares</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
<span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">points</span><span class="p">:</span>
<span class="n">subsquares</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">point</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
<span class="c1"># Groups: rows + columns + subsquares </span>
<span class="n">groups</span> <span class="o">=</span> <span class="n">table</span><span class="p">[:]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">table</span><span class="p">))</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">subsquares</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="k">del</span> <span class="n">grid</span><span class="p">,</span> <span class="n">subsquares</span><span class="p">,</span> <span class="n">table</span> <span class="c1"># analysis requires only: points, values, groups</span>
<span class="k">def</span> <span class="nf">comb</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
<span class="s1">'Format a fact (a value assigned to a given point)'</span>
<span class="k">return</span> <span class="n">intern</span><span class="p">(</span><span class="n">f</span><span class="s1">'</span><span class="si">{point}</span><span class="s1"> </span><span class="si">{value}</span><span class="s1">'</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">str_to_facts</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
<span class="s1">'Convert str in row major form to a list of facts'</span>
<span class="k">return</span> <span class="p">[</span><span class="n">comb</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span> <span class="k">for</span> <span class="n">point</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">if</span> <span class="n">value</span> <span class="o">!=</span> <span class="s1">' '</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">facts_to_str</span><span class="p">(</span><span class="n">facts</span><span class="p">):</span>
<span class="s1">'Convert a list of facts to a string in row major order with blanks for unknowns'</span>
<span class="n">point_to_value</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">str</span><span class="o">.</span><span class="n">split</span><span class="p">,</span> <span class="n">facts</span><span class="p">))</span>
<span class="k">return</span> <span class="s1">''</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">point_to_value</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="s1">' '</span><span class="p">)</span> <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">points</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">show</span><span class="p">(</span><span class="n">flatline</span><span class="p">):</span>
<span class="s1">'Display grid from a string (values in row major order with blanks for unknowns)'</span>
<span class="n">fmt</span> <span class="o">=</span> <span class="s1">'|'</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s1">'</span><span class="si">%s</span><span class="s1">'</span> <span class="o">*</span> <span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span>
<span class="n">sep</span> <span class="o">=</span> <span class="s1">'+'</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s1">'-'</span> <span class="o">*</span> <span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="n">offset</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">*</span> <span class="n">n</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">*</span> <span class="n">n</span><span class="o">**</span><span class="mi">2</span>
<span class="nb">print</span><span class="p">(</span><span class="n">fmt</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">flatline</span><span class="p">[</span><span class="n">offset</span><span class="p">:</span><span class="n">offset</span><span class="o">+</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">]))</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sep</span><span class="p">)</span>
<span class="k">for</span> <span class="n">given</span> <span class="ow">in</span> <span class="p">[</span>
<span class="s1">'53 7 6 195 98 6 8 6 34 8 3 17 2 6 6 28 419 5 8 79'</span><span class="p">,</span>
<span class="s1">' 75 4 5 8 17 6 36 2 7 1 5 1 1 5 8 96 1 82 3 4 9 48 '</span><span class="p">,</span>
<span class="s1">' 9 7 4 1 6 2 8 1 43 6 59 1 3 97 8 52 7 6 8 4 7 5 8 2 '</span><span class="p">,</span>
<span class="s1">'67 38 921 85 736 1 8 4 7 5 1 8 4 2 6 8 5 175 24 321 61 84'</span><span class="p">,</span>
<span class="s1">'27 15 8 3 7 4 7 5 1 7 9 2 6 2 5 8 6 5 4 8 59 41'</span><span class="p">,</span>
<span class="s1">'8 64 3 5 7 2 32 8 5 8 5 4 1 7 93 4 9 4 6 72 8'</span><span class="p">,</span>
<span class="p">]:</span>
<span class="n">cnf</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># each point assigned exactly one value</span>
<span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">points</span><span class="p">:</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">one_of</span><span class="p">(</span><span class="n">comb</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span> <span class="k">for</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">values</span><span class="p">)</span>
<span class="c1"># each value gets assigned to exactly one point in each group</span>
<span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
<span class="k">for</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">values</span><span class="p">:</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">one_of</span><span class="p">(</span><span class="n">comb</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span> <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">group</span><span class="p">)</span>
<span class="c1"># add facts for known values in a specific puzzle</span>
<span class="k">for</span> <span class="n">known</span> <span class="ow">in</span> <span class="n">str_to_facts</span><span class="p">(</span><span class="n">given</span><span class="p">):</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">basic_fact</span><span class="p">(</span><span class="n">known</span><span class="p">)</span>
<span class="c1"># solve it and display the results</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">facts_to_str</span><span class="p">(</span><span class="n">solve_one</span><span class="p">(</span><span class="n">cnf</span><span class="p">))</span>
<span class="n">show</span><span class="p">(</span><span class="n">given</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="n">show</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'=-'</span> <span class="o">*</span> <span class="mi">20</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="einstein-puzzle">
<h2>Einstein Puzzle<a class="headerlink" href="#einstein-puzzle" title="Permalink to this headline">¶</a></h2>
<p>Reportedly this puzzle is from Albert Einstein who is said
to have remarked that fewer than 2% of the population can
solve this puzzle (this is lore, neither fact is true).</p>
<div class="section" id="entities">
<h3>Entities<a class="headerlink" href="#entities" title="Permalink to this headline">¶</a></h3>
<ul class="simple">
<li>There are five houses in unique colors: Blue, green, red, white and yellow.</li>
<li>In each house lives a person of unique nationality: British, Danish, German, Norwegian and Swedish.</li>
<li>Each person drinks a unique beverage: Beer, coffee, milk, tea and water.</li>
<li>Each person smokes a unique cigar brand: Blue Master, Dunhill, Pall Mall, Prince and blend.</li>
<li>Each person keeps a unique pet: Cats, birds, dogs, fish and horses.</li>
</ul>
</div>
<div class="section" id="constraints">
<h3>Constraints<a class="headerlink" href="#constraints" title="Permalink to this headline">¶</a></h3>
<ul class="simple">
<li>The Brit lives in a red house.</li>
<li>The Swede keeps dogs as pets.</li>
<li>The Dane drinks tea.</li>
<li>The green house is on the left of the white, next to it.</li>
<li>The green house owner drinks coffee.</li>
<li>The person who smokes Pall Mall rears birds.</li>
<li>The owner of the yellow house smokes Dunhill.</li>
<li>The man living in the house right in the center drinks milk.</li>
<li>The Norwegian lives in the first house.</li>
<li>The man who smokes blend lives next to the one who keeps cats.</li>
<li>The man who keeps horses lives next to the man who smokes Dunhill.</li>
<li>The owner who smokes Blue Master drinks beer.</li>
<li>The German smokes Prince.</li>
<li>The Norwegian lives next to the blue house.</li>
<li>The man who smokes blend has a neighbor who drinks water.</li>
</ul>
</div>
<div class="section" id="goal">
<h3>Goal<a class="headerlink" href="#goal" title="Permalink to this headline">¶</a></h3>
<p>The question you need to answer is: “Who keeps fish?”</p>
</div>
</div>
<div class="section" id="solution-techniques">
<h2>Solution Techniques<a class="headerlink" href="#solution-techniques" title="Permalink to this headline">¶</a></h2>
<p>Christian Stigen Larsen has a
<a class="reference external" href="https://csl.name/post/einsteins-puzzle/">nice write-up</a>
on how to solve this problem by hand</p>
<p>Instead, we’ll use a computer to determine all of the relationships:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="s1">'1 norwegian'</span><span class="p">,</span> <span class="s1">'1 yellow'</span><span class="p">,</span> <span class="s1">'1 cat'</span><span class="p">,</span> <span class="s1">'1 water'</span><span class="p">,</span> <span class="s1">'1 dunhill'</span><span class="p">,</span>
<span class="s1">'2 dane'</span><span class="p">,</span> <span class="s1">'2 blue'</span><span class="p">,</span> <span class="s1">'2 horse'</span><span class="p">,</span> <span class="s1">'2 tea'</span><span class="p">,</span> <span class="s1">'2 blends'</span><span class="p">,</span>
<span class="s1">'3 brit'</span><span class="p">,</span> <span class="s1">'3 red'</span><span class="p">,</span> <span class="s1">'3 bird'</span><span class="p">,</span> <span class="s1">'3 milk'</span><span class="p">,</span> <span class="s1">'3 pall mall'</span><span class="p">,</span>
<span class="s1">'4 german'</span><span class="p">,</span> <span class="s1">'4 green'</span><span class="p">,</span> <span class="s1">'4 fish'</span><span class="p">,</span> <span class="s1">'4 coffee'</span><span class="p">,</span> <span class="s1">'4 prince'</span><span class="p">,</span>
<span class="s1">'5 swede'</span><span class="p">,</span> <span class="s1">'5 white'</span><span class="p">,</span> <span class="s1">'5 dog'</span><span class="p">,</span> <span class="s1">'5 root beer'</span><span class="p">,</span> <span class="s1">'5 blue master'</span><span class="p">]</span>
</pre></div>
</div>
<p>The <em>fish</em> is in <em>green</em> house owned by the <em>german</em> who drinks <em>coffee</em>
and smokes <em>Prince</em>.</p>
</div>
<div class="section" id="code-for-the-einstein-puzzle">
<h2>Code for the Einstein Puzzle<a class="headerlink" href="#code-for-the-einstein-puzzle" title="Permalink to this headline">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sat_utils</span> <span class="k">import</span> <span class="n">solve_one</span><span class="p">,</span> <span class="n">from_dnf</span><span class="p">,</span> <span class="n">one_of</span>
<span class="kn">from</span> <span class="nn">sys</span> <span class="k">import</span> <span class="n">intern</span>
<span class="kn">from</span> <span class="nn">pprint</span> <span class="k">import</span> <span class="n">pprint</span>
<span class="n">houses</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'1'</span><span class="p">,</span> <span class="s1">'2'</span><span class="p">,</span> <span class="s1">'3'</span><span class="p">,</span> <span class="s1">'4'</span><span class="p">,</span> <span class="s1">'5'</span> <span class="p">]</span>
<span class="n">groups</span> <span class="o">=</span> <span class="p">[</span>
<span class="p">[</span><span class="s1">'dane'</span><span class="p">,</span> <span class="s1">'brit'</span><span class="p">,</span> <span class="s1">'swede'</span><span class="p">,</span> <span class="s1">'norwegian'</span><span class="p">,</span> <span class="s1">'german'</span> <span class="p">],</span>
<span class="p">[</span><span class="s1">'yellow'</span><span class="p">,</span> <span class="s1">'red'</span><span class="p">,</span> <span class="s1">'white'</span><span class="p">,</span> <span class="s1">'green'</span><span class="p">,</span> <span class="s1">'blue'</span> <span class="p">],</span>
<span class="p">[</span><span class="s1">'horse'</span><span class="p">,</span> <span class="s1">'cat'</span><span class="p">,</span> <span class="s1">'bird'</span><span class="p">,</span> <span class="s1">'fish'</span><span class="p">,</span> <span class="s1">'dog'</span> <span class="p">],</span>
<span class="p">[</span><span class="s1">'water'</span><span class="p">,</span> <span class="s1">'tea'</span><span class="p">,</span> <span class="s1">'milk'</span><span class="p">,</span> <span class="s1">'coffee'</span><span class="p">,</span> <span class="s1">'root beer'</span><span class="p">],</span>
<span class="p">[</span><span class="s1">'pall mall'</span><span class="p">,</span> <span class="s1">'prince'</span><span class="p">,</span> <span class="s1">'blue master'</span><span class="p">,</span> <span class="s1">'dunhill'</span><span class="p">,</span> <span class="s1">'blends'</span> <span class="p">],</span>
<span class="p">]</span>
<span class="n">values</span> <span class="o">=</span> <span class="p">[</span><span class="n">value</span> <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span> <span class="k">for</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">group</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">comb</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">house</span><span class="p">):</span>
<span class="s1">'Format how a value is shown at a given house'</span>
<span class="k">return</span> <span class="n">intern</span><span class="p">(</span><span class="n">f</span><span class="s1">'</span><span class="si">{value}</span><span class="s1"> </span><span class="si">{house}</span><span class="s1">'</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">found_at</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">house</span><span class="p">):</span>
<span class="s1">'Value known to be at a specific house'</span>
<span class="k">return</span> <span class="p">[(</span><span class="n">comb</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">house</span><span class="p">),)]</span>
<span class="k">def</span> <span class="nf">same_house</span><span class="p">(</span><span class="n">value1</span><span class="p">,</span> <span class="n">value2</span><span class="p">):</span>
<span class="s1">'The two values occur in the same house: brit1 & red1 | brit2 & red2 ...'</span>
<span class="k">return</span> <span class="n">from_dnf</span><span class="p">([(</span><span class="n">comb</span><span class="p">(</span><span class="n">value1</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">comb</span><span class="p">(</span><span class="n">value2</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">houses</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">consecutive</span><span class="p">(</span><span class="n">value1</span><span class="p">,</span> <span class="n">value2</span><span class="p">):</span>
<span class="s1">'The values are in consecutive houses: green1 & white2 | green2 & white3 ...'</span>
<span class="k">return</span> <span class="n">from_dnf</span><span class="p">([(</span><span class="n">comb</span><span class="p">(</span><span class="n">value1</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">comb</span><span class="p">(</span><span class="n">value2</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">houses</span><span class="p">,</span> <span class="n">houses</span><span class="p">[</span><span class="mi">1</span><span class="p">:])])</span>
<span class="k">def</span> <span class="nf">beside</span><span class="p">(</span><span class="n">value1</span><span class="p">,</span> <span class="n">value2</span><span class="p">):</span>
<span class="s1">'The values occur side-by-side: blends1 & cat2 | blends2 & cat1 ...'</span>
<span class="k">return</span> <span class="n">from_dnf</span><span class="p">([(</span><span class="n">comb</span><span class="p">(</span><span class="n">value1</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">comb</span><span class="p">(</span><span class="n">value2</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">houses</span><span class="p">,</span> <span class="n">houses</span><span class="p">[</span><span class="mi">1</span><span class="p">:])]</span> <span class="o">+</span>
<span class="p">[(</span><span class="n">comb</span><span class="p">(</span><span class="n">value2</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">comb</span><span class="p">(</span><span class="n">value1</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">houses</span><span class="p">,</span> <span class="n">houses</span><span class="p">[</span><span class="mi">1</span><span class="p">:])]</span>
<span class="p">)</span>
<span class="n">cnf</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># each house gets exactly one value from each attribute group</span>
<span class="k">for</span> <span class="n">house</span> <span class="ow">in</span> <span class="n">houses</span><span class="p">:</span>
<span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">one_of</span><span class="p">(</span><span class="n">comb</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">house</span><span class="p">)</span> <span class="k">for</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">group</span><span class="p">)</span>
<span class="c1"># each value gets assigned to exactly one house</span>
<span class="k">for</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">values</span><span class="p">:</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">one_of</span><span class="p">(</span><span class="n">comb</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">house</span><span class="p">)</span> <span class="k">for</span> <span class="n">house</span> <span class="ow">in</span> <span class="n">houses</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'brit'</span><span class="p">,</span> <span class="s1">'red'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'swede'</span><span class="p">,</span> <span class="s1">'dog'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'dane'</span><span class="p">,</span> <span class="s1">'tea'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">consecutive</span><span class="p">(</span><span class="s1">'green'</span><span class="p">,</span> <span class="s1">'white'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'green'</span><span class="p">,</span> <span class="s1">'coffee'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'pall mall'</span><span class="p">,</span> <span class="s1">'bird'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'yellow'</span><span class="p">,</span> <span class="s1">'dunhill'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">found_at</span><span class="p">(</span><span class="s1">'milk'</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">found_at</span><span class="p">(</span><span class="s1">'norwegian'</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">beside</span><span class="p">(</span><span class="s1">'blends'</span><span class="p">,</span> <span class="s1">'cat'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">beside</span><span class="p">(</span><span class="s1">'horse'</span><span class="p">,</span> <span class="s1">'dunhill'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'blue master'</span><span class="p">,</span> <span class="s1">'root beer'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">same_house</span><span class="p">(</span><span class="s1">'german'</span><span class="p">,</span> <span class="s1">'prince'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">beside</span><span class="p">(</span><span class="s1">'norwegian'</span><span class="p">,</span> <span class="s1">'blue'</span><span class="p">)</span>
<span class="n">cnf</span> <span class="o">+=</span> <span class="n">beside</span><span class="p">(</span><span class="s1">'blends'</span><span class="p">,</span> <span class="s1">'water'</span><span class="p">)</span>
<span class="n">pprint</span><span class="p">(</span><span class="n">solve_one</span><span class="p">(</span><span class="n">cnf</span><span class="p">))</span>
</pre></div>
</div>
</div>
<div class="section" id="essential-utilities-for-humanization">
<h2>Essential Utilities for Humanization<a class="headerlink" href="#essential-utilities-for-humanization" title="Permalink to this headline">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="s1">'Utility functions to humanize interaction with pycosat'</span>
<span class="n">__author__</span> <span class="o">=</span> <span class="s1">'Raymond Hettinger'</span>
<span class="kn">import</span> <span class="nn">pycosat</span> <span class="c1"># https://pypi.python.org/pypi/pycosat</span>
<span class="kn">from</span> <span class="nn">itertools</span> <span class="k">import</span> <span class="n">combinations</span>
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">lru_cache</span>
<span class="kn">from</span> <span class="nn">sys</span> <span class="k">import</span> <span class="n">intern</span>
<span class="k">def</span> <span class="nf">make_translate</span><span class="p">(</span><span class="n">cnf</span><span class="p">):</span>
<span class="sd">"""Make translator from symbolic CNF to PycoSat's numbered clauses.</span>
<span class="sd"> Return a literal to number dictionary and reverse lookup dict</span>
<span class="sd"> >>> make_translate([['~a', 'b', '~c'], ['a', '~c']])</span>
<span class="sd"> ({'a': 1, 'c': 3, 'b': 2, '~a': -1, '~b': -2, '~c': -3},</span>
<span class="sd"> {1: 'a', 2: 'b', 3: 'c', -1: '~a', -3: '~c', -2: '~b'})</span>
<span class="sd"> """</span>
<span class="n">lit2num</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">clause</span> <span class="ow">in</span> <span class="n">cnf</span><span class="p">:</span>
<span class="k">for</span> <span class="n">literal</span> <span class="ow">in</span> <span class="n">clause</span><span class="p">:</span>
<span class="k">if</span> <span class="n">literal</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">lit2num</span><span class="p">:</span>
<span class="n">var</span> <span class="o">=</span> <span class="n">literal</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="k">if</span> <span class="n">literal</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'~'</span> <span class="k">else</span> <span class="n">literal</span>
<span class="n">num</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">lit2num</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">lit2num</span><span class="p">[</span><span class="n">intern</span><span class="p">(</span><span class="n">var</span><span class="p">)]</span> <span class="o">=</span> <span class="n">num</span>
<span class="n">lit2num</span><span class="p">[</span><span class="n">intern</span><span class="p">(</span><span class="s1">'~'</span> <span class="o">+</span> <span class="n">var</span><span class="p">)]</span> <span class="o">=</span> <span class="o">-</span><span class="n">num</span>
<span class="n">num2var</span> <span class="o">=</span> <span class="p">{</span><span class="n">num</span><span class="p">:</span><span class="n">lit</span> <span class="k">for</span> <span class="n">lit</span><span class="p">,</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">lit2num</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span>
<span class="k">return</span> <span class="n">lit2num</span><span class="p">,</span> <span class="n">num2var</span>
<span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="n">cnf</span><span class="p">,</span> <span class="n">uniquify</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="s1">'Translate a symbolic cnf to a numbered cnf and return a reverse mapping'</span>
<span class="c1"># DIMACS CNF file format:</span>
<span class="c1"># http://people.sc.fsu.edu/~jburkardt/data/cnf/cnf.html</span>
<span class="k">if</span> <span class="n">uniquify</span><span class="p">:</span>
<span class="n">cnf</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">dict</span><span class="o">.</span><span class="n">fromkeys</span><span class="p">(</span><span class="n">cnf</span><span class="p">))</span>
<span class="n">lit2num</span><span class="p">,</span> <span class="n">num2var</span> <span class="o">=</span> <span class="n">make_translate</span><span class="p">(</span><span class="n">cnf</span><span class="p">)</span>
<span class="n">numbered_cnf</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">([</span><span class="n">lit2num</span><span class="p">[</span><span class="n">lit</span><span class="p">]</span> <span class="k">for</span> <span class="n">lit</span> <span class="ow">in</span> <span class="n">clause</span><span class="p">])</span> <span class="k">for</span> <span class="n">clause</span> <span class="ow">in</span> <span class="n">cnf</span><span class="p">]</span>
<span class="k">return</span> <span class="n">numbered_cnf</span><span class="p">,</span> <span class="n">num2var</span>
<span class="k">def</span> <span class="nf">itersolve</span><span class="p">(</span><span class="n">symbolic_cnf</span><span class="p">,</span> <span class="n">include_neg</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="n">numbered_cnf</span><span class="p">,</span> <span class="n">num2var</span> <span class="o">=</span> <span class="n">translate</span><span class="p">(</span><span class="n">symbolic_cnf</span><span class="p">)</span>
<span class="k">for</span> <span class="n">solution</span> <span class="ow">in</span> <span class="n">pycosat</span><span class="o">.</span><span class="n">itersolve</span><span class="p">(</span><span class="n">numbered_cnf</span><span class="p">):</span>
<span class="k">yield</span> <span class="p">[</span><span class="n">num2var</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">solution</span> <span class="k">if</span> <span class="n">include_neg</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">></span> <span class="mi">0</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">solve_all</span><span class="p">(</span><span class="n">symcnf</span><span class="p">,</span> <span class="n">include_neg</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">itersolve</span><span class="p">(</span><span class="n">symcnf</span><span class="p">,</span> <span class="n">include_neg</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">solve_one</span><span class="p">(</span><span class="n">symcnf</span><span class="p">,</span> <span class="n">include_neg</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">next</span><span class="p">(</span><span class="n">itersolve</span><span class="p">(</span><span class="n">symcnf</span><span class="p">,</span> <span class="n">include_neg</span><span class="p">))</span>
<span class="c1">############### Support for Building CNFs ##########################</span>
<span class="nd">@lru_cache</span><span class="p">(</span><span class="n">maxsize</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">neg</span><span class="p">(</span><span class="n">element</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'element'</span><span class="p">:</span>
<span class="s1">'Negate a single element'</span>
<span class="k">return</span> <span class="n">intern</span><span class="p">(</span><span class="n">element</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="k">if</span> <span class="n">element</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s1">'~'</span><span class="p">)</span> <span class="k">else</span> <span class="s1">'~'</span> <span class="o">+</span> <span class="n">element</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">from_dnf</span><span class="p">(</span><span class="n">groups</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="s1">'Convert from or-of-ands to and-of-ors'</span>
<span class="n">cnf</span> <span class="o">=</span> <span class="p">{</span><span class="nb">frozenset</span><span class="p">()}</span>
<span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
<span class="n">nl</span> <span class="o">=</span> <span class="p">{</span><span class="nb">frozenset</span><span class="p">([</span><span class="n">literal</span><span class="p">])</span> <span class="p">:</span> <span class="n">neg</span><span class="p">(</span><span class="n">literal</span><span class="p">)</span> <span class="k">for</span> <span class="n">literal</span> <span class="ow">in</span> <span class="n">group</span><span class="p">}</span>
<span class="c1"># The "clause | literal" prevents dup lits: {x, x, y} -> {x, y}</span>
<span class="c1"># The nl check skips over identities: {x, ~x, y} -> True</span>
<span class="n">cnf</span> <span class="o">=</span> <span class="p">{</span><span class="n">clause</span> <span class="o">|</span> <span class="n">literal</span> <span class="k">for</span> <span class="n">literal</span> <span class="ow">in</span> <span class="n">nl</span> <span class="k">for</span> <span class="n">clause</span> <span class="ow">in</span> <span class="n">cnf</span>
<span class="k">if</span> <span class="n">nl</span><span class="p">[</span><span class="n">literal</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">clause</span><span class="p">}</span>
<span class="c1"># The sc check removes clauses with superfluous terms:</span>
<span class="c1"># {{x}, {x, z}, {y, z}} -> {{x}, {y, z}}</span>
<span class="c1"># Should this be left until the end?</span>
<span class="n">sc</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">cnf</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">len</span><span class="p">)</span> <span class="c1"># XXX not deterministic</span>
<span class="n">cnf</span> <span class="o">-=</span> <span class="p">{</span><span class="n">clause</span> <span class="k">for</span> <span class="n">clause</span> <span class="ow">in</span> <span class="n">cnf</span> <span class="k">if</span> <span class="n">clause</span> <span class="o">></span> <span class="n">sc</span><span class="p">}</span>
<span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">cnf</span><span class="p">))</span>
<span class="k">class</span> <span class="nc">Q</span><span class="p">:</span>
<span class="s1">'Quantifier for the number of elements that are true'</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elements</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">elements</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">combinations</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">neg</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">elements</span><span class="p">),</span> <span class="n">n</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">self</span> <span class="o"><</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">combinations</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">elements</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">elements</span><span class="p">)</span><span class="o">-</span><span class="n">n</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">self</span> <span class="o">></span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="bp">self</span> <span class="o"><=</span> <span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span> <span class="o">>=</span> <span class="n">n</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">NotImplementedError</span>
<span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-></span> <span class="nb">str</span><span class="p">:</span>
<span class="k">return</span> <span class="n">f</span><span class="s1">'</span><span class="si">{self.__class__.__name__}</span><span class="s1">(elements=</span><span class="si">{self.elements!r}</span><span class="s1">)'</span>
<span class="k">def</span> <span class="nf">all_of</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="s1">'Forces inclusion of matching rows on a truth table'</span>
<span class="k">return</span> <span class="n">Q</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">some_of</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="s1">'At least one of the elements must be true'</span>
<span class="k">return</span> <span class="n">Q</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">>=</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">one_of</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="s1">'Exactly one of the elements is true'</span>
<span class="k">return</span> <span class="n">Q</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">basic_fact</span><span class="p">(</span><span class="n">element</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="s1">'Assert that this one element always matches'</span>
<span class="k">return</span> <span class="n">Q</span><span class="p">([</span><span class="n">element</span><span class="p">])</span> <span class="o">==</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">none_of</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">-></span> <span class="s1">'cnf'</span><span class="p">:</span>
<span class="s1">'Forces exclusion of matching rows on a truth table'</span>
<span class="k">return</span> <span class="n">Q</span><span class="p">(</span><span class="n">elements</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
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