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Strictly speaking, fair backtracking is not required for all search problems. A fair strategy guarantees all branches make progress, preventing any single branch from starving the others. The List monad handles non-deterministic computation well, and within a finite search space it produces the same results as a fair stream. When the search space is infinite, or when one branch may produce unbounded results, fairness becomes essential to ensure completeness.
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Strictly speaking, fair backtracking is not required for all search problems. A fair strategy guarantees all branches make progress, preventing any single branch from starving the others. The List monad handles non-deterministic computation well, and within a finite search space it produces the same results as a fair stream. When the search space is infinite, or when one branch may produce unbounded results, fairness becomes essential to ensure completeness.
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<a href="https://blog.gluegadget.com/post/fairstream/">Fairstream</a>
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<div class="post-meta"><time class="post-date">2026-02-17</time></div>
<div class="post-content"><div>
<p><a href="https://en.wikipedia.org/wiki/Backtracking">Backtracking</a> is a versatile approach for solving search problems by building solutions incrementally. If a partial solution cannot be extended, it is discarded and the process returns to a previous step to explore an alternative path. This method is generally more efficient than brute-force searching due to pruning: stopping exploration of a branch as soon as it violates a constraint, which eliminates entire sections of the search space.</p>
<p>Strictly speaking, fair backtracking is not required for all search problems. A fair strategy guarantees all branches make progress, preventing any single branch from starving the others. The List monad handles non-deterministic computation well, and within a finite search space it produces the same results as a fair stream. When the search space is infinite, or when one branch may produce unbounded results, fairness becomes essential to ensure completeness.</p>
<p><a href="https://github.com/codiff/fairstream"><code>fairstream</code></a> is a Scala implementation of fair backtracking based on the <a href="https://okmij.org/ftp/Computation/monads.html#fair-bt-stream">work</a> of Oleg Kiselyov.</p>
<h2 id="the-problem-with-depth-first-search">The problem with depth-first search<a href="#the-problem-with-depth-first-search" class="hanchor" ariaLabel="Anchor">#</a> </h2>
<p>Consider Pythagorean triples: tuples $(i, j, k)$ such that $i^2+j^2=k^2$, with $(3, 4, 5)$ as the canonical first example. If we try to generate all such triples using nested infinite generators, a conventional stream composition based on sequential <code>flatMap</code> gets stuck exploring an infinite branch that contains no solution and never comes back up to try a larger value, so it may fail to produce any triples at all despite solutions existing.</p>
<p>The following example demonstrates how <a href="https://fs2.io/">fs2</a>.Stream’s depth-first approach can fail to find Pythagorean triples, even though the result set is non-empty:</p>
<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-scala" data-lang="scala"><span class="line"><span class="cl"><span class="k">val</span> <span class="n">number</span><span class="k">:</span> <span class="kt">fs2.Stream</span><span class="o">[</span><span class="kt">IO</span>, <span class="kt">Int</span><span class="o">]</span> <span class="k">=</span> <span class="n">fs2</span><span class="o">.</span><span class="nc">Stream</span><span class="o">.</span><span class="n">iterate</span><span class="o">(</span><span class="mi">1</span><span class="o">)(</span><span class="k">_</span> <span class="o">+</span> <span class="mi">1</span><span class="o">)</span>
</span></span><span class="line"><span class="cl">
</span></span><span class="line"><span class="cl"><span class="k">val</span> <span class="n">triples</span> <span class="k">=</span> <span class="k">for</span> <span class="o">{</span>
</span></span><span class="line"><span class="cl"> <span class="n">i</span> <span class="k"><-</span> <span class="n">number</span>
</span></span><span class="line"><span class="cl"> <span class="n">j</span> <span class="k"><-</span> <span class="n">number</span>
</span></span><span class="line"><span class="cl"> <span class="n">k</span> <span class="k"><-</span> <span class="n">number</span>
</span></span><span class="line"><span class="cl"> <span class="k">if</span> <span class="n">i</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">*</span> <span class="n">j</span> <span class="o">==</span> <span class="n">k</span> <span class="o">*</span> <span class="n">k</span>
</span></span><span class="line"><span class="cl"><span class="o">}</span> <span class="k">yield</span> <span class="o">(</span><span class="n">i</span><span class="o">,</span> <span class="n">j</span><span class="o">,</span> <span class="n">k</span><span class="o">)</span>
</span></span><span class="line"><span class="cl">
</span></span><span class="line"><span class="cl"><span class="n">triples</span><span class="o">.</span><span class="n">take</span><span class="o">(</span><span class="mi">7</span><span class="o">).</span><span class="n">compile</span><span class="o">.</span><span class="n">toList</span>
</span></span></code></pre></div><p>The <code>for</code> comprehension desugars into nested <code>flatMap</code> calls. Since <code>number</code> is infinite, the innermost generator tries $k = 1, 2, 3, \ldots$ forever for $i = 1, j = 1$ before it ever consider $j = 2$. No Pythagorean triple exists for $i = 1, j = 1$, so the stream never produces a result.</p>
<p>This is not a quirk of ofs2. Any depth-first <code>flatMap</code> over infinite collections has this problem, including Scala’s standard library <code>LazyList</code>.</p>
<h2 id="fair-interleaving-with-fairstream">Fair interleaving with <code>fairstream</code><a href="#fair-interleaving-with-fairstream" class="hanchor" ariaLabel="Anchor">#</a> </h2>
<p><code>fairstream</code> solves this by replacing sequential concatenation with fair disjuction (<code>mplus</code>), which interleaves branches so that every candidate is eventually reached.</p>
<p>At the time of writing, the library is published as snapshots:</p>
<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-scala" data-lang="scala"><span class="line"><span class="cl"><span class="n">resolvers</span> <span class="o">+=</span> <span class="nc">Resolver</span><span class="o">.</span><span class="n">sonatypeCentralSnapshots</span>
</span></span><span class="line"><span class="cl">
</span></span><span class="line"><span class="cl"><span class="n">libraryDependencies</span> <span class="o">+=</span> <span class="s">"com.codiff"</span> <span class="o">%%</span> <span class="s">"fairstream"</span> <span class="o">%</span> <span class="s">"0.0-9f9db42-SNAPSHOT"</span>
</span></span></code></pre></div><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-scala" data-lang="scala"><span class="line"><span class="cl"><span class="k">import</span> <span class="nn">com.codiff.fairstream.Fair</span>
</span></span><span class="line"><span class="cl"><span class="k">import</span> <span class="nn">com.codiff.fairstream.Fair._</span>
</span></span><span class="line"><span class="cl">
</span></span><span class="line"><span class="cl"><span class="k">lazy</span> <span class="k">val</span> <span class="n">number</span><span class="k">:</span> <span class="kt">Fair</span><span class="o">[</span><span class="kt">Int</span><span class="o">]</span> <span class="k">=</span> <span class="n">mplus</span><span class="o">(</span><span class="n">unit</span><span class="o">(</span><span class="mi">0</span><span class="o">),</span> <span class="n">number</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="k">_</span> <span class="o">+</span> <span class="mi">1</span><span class="o">))</span>
</span></span><span class="line"><span class="cl">
</span></span><span class="line"><span class="cl"><span class="k">val</span> <span class="n">triples</span> <span class="k">=</span> <span class="k">for</span> <span class="o">{</span>
</span></span><span class="line"><span class="cl"> <span class="n">i</span> <span class="k"><-</span> <span class="n">number</span>
</span></span><span class="line"><span class="cl"> <span class="k">_</span> <span class="k"><-</span> <span class="n">guard</span><span class="o">(</span><span class="n">i</span> <span class="o">></span> <span class="mi">0</span><span class="o">)</span>
</span></span><span class="line"><span class="cl"> <span class="n">j</span> <span class="k"><-</span> <span class="n">number</span>
</span></span><span class="line"><span class="cl"> <span class="k">_</span> <span class="k"><-</span> <span class="n">guard</span><span class="o">(</span><span class="n">j</span> <span class="o">></span> <span class="mi">0</span><span class="o">)</span>
</span></span><span class="line"><span class="cl"> <span class="n">k</span> <span class="k"><-</span> <span class="n">number</span>
</span></span><span class="line"><span class="cl"> <span class="k">_</span> <span class="k"><-</span> <span class="n">guard</span><span class="o">(</span><span class="n">k</span> <span class="o">></span> <span class="mi">0</span><span class="o">)</span>
</span></span><span class="line"><span class="cl"> <span class="k">_</span> <span class="k"><-</span> <span class="n">guard</span><span class="o">(</span><span class="n">i</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">*</span> <span class="n">j</span> <span class="o">==</span> <span class="n">k</span> <span class="o">*</span> <span class="n">k</span><span class="o">)</span>
</span></span><span class="line"><span class="cl"><span class="o">}</span> <span class="k">yield</span> <span class="o">(</span><span class="n">i</span><span class="o">,</span> <span class="n">j</span><span class="o">,</span> <span class="n">k</span><span class="o">)</span>
</span></span><span class="line"><span class="cl">
</span></span><span class="line"><span class="cl"><span class="nc">Fair</span><span class="o">.</span><span class="n">runM</span><span class="o">(</span><span class="nc">None</span><span class="o">,</span> <span class="nc">Some</span><span class="o">(</span><span class="mi">7</span><span class="o">),</span> <span class="n">triples</span><span class="o">)</span>
</span></span></code></pre></div><h2 id="fairt-effectful-fair-backtracking"><code>FairT</code>: effectful fair backtracking<a href="#fairt-effectful-fair-backtracking" class="hanchor" ariaLabel="Anchor">#</a> </h2>
<p><code>Fair[A]</code> is a pure computation. But what if your search branches need to perform effects, like reading from a database, calling an API, or logging? That’s where <code>FairT[F[_], A]</code> comes in.</p>
<p><code>FairT</code> is a monad transformer that layers fair backtracking on top of any effect <code>F</code>. This means you can interleave non-deterministic search with <code>IO</code>, <code>Task</code>, or any other cats-effect compatible monad.</p>
<h2 id="fs2-integration">fs2 integration<a href="#fs2-integration" class="hanchor" ariaLabel="Anchor">#</a> </h2>
<p>If you want to stay in the fs2 ecosystem, <code>fairstream-fs2</code> provides a conversion from <code>Fair</code> and <code>FairT</code> to <code>fs2.Stream</code>:</p>
<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-scala" data-lang="scala"><span class="line"><span class="cl"><span class="n">libraryDependencies</span> <span class="o">+=</span> <span class="s">"com.codiff"</span> <span class="o">%%</span> <span class="s">"fairstream-fs2"</span> <span class="o">%</span> <span class="s">"0.0-9f9db42-SNAPSHOT"</span>
</span></span></code></pre></div><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-scala" data-lang="scala"><span class="line"><span class="cl"><span class="k">import</span> <span class="nn">com.codiff.fairstream.fs2.syntax._</span>
</span></span><span class="line"><span class="cl">
</span></span><span class="line"><span class="cl"><span class="n">triples</span><span class="o">.</span><span class="n">toFs2</span><span class="o">.</span><span class="n">take</span><span class="o">(</span><span class="mi">7</span><span class="o">).</span><span class="n">compile</span><span class="o">.</span><span class="n">toList</span>
</span></span></code></pre></div><p>This should let you compose fair backtracking with all the stream processing, concurrency, and resource management that fs2 provides. You define your search logic using <code>Fair</code> or <code>FairT</code>, then convert to <code>fs2.Stream</code> at the boundary where you need to integrate with the rest of your application.</p>
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