implement TODO

.github/workflows/test.yml

@@ -0,0 +1,24 @@
+name: Go Test
+
+on:
+  push:
+    branches: [ main ]
+  pull_request:
+    branches: [ main ]
+
+jobs:
+  test:
+    runs-on: ubuntu-latest
+    steps:
+    - uses: actions/checkout@v4
+
+    - name: Set up Go
+      uses: actions/setup-go@v5
+      with:
+        go-version: '1.21'
+
+    - name: Install dependencies
+      run: go mod download
+
+    - name: Run tests
+      run: go test -v ./...

GUIDE.md

@@ -414,7 +414,39 @@ Schönfinkel now provides us with the methodology of how to move the the argumen
 
 As far as our implementation goes, we can treat `U` as a constant as it has no meaning that is compatible with our tree definition. Maybe some day I will come back and implement the conversion of first-order logic to combinatory logic and vice versa.
 
-The modern-day interpretation of this section is probably that combinatory logic is useful as an encoding mechanism to get as little syntax overhead as possible (and first-order logic is one such example). I find systems that build on top of the system (like the [Church encoding](https://en.wikipedia.org/wiki/Church_encoding)) much more interesting. Church encoding can be found in [combinator.go](./church.go) and [combinator_test.go](./church_test.go).
+The modern-day interpretation of this section is probably that combinatory logic is useful as an encoding mechanism to get as little syntax overhead as possible (and first-order logic is one such example). I find systems that build on top of the system (like the [Church encoding](https://en.wikipedia.org/wiki/Church_encoding)) much more interesting. Church encoding can be found in [combinator.go](./combinator.go) and [combinator_test.go](./combinator_test.go).
+
+### Fixed-point Combinator
+
+A [fixed-point combinator](https://en.wikipedia.org/wiki/Fixed-point_combinator) is a combinator that can be used to implement recursion in a system that doesn't have it built-in. The most famous of these is the **Y Combinator**.
+
+The Y combinator has the property that:
+
+$$Yf = f(Yf)$$
+
+This means that $Yf$ is a fixed point of $f$. In our SKI basis, we can define $Y$ as:
+
+$$Y = S (K (S I I)) (S (S (K S) K) (K (S I I)))$$
+
+(Note: $S(S(KS)K)$ is $Z$ in the Schönfinkel basis, or $B$ in the BCKW basis).
+
+Because $Yf$ expands infinitely, our reduction engine will eventually hit a loop detector or timeout when attempting to fully reduce it. You can see this in action in `TestY` in [combinator_test.go](./combinator_test.go).
+
+### Universal Combinator
+
+A [universal combinator](https://en.wikipedia.org/wiki/Universal_combinator) is a combinator that can represent any other combinator through a specific encoding. Similar to [John Tromp's Binary Lambda Calculus](https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861), we can define an encoding for SKI terms and a universal combinator $U$ to execute them.
+
+Our encoding represents each SKI combinator as a higher-order combinator:
+- $enc(S) = \lambda s k i a. s$ (represented as `A` in our implementation)
+- $enc(K) = \lambda s k i a. k$ (represented as `B` in our implementation)
+- $enc(I) = \lambda s k i a. i$ (represented as `C` in our implementation)
+- $enc(MN) = \lambda s k i a. a \ enc(M) \ enc(N)$ (represented as `D` in our implementation)
+
+The universal combinator $U$ is defined such that it decodes these terms and applies the resulting SKI combinators:
+
+$$U = Y (\lambda u e. e \ S \ K \ I \ (\lambda m n. u \ m \ (u \ n)))$$
+
+In our `Universal` basis, you can transform encoded terms back into their SKI counterparts. For example, `ASKII` will reduce to `S`. You can find these examples in `TestUniversal` in [combinator_test.go](./combinator_test.go).
 
 ## Section 6
 

TODO.md

@@ -1,2 +1,2 @@
-1. Get [Y Combinator](https://en.wikipedia.org/wiki/Fixed-point_combinator) example
-1. Create a Universal Combinator given some encoding, similar to [Tromp's Binary Lambda Calculus universal machine](https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861).
+1. [x] Get [Y Combinator](https://en.wikipedia.org/wiki/Fixed-point_combinator) example
+1. [x] Create a Universal Combinator given some encoding, similar to [Tromp's Binary Lambda Calculus universal machine](https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861).

combinator.go

@@ -56,10 +56,60 @@ var (
 	Schonfinkel = Basis{I, K, T, Z, S}
 )
 
+// Fixed-point (https://en.wikipedia.org/wiki/Fixed-point_combinator#Y_combinator)
+var (
+	Y = Combinator{
+		Name:       "Y",
+		Arguments:  []string{"f"},
+		Definition: "S(K(SII))(S(S(KS)K)(K(SII)))f",
+	}
+)
+
 // SK and SKI (https://en.wikipedia.org/wiki/SKI_combinator_calculus)
 var (
 	SK  = Basis{S, K}
-	SKI = Basis{S, K, I}
+	SKI = Basis{S, K, I, Y}
+)
+
+// Universal Combinator (https://en.wikipedia.org/wiki/Universal_combinator)
+// Based on a higher-order encoding of SKI terms.
+var (
+	// Encoded S: λs k i a. s
+	ES = Combinator{
+		Name:       "A",
+		Arguments:  []string{"s", "k", "i", "a"},
+		Definition: "s",
+	}
+
+	// Encoded K: λs k i a. k
+	EK = Combinator{
+		Name:       "B",
+		Arguments:  []string{"s", "k", "i", "a"},
+		Definition: "k",
+	}
+
+	// Encoded I: λs k i a. i
+	EI = Combinator{
+		Name:       "C",
+		Arguments:  []string{"s", "k", "i", "a"},
+		Definition: "i",
+	}
+
+	// Encoded Application: λm n s k i a. a m n
+	EA = Combinator{
+		Name:       "D",
+		Arguments:  []string{"m", "n", "s", "k", "i", "a"},
+		Definition: "amn",
+	}
+
+	// Universal Combinator U: Y (λu e. e S K I (λm n. u m (u n)))
+	U = Combinator{
+		Name:       "U",
+		Arguments:  []string{"e"},
+		Definition: "Y(S(K(S(S(S(SI(KS))(KK))(KI))))(S(KK)(S(S(KS)(S(K(S(KS)))(S(KK))))K)))e",
+	}
+
+	Universal = Basis{S, K, I, Y, ES, EK, EI, EA, U}
 )
 
 // BCKW (https://en.wikipedia.org/wiki/B,_C,_K,_W_system)
@@ -95,6 +145,9 @@ var (
 			Arguments: []string{"x"},
 			// Note the use of other combinators in the definition
 			// makes Iota "improper"
+			//
+			// i x = x S K
+			// i = S (S I (K S)) (K K)
 			Definition: "xSK",
 		},
 	}

combinator_test.go

@@ -121,3 +121,43 @@ func TestFullyImproperCombinators(t *testing.T) {
 		})
 	}
 }
+
+func TestY(t *testing.T) {
+	// Use a small frame limit locally for this test to avoid stack overflow
+	oldMax := MaxFrames
+	MaxFrames = 100
+	defer func() { MaxFrames = oldMax }()
+
+	_, err := SKI.Transform(context.Background(), "Yf")
+	if err == nil {
+		t.Error("expected error")
+	}
+	if err.Error() != "loop detected" {
+		t.Errorf("expected error loop detected, got %s", err.Error())
+	}
+}
+
+func TestUniversal(t *testing.T) {
+	tests := []struct {
+		expr     string
+		expected string
+	}{
+		{"ASKII", "S"},
+		{"BSKII", "K"},
+		{"CSKII", "I"},
+	}
+
+    basis := Universal
+
+	for _, tc := range tests {
+		t.Run(tc.expr, func(t *testing.T) {
+			actualResult, err := basis.Transform(context.Background(), tc.expr)
+			if err != nil {
+				t.Fatalf("Transform failed: %v", err)
+			}
+			if tc.expected != actualResult {
+				t.Errorf("transformed %s incorrectly, expected %s but got %s", tc.expr, tc.expected, actualResult)
+			}
+		})
+	}
+}