From 04f9412a0b4a2596da1ac6f2ebb8a10ec51dffb4 Mon Sep 17 00:00:00 2001
From: Harshit <harshit.lalwani@research.iiit.ac.in>
Date: Sun, 16 Nov 2025 17:49:41 +0000
Subject: [PATCH] added parity-check, digit-reversal

---
 algorithms/DigitReversal.typ | 117 +++++++++++++++++++++++++++++++++++
 algorithms/parity.typ        |  91 +++++++++++++++++++++++++++
 main.typ                     |   4 ++
 3 files changed, 212 insertions(+)
 create mode 100644 algorithms/DigitReversal.typ
 create mode 100644 algorithms/parity.typ

diff --git a/algorithms/DigitReversal.typ b/algorithms/DigitReversal.typ
new file mode 100644
index 0000000..862848f
--- /dev/null
+++ b/algorithms/DigitReversal.typ
@@ -0,0 +1,117 @@
+#import "../lib/style.typ": *
+#import "../lib/mapcode.typ": *
+
+== Digit Reversal
+
+Reverse the digits of a non-negative integer $n$ using tail-recursive computation. This algorithm demonstrates efficient fixed-point computation with a 2-element state pair $(n_p, a_p)$ representing remaining digits and accumulator.
+
+Formal definition:
+$
+"reverse"(n) = cases(
+  0 & "if " n = 0,
+  10 dot "reverse"(n div 10) + (n mod 10) & "otherwise"
+)
+$
+
+Examples:
+- $"reverse"(123) = 321$
+- $"reverse"(1000) = 1$
+- $"reverse"(12345) = 54321$
+
+*As mapcode (2-state pair):*
+
+$ I = n : NN quad X = NN times NN quad A = NN $
+
+$ rho(n) = (n, 0) $
+
+$ F((n_p, a_p)) = cases(
+  (n_p div 10, a_p dot 10 + n_p mod 10) & "if " n_p > 0,
+  (0, a_p) & "if " n_p = 0
+) $
+
+$ pi((n_p, a_p)) = a_p $
+
+#let inst = 123;
+#figure(
+  caption: [Digit reversal using mapcode for reverse(#inst) = 321],
+$
+#{
+  let rho = (inst) => {
+    (inst, 0)
+  }
+
+  let F = (x) => {
+    let (n_p, a_p) = x
+    if n_p == 0 {
+      (0, a_p)
+    } else {
+      let n_i = int(n_p / 10)
+      let a_i = a_p * 10 + int(n_p) - n_i * 10
+      (n_i, a_i)
+    }
+  }
+
+  let pi = (x) => {
+    x.at(1)
+  }
+
+  let X_h = (x, diff_mask: none) => {
+    let (n_p, a_p) = x
+    let n_box = if diff_mask != none and diff_mask.at(0) {
+      rect(fill: yellow.transparentize(70%), inset: 1pt)[$ #n_p $]
+    } else {
+      rect(stroke: none, inset: 1pt)[$ #n_p $]
+    }
+    let a_box = if diff_mask != none and diff_mask.at(1) {
+      rect(fill: yellow.transparentize(70%), inset: 1pt)[$ #a_p $]
+    } else {
+      rect(stroke: none, inset: 1pt)[$ #a_p $]
+    }
+    $lr((#n_box, #a_box))$
+  }
+
+  // Build history
+  let x0 = rho(inst)
+  let xs = (x0,)
+  let xn = x0
+  let prev = none
+  let iters = 0
+  while prev != xn and iters < 10 {
+    prev = xn
+    xn = F(xn)
+    if xn != xs.at(-1) {
+      xs.push(xn)
+    }
+    iters += 1
+  }
+  let ans = pi(xn)
+
+  // Compute diffs
+  let get_diff = (a, b) => {
+    let (n_a, a_a) = a
+    let (n_b, a_b) = b
+    (n_a != n_b, a_a != a_b)
+  }
+  
+  let diffs = xs.windows(2).map(((prev, next)) => {
+    get_diff(prev, next)
+  })
+  diffs.insert(0, (false, false))
+
+  // Format trace
+  let trace_items = xs.enumerate().map(((idx, x_i)) => {
+    let label = if idx == 0 { $rho$ } else { $F$ }
+    let state = X_h(x_i, diff_mask: diffs.at(idx))
+    $(#label): #state$
+  })
+
+  box(inset: 5pt)[
+    Trace:
+    #h(1em)
+    #trace_items.join([ $arrow.r$ ])
+    #h(1em)
+    $pi = #ans$
+  ]
+}
+$
+)
diff --git a/algorithms/parity.typ b/algorithms/parity.typ
new file mode 100644
index 0000000..3362d60
--- /dev/null
+++ b/algorithms/parity.typ
@@ -0,0 +1,91 @@
+#import "../lib/style.typ": *
+#import "../lib/mapcode.typ": *
+
+== Parity
+
+Determine whether a non-negative integer $n$ is even or odd using mutually recursive functions.
+
+Formal definition:
+$
+"is_even"(0) &= "true",\
+"is_odd"(0) &= "false",\
+"is_even"(n) &= "is_odd"(n-1) quad "for " n > 0,\
+"is_odd"(n) &= "is_even"(n-1) quad "for " n > 0
+$
+
+Examples:
+- $"parity"(0) -> ("T", "F")$
+- $"parity"(1) -> ("F", "T")$
+- $"parity"(3) -> ("F", "T")$
+
+*As mapcode:*
+
+$ I = n : NN quad X_n = [0..n] -> (BB times BB)_bot quad A = BB times BB $
+
+$ rho(n) = {i |-> bot | i in {0 dots n}} $
+
+$ F(x_n) = cases(
+  ("T", "F") & "if " n = 0,
+  (x_(n-1)[1], x_(n-1)[0]) & "if " n > 0 "and" x_(n-1) != bot,
+  bot & "otherwise"
+) $
+
+$ pi(x) = x_(|x| - 1) $
+
+#let inst = 3;
+#figure(
+  caption: [Parity computation using mapcode for $n = #inst$],
+$
+#{
+  let rho = (inst) => {
+    let x = ()
+    for i in range(0, inst + 1) {
+      x.push(none)
+    }
+    x
+  }
+
+  let F_i = (x) => ((i,)) => {
+    if i == 0 {
+      (true, false)
+    } else if x.at(i - 1) != none {
+      let (even_prev, odd_prev) = x.at(i - 1)
+      (odd_prev, even_prev)
+    } else {
+      none
+    }
+  }
+  let F = map_tensor(F_i, dim: 1)
+
+  let pi = (i) => (x) => x.at(i)
+
+  let X_h = (x, diff_mask: none) => {
+    let cells = x.enumerate().map(((i, x_i)) => {
+      let val = if x_i != none {
+        let (e, o) = x_i
+        let e_str = if e { "T" } else { "F" }
+        let o_str = if o { "T" } else { "F" }
+        $#e_str#o_str$
+      } else {
+        $bot$
+      }
+      if diff_mask != none and diff_mask.at(i) {
+        rect(fill: yellow.transparentize(70%), inset: 1pt)[$#val$]
+      } else {
+        rect(stroke: none, inset: 1pt)[$#val$]
+      }
+    })
+    $vec(delim: "[", ..cells)$
+  }
+
+  mapcode-viz(
+    rho, F, pi(inst),
+    X_h: X_h,
+    pi_name: [$pi(#inst)$],
+    group-size: 4,
+    cell-size: 8mm,
+    scale-fig: 75%
+  )(inst)
+}
+$
+)
diff --git a/main.typ b/main.typ
index 9952d9b..1882467 100644
--- a/main.typ
+++ b/main.typ
@@ -47,6 +47,10 @@ All primitives are _strict_ meaning they do not allow for undefined values (i.e.
 #pagebreak()
 #include "algorithms/fibonacci.typ"
 #pagebreak()
+#include "algorithms/parity.typ"
+#pagebreak()
+#include "algorithms/DigitReversal.typ"
+#pagebreak()
 #include "algorithms/BinomialCoefficients.typ"
 #pagebreak()
 #include "algorithms/LongestCommonSubsequence.typ"