diff --git a/javascript/ql/lib/semmle/javascript/security/performance/ExponentialBackTracking.qll b/javascript/ql/lib/semmle/javascript/security/performance/ExponentialBackTracking.qll
index 99b4062dfdc..8ba5a75d431 100644
--- a/javascript/ql/lib/semmle/javascript/security/performance/ExponentialBackTracking.qll
+++ b/javascript/ql/lib/semmle/javascript/security/performance/ExponentialBackTracking.qll
@@ -141,26 +141,28 @@ private class StatePair extends TStatePair {
 }
 
 /**
- * Holds for all constructed state pairs.
+ * Holds for `(fork, fork)` state pairs when `isFork(fork, _, _, _, _)` holds.
  *
- * Used in `statePairDist`
+ * Used in `statePairDistToFork`
  */
-private predicate isStatePair(StatePair p) { any() }
+private predicate isStatePairFork(StatePair p) {
+  exists(State fork | p = MkStatePair(fork, fork) and isFork(fork, _, _, _, _))
+}
 
 /**
  * Holds if there are transitions from the components of `q` to the corresponding
  * components of `r`.
  *
- * Used in `statePairDist`
+ * Used in `statePairDistToFork`
  */
-private predicate delta2(StatePair q, StatePair r) { step(q, _, _, r) }
+private predicate reverseStep(StatePair r, StatePair q) { step(q, _, _, r) }
 
 /**
  * Gets the minimum length of a path from `q` to `r` in the
  * product automaton.
  */
-private int statePairDist(StatePair q, StatePair r) =
-  shortestDistances(isStatePair/1, delta2/2)(q, r, result)
+private int statePairDistToFork(StatePair q, StatePair r) =
+  shortestDistances(isStatePairFork/1, reverseStep/2)(r, q, result)
 
 /**
  * Holds if there are transitions from `q` to `r1` and from `q` to `r2`
@@ -255,14 +257,7 @@ private predicate step(StatePair q, InputSymbol s1, InputSymbol s2, State r1, St
 
 private newtype TTrace =
   Nil() or
-  Step(InputSymbol s1, InputSymbol s2, TTrace t) {
-    exists(StatePair p |
-      isReachableFromFork(_, p, t, _) and
-      step(p, s1, s2, _)
-    )
-    or
-    t = Nil() and isFork(_, s1, s2, _, _)
-  }
+  Step(InputSymbol s1, InputSymbol s2, TTrace t) { isReachableFromFork(_, _, s1, s2, t, _) }
 
 /**
  * A list of pairs of input symbols that describe a path in the product automaton
@@ -284,20 +279,28 @@ private class Trace extends TTrace {
  * a path from `r` back to `(fork, fork)` with `rem` steps.
  */
 private predicate isReachableFromFork(State fork, StatePair r, Trace w, int rem) {
+  exists(InputSymbol s1, InputSymbol s2, Trace v |
+    isReachableFromFork(fork, r, s1, s2, v, rem) and
+    w = Step(s1, s2, v)
+  )
+}
+
+private predicate isReachableFromFork(
+  State fork, StatePair r, InputSymbol s1, InputSymbol s2, Trace v, int rem
+) {
   // base case
-  exists(InputSymbol s1, InputSymbol s2, State q1, State q2 |
+  exists(State q1, State q2 |
     isFork(fork, s1, s2, q1, q2) and
     r = MkStatePair(q1, q2) and
-    w = Step(s1, s2, Nil()) and
-    rem = statePairDist(r, MkStatePair(fork, fork))
+    v = Nil() and
+    rem = statePairDistToFork(r, MkStatePair(fork, fork))
   )
   or
   // recursive case
-  exists(StatePair p, Trace v, InputSymbol s1, InputSymbol s2 |
+  exists(StatePair p |
     isReachableFromFork(fork, p, v, rem + 1) and
     step(p, s1, s2, r) and
-    w = Step(s1, s2, v) and
-    rem >= statePairDist(r, MkStatePair(fork, fork))
+    rem = statePairDistToFork(r, MkStatePair(fork, fork))
   )
 }
 
diff --git a/python/ql/lib/semmle/python/security/performance/ExponentialBackTracking.qll b/python/ql/lib/semmle/python/security/performance/ExponentialBackTracking.qll
index 99b4062dfdc..8ba5a75d431 100644
--- a/python/ql/lib/semmle/python/security/performance/ExponentialBackTracking.qll
+++ b/python/ql/lib/semmle/python/security/performance/ExponentialBackTracking.qll
@@ -141,26 +141,28 @@ private class StatePair extends TStatePair {
 }
 
 /**
- * Holds for all constructed state pairs.
+ * Holds for `(fork, fork)` state pairs when `isFork(fork, _, _, _, _)` holds.
  *
- * Used in `statePairDist`
+ * Used in `statePairDistToFork`
  */
-private predicate isStatePair(StatePair p) { any() }
+private predicate isStatePairFork(StatePair p) {
+  exists(State fork | p = MkStatePair(fork, fork) and isFork(fork, _, _, _, _))
+}
 
 /**
  * Holds if there are transitions from the components of `q` to the corresponding
  * components of `r`.
  *
- * Used in `statePairDist`
+ * Used in `statePairDistToFork`
  */
-private predicate delta2(StatePair q, StatePair r) { step(q, _, _, r) }
+private predicate reverseStep(StatePair r, StatePair q) { step(q, _, _, r) }
 
 /**
  * Gets the minimum length of a path from `q` to `r` in the
  * product automaton.
  */
-private int statePairDist(StatePair q, StatePair r) =
-  shortestDistances(isStatePair/1, delta2/2)(q, r, result)
+private int statePairDistToFork(StatePair q, StatePair r) =
+  shortestDistances(isStatePairFork/1, reverseStep/2)(r, q, result)
 
 /**
  * Holds if there are transitions from `q` to `r1` and from `q` to `r2`
@@ -255,14 +257,7 @@ private predicate step(StatePair q, InputSymbol s1, InputSymbol s2, State r1, St
 
 private newtype TTrace =
   Nil() or
-  Step(InputSymbol s1, InputSymbol s2, TTrace t) {
-    exists(StatePair p |
-      isReachableFromFork(_, p, t, _) and
-      step(p, s1, s2, _)
-    )
-    or
-    t = Nil() and isFork(_, s1, s2, _, _)
-  }
+  Step(InputSymbol s1, InputSymbol s2, TTrace t) { isReachableFromFork(_, _, s1, s2, t, _) }
 
 /**
  * A list of pairs of input symbols that describe a path in the product automaton
@@ -284,20 +279,28 @@ private class Trace extends TTrace {
  * a path from `r` back to `(fork, fork)` with `rem` steps.
  */
 private predicate isReachableFromFork(State fork, StatePair r, Trace w, int rem) {
+  exists(InputSymbol s1, InputSymbol s2, Trace v |
+    isReachableFromFork(fork, r, s1, s2, v, rem) and
+    w = Step(s1, s2, v)
+  )
+}
+
+private predicate isReachableFromFork(
+  State fork, StatePair r, InputSymbol s1, InputSymbol s2, Trace v, int rem
+) {
   // base case
-  exists(InputSymbol s1, InputSymbol s2, State q1, State q2 |
+  exists(State q1, State q2 |
     isFork(fork, s1, s2, q1, q2) and
     r = MkStatePair(q1, q2) and
-    w = Step(s1, s2, Nil()) and
-    rem = statePairDist(r, MkStatePair(fork, fork))
+    v = Nil() and
+    rem = statePairDistToFork(r, MkStatePair(fork, fork))
   )
   or
   // recursive case
-  exists(StatePair p, Trace v, InputSymbol s1, InputSymbol s2 |
+  exists(StatePair p |
     isReachableFromFork(fork, p, v, rem + 1) and
     step(p, s1, s2, r) and
-    w = Step(s1, s2, v) and
-    rem >= statePairDist(r, MkStatePair(fork, fork))
+    rem = statePairDistToFork(r, MkStatePair(fork, fork))
   )
 }
 
diff --git a/ruby/ql/lib/codeql/ruby/security/performance/ExponentialBackTracking.qll b/ruby/ql/lib/codeql/ruby/security/performance/ExponentialBackTracking.qll
index 99b4062dfdc..8ba5a75d431 100644
--- a/ruby/ql/lib/codeql/ruby/security/performance/ExponentialBackTracking.qll
+++ b/ruby/ql/lib/codeql/ruby/security/performance/ExponentialBackTracking.qll
@@ -141,26 +141,28 @@ private class StatePair extends TStatePair {
 }
 
 /**
- * Holds for all constructed state pairs.
+ * Holds for `(fork, fork)` state pairs when `isFork(fork, _, _, _, _)` holds.
  *
- * Used in `statePairDist`
+ * Used in `statePairDistToFork`
  */
-private predicate isStatePair(StatePair p) { any() }
+private predicate isStatePairFork(StatePair p) {
+  exists(State fork | p = MkStatePair(fork, fork) and isFork(fork, _, _, _, _))
+}
 
 /**
  * Holds if there are transitions from the components of `q` to the corresponding
  * components of `r`.
  *
- * Used in `statePairDist`
+ * Used in `statePairDistToFork`
  */
-private predicate delta2(StatePair q, StatePair r) { step(q, _, _, r) }
+private predicate reverseStep(StatePair r, StatePair q) { step(q, _, _, r) }
 
 /**
  * Gets the minimum length of a path from `q` to `r` in the
  * product automaton.
  */
-private int statePairDist(StatePair q, StatePair r) =
-  shortestDistances(isStatePair/1, delta2/2)(q, r, result)
+private int statePairDistToFork(StatePair q, StatePair r) =
+  shortestDistances(isStatePairFork/1, reverseStep/2)(r, q, result)
 
 /**
  * Holds if there are transitions from `q` to `r1` and from `q` to `r2`
@@ -255,14 +257,7 @@ private predicate step(StatePair q, InputSymbol s1, InputSymbol s2, State r1, St
 
 private newtype TTrace =
   Nil() or
-  Step(InputSymbol s1, InputSymbol s2, TTrace t) {
-    exists(StatePair p |
-      isReachableFromFork(_, p, t, _) and
-      step(p, s1, s2, _)
-    )
-    or
-    t = Nil() and isFork(_, s1, s2, _, _)
-  }
+  Step(InputSymbol s1, InputSymbol s2, TTrace t) { isReachableFromFork(_, _, s1, s2, t, _) }
 
 /**
  * A list of pairs of input symbols that describe a path in the product automaton
@@ -284,20 +279,28 @@ private class Trace extends TTrace {
  * a path from `r` back to `(fork, fork)` with `rem` steps.
  */
 private predicate isReachableFromFork(State fork, StatePair r, Trace w, int rem) {
+  exists(InputSymbol s1, InputSymbol s2, Trace v |
+    isReachableFromFork(fork, r, s1, s2, v, rem) and
+    w = Step(s1, s2, v)
+  )
+}
+
+private predicate isReachableFromFork(
+  State fork, StatePair r, InputSymbol s1, InputSymbol s2, Trace v, int rem
+) {
   // base case
-  exists(InputSymbol s1, InputSymbol s2, State q1, State q2 |
+  exists(State q1, State q2 |
     isFork(fork, s1, s2, q1, q2) and
     r = MkStatePair(q1, q2) and
-    w = Step(s1, s2, Nil()) and
-    rem = statePairDist(r, MkStatePair(fork, fork))
+    v = Nil() and
+    rem = statePairDistToFork(r, MkStatePair(fork, fork))
   )
   or
   // recursive case
-  exists(StatePair p, Trace v, InputSymbol s1, InputSymbol s2 |
+  exists(StatePair p |
     isReachableFromFork(fork, p, v, rem + 1) and
     step(p, s1, s2, r) and
-    w = Step(s1, s2, v) and
-    rem >= statePairDist(r, MkStatePair(fork, fork))
+    rem = statePairDistToFork(r, MkStatePair(fork, fork))
   )
 }