T0-3 Formal Specification: Zeckendorf Constraint Emergence
1. Type Definitions
-- Binary string type
type BinaryString = [Bit] where Bit ∈ {0, 1}
type Length = ℕ
-- Constraint specification
type Constraint = BinaryString → Bool
-- Encoding scheme with bijection requirement
type BijectiveEncoding = {
weights: [ℕ],
constraint: Constraint,
encode: ℕ → BinaryString,
decode: BinaryString → ℕ,
bijective: ∀n. decode(encode(n)) = n ∧
∀s. constraint(s) → encode(decode(s)) = s
}
2. Correct Fibonacci Definition
2.1 Standard Fibonacci Sequence
fibonacci : ℕ → ℕ
fibonacci(1) = 1
fibonacci(2) = 2
fibonacci(n) = fibonacci(n-1) + fibonacci(n-2) for n > 2
-- Sequence: F₁=1, F₂=2, F₃=3, F₄=5, F₅=8, F₆=13, F₇=21, ...
2.2 Counting Formula
count_no11_strings : ℕ → ℕ
count_no11_strings(0) = 1 -- empty string
count_no11_strings(1) = 2 -- "0", "1"
count_no11_strings(n) = count_no11_strings(n-1) + count_no11_strings(n-2)
-- THEOREM: count_no11_strings(n) = fibonacci(n+1)
3. Uniqueness vs Density
3.1 Bijection Requirement
is_bijective : Encoding → Bool
is_bijective(E) =
(∀n ∈ ℕ. ∃!s. E.constraint(s) ∧ E.decode(s) = n) ∧
(∀s. E.constraint(s) → ∃!n. E.decode(s) = n)
3.2 Information Density
information_density : Constraint × Length → ℝ
information_density(Γ, n) = log₂(count_valid(Γ, n)) / n
where count_valid(Γ, n) = |{s : |s| = n ∧ Γ(s)}|
3.3 Density Comparison
-- Verified densities for different constraints:
density_no_constraint : ℕ → ℝ
density_no_constraint(n) = 1.0 -- 2ⁿ strings
density_no111 : ℕ → ℝ
density_no111(n) ≈ 0.755 -- ~1.465ⁿ strings
density_no11 : ℕ → ℝ
density_no11(n) → log₂(φ) ≈ 0.694 -- fibonacci(n+1) strings
-- FACT: density_no11 < density_no111 < density_no_constraint
4. Constraint Specifications
4.1 No-11 Constraint
no_consecutive_ones : Constraint
no_consecutive_ones(s) = ∀i ∈ [0..|s|-2]. ¬(s[i] = 1 ∧ s[i+1] = 1)
4.2 Alternative Constraints (Non-Bijective)
-- No three consecutive ones (NOT BIJECTIVE)
no_111 : Constraint
no_111(s) = ∀i ∈ [0..|s|-3]. ¬(s[i] = 1 ∧ s[i+1] = 1 ∧ s[i+2] = 1)
-- Allows "11" patterns which break uniqueness
5. Zeckendorf Encoding
5.1 Decoder with Correct Indexing
zeckendorf_decode : BinaryString → ℕ
zeckendorf_decode([]) = 0
zeckendorf_decode(s) = Σᵢ s[i] × fibonacci(i+1)
where i ∈ [0..|s|-1]
-- Uses F₁, F₂, F₃, ... (starts from F₁)
5.2 Encoder Algorithm
zeckendorf_encode : ℕ → BinaryString
zeckendorf_encode(0) = []
zeckendorf_encode(n) =
let fibs = [fibonacci(i) | i <- [1..], fibonacci(i) ≤ n]
result = greedy_select(n, reverse(fibs))
in ensure_no_consecutive_ones(result)
greedy_select : ℕ × [ℕ] → BinaryString
-- Greedy algorithm selecting largest Fibonacci numbers
-- Ensures no consecutive selections
6. Key Theorems (Corrected)
6.1 Uniqueness Theorem
theorem_zeckendorf_uniqueness:
∀n : ℕ. ∃!s : BinaryString.
no_consecutive_ones(s) ∧
zeckendorf_decode(s) = n
proof_sketch:
- Existence: by greedy algorithm
- Uniqueness: by Fibonacci growth property
6.2 No-111 Lacks Uniqueness
theorem_no111_not_bijective:
∃n : ℕ. ∃s₁,s₂ : BinaryString.
s₁ ≠ s₂ ∧
no_111(s₁) ∧ no_111(s₂) ∧
decode_fibonacci(s₁) = decode_fibonacci(s₂)
proof:
-- "11" patterns create redundancy via F_i + F_{i+1} = F_{i+2}
-- Example: different representations for same value
6.3 Counting Formula Correctness
theorem_fibonacci_counting:
∀n : ℕ. count_no11_strings(n) = fibonacci(n+1)
proof:
by induction on n
base: n=0 → 1 = fibonacci(1) ✓
n=1 → 2 = fibonacci(2) ✓
step: count(n) = count(n-1) + count(n-2)
= fibonacci(n) + fibonacci(n-1)
= fibonacci(n+1) ✓
7. Trade-off Analysis
7.1 Uniqueness-Density Trade-off
constraint_properties : Constraint → (Bool, ℝ)
constraint_properties(Γ) = (is_bijective(Γ), average_density(Γ))
-- No-11: (true, 0.694) - bijective but lower density
-- No-111: (false, 0.755) - higher density but not bijective
-- None: (false, 1.0) - maximum density but not bijective
7.2 Optimality Criterion
optimal_for_uniqueness : Constraint
optimal_for_uniqueness = argmax_{Γ} density(Γ)
subject to is_bijective(Γ)
-- RESULT: optimal_for_uniqueness = no_consecutive_ones
8. Verification Algorithms
8.1 Uniqueness Checker
verify_uniqueness : Constraint × Length → Bool
verify_uniqueness(Γ, n) =
let valid = [s | s <- all_strings(n), Γ(s)]
values = map decode valid
in length(values) = length(nub(values)) -- no duplicates
8.2 Bijection Verifier
verify_bijection : Encoding × ℕ → Bool
verify_bijection(E, max_n) =
∀n ∈ [0..max_n].
E.decode(E.encode(n)) = n ∧
E.constraint(E.encode(n))
8.3 Density Calculator
compute_density : Constraint → Length → ℝ
compute_density(Γ, n) =
let valid_count = count_valid(Γ, n)
in log₂(valid_count) / n
-- For no-11: valid_count = fibonacci(n+1)
9. Computational Complexity
-- Zeckendorf encoding complexity
encode_complexity : ℕ → Complexity
encode_complexity(n) = O(log n)
-- Uniqueness verification
uniqueness_check : ℕ → Complexity
uniqueness_check(n) = O(fibonacci(n+1)) = O(φⁿ)
-- Density computation
density_computation : ℕ → Complexity
density_computation(n) = O(n) -- via recursion
10. Formal Properties
10.1 Complete Coverage
property_complete_coverage:
∀n : ℕ. ∃s : BinaryString.
no_consecutive_ones(s) ∧
zeckendorf_decode(s) = n
10.2 Unique Representation
property_unique_representation:
∀n : ℕ. ∀s₁,s₂ : BinaryString.
no_consecutive_ones(s₁) ∧
no_consecutive_ones(s₂) ∧
zeckendorf_decode(s₁) = zeckendorf_decode(s₂)
→ s₁ = s₂
10.3 No Other Bijective Constraint
property_uniqueness_of_no11:
∀Γ : Constraint.
(Γ ≠ no_consecutive_ones ∧ is_bijective_with_fibonacci(Γ))
→ false
11. Machine-Verifiable Assertions
-- Correct counting formula
assert_counting:
assert(∀n ≤ 20. count_no11_strings(n) = fibonacci(n+1))
-- Zeckendorf uniqueness
assert_uniqueness:
assert(∀n ≤ 1000. unique_representation(n))
-- Density calculation
assert_density:
assert(|density_no11(20) - log₂(φ)| < 0.01)
-- No-111 lacks uniqueness
assert_no111_redundancy:
assert(∃n. multiple_representations_no111(n))
-- Trade-off verification
assert_tradeoff:
assert(density_no11 < density_no111 ∧
is_bijective(no11) ∧
¬is_bijective(no111))
12. Test Generation
-- Generate uniqueness test cases
test_uniqueness : Length → [(ℕ, BinaryString)]
test_uniqueness(n) =
[(i, zeckendorf_encode(i)) | i <- [0..fibonacci(n+1)-1]]
-- Generate non-uniqueness examples for no-111
test_no111_redundancy : Length → [(BinaryString, BinaryString, ℕ)]
test_no111_redundancy(n) =
find_redundant_pairs(no_111, n)
-- Verify counting formula
test_counting : Length → [(ℕ, ℕ, ℕ)]
test_counting(max_n) =
[(n, count_no11_strings(n), fibonacci(n+1)) | n <- [0..max_n]]
This formal specification correctly represents:
- The actual Fibonacci sequence (F₁=1, F₂=1, ...)
- The correct counting formula (n-bit strings = F_{n+1})
- The true nature of the constraint (uniqueness, not density maximization)
- The fundamental trade-off between uniqueness and density
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