diff --git a/src/commonMain/kotlin/ai/hypergraph/kaliningraph/parsing/SeqValiant.kt b/src/commonMain/kotlin/ai/hypergraph/kaliningraph/parsing/SeqValiant.kt
new file mode 100644
index 00000000..b94a9519
--- /dev/null
+++ b/src/commonMain/kotlin/ai/hypergraph/kaliningraph/parsing/SeqValiant.kt
@@ -0,0 +1,72 @@
+package ai.hypergraph.kaliningraph.parsing
+
+import ai.hypergraph.kaliningraph.tensor.UTMatrix
+import ai.hypergraph.kaliningraph.types.*
+
+typealias PForest = Set<PTree>
+operator fun PForest.contains(v: Σᐩ) = PTree(v) in this
+fun PSingleton(v: Σᐩ): List<Π2A<PTree>> = listOf(PTree(v) to PTree())
+
+// Algebraic data type / polynomial functor for parse forests
+data class PTree(val root: Σᐩ = "ε", val children: List<Π2A<PTree>> = listOf()) {
+  // Returns the set of all strings derivable from the given PTree
+  fun choose(): Sequence<Σᐩ> =
+    if (children.isEmpty()) sequenceOf(if("ε" in root) "" else root)
+    else children.asSequence().flatMap { (l, r) ->
+      // TODO: Use weighted choice mechanism
+      (l.choose() * r.choose()).map { (a, b) ->
+        if (a == "") b else if (b == "") a else "$a $b"
+      }
+    }
+
+  override fun hashCode(): Int = root.hashCode()
+  override fun equals(other: Any?) = other is PTree && root == other.root
+}
+
+// Lazily computes all syntactically strings compatible with the given template
+fun CFG.solveSeq(s: Σᐩ): Sequence<Σᐩ> = solveSeq(s.tokenizeByWhitespace())
+
+fun CFG.solveSeq(s: List<Σᐩ>): Sequence<Σᐩ> =
+  try { solvePTreeFPSeq(s) }
+  catch (e: Exception) { e.printStackTrace(); null } ?: sequenceOf()
+
+fun CFG.solvePTreeFPSeq(
+  tokens: List<Σᐩ>,
+  utMatrix: UTMatrix<PForest> = initPForestMatrix(tokens, pforestAlgebra()),
+) =
+  utMatrix.seekFixpoint().toFullMatrix()[0].last()
+    .firstOrNull { it.root == START_SYMBOL }?.choose() ?: emptySequence()
+
+fun CFG.initPForestMatrix(
+  tokens: List<Σᐩ>,
+  algebra: Ring<PForest>
+): UTMatrix<PForest> =
+  UTMatrix(
+    ts = tokens.map { token ->
+      (if (token != HOLE_MARKER) bimap[listOf(token)] else unitNonterminals)
+        .associateWith { nt ->
+          if (token != HOLE_MARKER) PSingleton(token)
+          else bimap.UNITS[nt]?.map { PSingleton(it) }?.flatten()?.toSet()?.toList() ?: listOf()
+        }.map { (k, v) -> PTree(k, v) }.toSet()
+    }.toTypedArray(),
+    algebra = algebra
+  )
+
+// Maintains a sorted list of nonterminal roots and their leaves
+fun CFG.pforestAlgebra(): Ring<PForest> =
+  Ring.of(
+    nil = emptySet(),
+    plus = { x, y -> x union y },
+    times = { x, y -> joinSeq(x, y) },
+  )
+
+// X ⊗ Z := { w | <x, z> ∈ X × Z, (w -> xz) ∈ P }
+fun CFG.joinSeq(X: PForest, Z: PForest): PForest =
+  bimap.TRIPL.filter { (_, x, z) -> x in X && z in Z }
+    .groupingBy { it.first }.aggregate { _, acc: List<Π2A<PTree>>?, it, _-> 
+      val (w, x, z) = it
+      val ptreeX = X.first { it.root == x }
+      val ptreeZ = Z.first { it.root == z }
+      val pair = ptreeX to ptreeZ
+      if (acc == null) listOf(pair) else acc + pair
+    }.map { (k, v) -> PTree(k, v) }.toSet()
\ No newline at end of file
diff --git a/src/commonMain/kotlin/ai/hypergraph/kaliningraph/parsing/SortValiant.kt b/src/commonMain/kotlin/ai/hypergraph/kaliningraph/parsing/SortValiant.kt
index 33bf5652..9eba9322 100644
--- a/src/commonMain/kotlin/ai/hypergraph/kaliningraph/parsing/SortValiant.kt
+++ b/src/commonMain/kotlin/ai/hypergraph/kaliningraph/parsing/SortValiant.kt
@@ -6,6 +6,12 @@ import ai.hypergraph.kaliningraph.levenshtein
 import ai.hypergraph.kaliningraph.tensor.UTMatrix
 import ai.hypergraph.kaliningraph.types.*
 
+// The main issue with SortValiant is we eagerly compute the Cartesian product
+// and this blows up very quickly, so we need to sort and prune aggressively.
+// We can instead use a lazy Cartesian product, which is what SeqValiant does.
+// The downside is that we lose the ability to sort the results while parsing,
+// but we can still use a metric to sort the results after the fact.
+
 // Returns all syntactically strings ordered by distance to withRespect
 fun CFG.solve(s: Σᐩ, metric: ChoiceMetric): Set<Σᐩ> =
   solve(s.tokenizeByWhitespace(), metric)
@@ -49,6 +55,9 @@ const val MAX_CAPACITY = 100
 fun CFG.join(X: Sort, Z: Sort, metric: ChoiceMetric = { it.weight }): Sort =
   bimap.TRIPL.filter { (_, x, z) -> x in X && z in Z }
   .map { (w, x, z) ->
+    // This Cartesian product becomes expensive quickly so MAX_CAPACITY is used
+    // to limit the number of elements in the product. This is a greedy approach
+    // and we always take the top MAX_CAPACITY-elements by the provided metric.
     ((X[x] ?: setOf()) * (Z[z] ?: setOf()))
       .map { (q, r) -> w to (q + r) }
   }.flatten().groupingBy { it.first }
diff --git a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SetValiantTest.kt b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SetValiantTest.kt
index 4232dfff..0f611f83 100644
--- a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SetValiantTest.kt
+++ b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SetValiantTest.kt
@@ -369,6 +369,16 @@ class SetValiantTest {
     }.also { println("Finished in ${it.inWholeMilliseconds}ms.") }
   }
 
+/*
+./gradlew jvmTest --tests "ai.hypergraph.kaliningraph.parsing.SetValiantTest.testSeqValiant"
+*/
+  @Test
+  fun testSeqValiant() {
+    val allSols = seq2parsePythonCFG.solveSeq("_ _ _ _ _").sortedBy { it.length }.toList()
+    allSols.forEach { println(it); assertTrue("\"$it\" was invalid!") { it in seq2parsePythonCFG.language } }
+    println("Found ${allSols.size} solutions, all were valid!")
+  }
+
   val seq2parsePythonCFG: CFG =
     """
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