Gödel's Incompleteness Of ZKPs

Application of Gödel's Incompleteness On Zero Knowledge Proofs

Abstract

Gödel's Incompleteness Theorems and zero-knowledge proofs (ZKPs) both reveal fundamental limitations in formal systems and cryptographic protocols. Gödel's theorems establish that in any sufficiently powerful formal system, there exist true statements that cannot be proven within that system. Zero-knowledge proofs, in a parallel fashion, enable one party (the prover) to convince another party (the verifier) of the truth of a statement without revealing the proof itself. This paper explores the intersection of these concepts, demonstrating that just as Gödel's theorems highlight the existence of true but unprovable statements, ZKPs allow the validation of truths while keeping the proof hidden.

Overview

Gödel's Incompleteness Theorems are two fundamental theorems in mathematical logic that have profound implications for the limits of formal systems. Here is a concise explanation of these theorems as they pertain to proofs that may be proven but cannot be derived from the system:

1. First Incompleteness Theorem:

   • Statement: In any consistent formal system that is capable of expressing elementary arithmetic, there exist propositions that are true but cannot be proven within the system.

   • Implication: This theorem reveals that no matter how comprehensive a formal system is, there will always be true statements about the natural numbers that the system cannot prove. In other words, there are mathematical truths that lie beyond the reach of formal proof within the system itself.

2. Second Incompleteness Theorem:

   • Statement: No consistent system can prove its own consistency.

   • Implication: If a formal system can prove its own consistency, it is, in fact, inconsistent. This means that a system cannot use its own axioms and inference rules to demonstrate that no contradictions can be derived from them.

Application to Proofs That Cannot Be Derived from the System

Gödel's Incompleteness Theorems imply that within any sufficiently powerful formal system (such as Peano arithmetic), there exist statements (Gödel sentences) that are true but unprovable within the system. These theorems can be understood through the lens of proof and derivation as follows:

• True but Unprovable Statements: There are certain statements ( G ) that, although true, cannot be derived using the axioms and rules of inference of the formal system. These statements ( G ) are constructed in such a way that if the system were to prove ( G ), it would lead to a contradiction, thus implying the system is inconsistent.

• Gödel Sentence: For a formal system ( S ), Gödel constructed a specific statement ( G_S ) which essentially says, "This statement is not provable in ( S )." If ( G_S ) were provable in ( S ), ( S ) would be inconsistent because it would prove a statement that asserts its own unprovability.

Fundamental Statement

In essence, Gödel's Incompleteness Theorems demonstrate that there are intrinsic limitations to what can be achieved with formal mathematical systems. There are truths that exist outside the boundaries of formal derivation, illustrating that the scope of formal proofs is inherently limited. This challenges the notion that mathematics, or any formal system, can be completely understood and proven solely through its own rules and axioms.

Exploration

By examining scenarios such as the Hamiltonian cycle problem, we illustrate how ZKPs embody the principles of incompleteness, showing that some truths can be acknowledged without formal derivation or disclosure. This relationship underscores the deep connections between mathematical logic and cryptographic protocols, emphasizing the intertwined nature of provability and secrecy.

Gödel's Incompleteness Theorems are embodied in the concept of zero-knowledge proofs involving probability within a formal system and the properties of zero-knowledge proofs (ZKPs). Zero-knowledge proofs allow one party (the prover) to convince another party (the verifier) that a statement is true without revealing any information beyond the validity of the statement itself.

1. Zero-Knowledge Proofs (ZKPs) Overview

Zero-knowledge proofs are a type of cryptographic protocol that ensures the following properties:

1. Completeness: If the statement is true, the honest verifier will be convinced of this fact by an honest prover.

2. Soundness: If the statement is false, no cheating prover can convince the honest verifier that it is true, except with some small probability.

3. Zero-Knowledge: If the statement is true, the verifier learns nothing other than the fact that the statement is true.

2. Application of Gödel's Incompleteness in Zero-Knowledge Proofs

Gödel's Incompleteness Theorems imply that within any sufficiently powerful formal system, there exist true statements that cannot be proven within the system. Applying this to zero-knowledge proofs, we can draw several parallels and insights:

1. Undecidable Statements: Gödel's theorems show that there are true statements that cannot be proven within a system. Similarly, in the context of ZKPs, there might exist statements whose validity can be demonstrated to a verifier without providing a traditional proof that can be verified within a formal system. This means the verifier is convinced of the truth without accessing the actual proof, aligning with the concept that certain truths might be acknowledged without formal derivation.

2. Proofs Beyond Formal Systems: In zero-knowledge proofs, the prover demonstrates the truth of a statement without revealing the proof itself. This mirrors Gödel's idea that certain truths exist beyond formal provability. A zero-knowledge proof ensures that the verifier is convinced of the statement's truth, much like how Gödel's theorems assure us of the existence of truths beyond formal systems, but without the verifier gaining knowledge about the underlying proof (which remains "undefined" or "hidden").

3. Secrecy and Provability: Zero-knowledge proofs emphasize secrecy; the verifier learns nothing beyond the truth of the statement. This aligns with Gödel's concept that certain truths cannot be fully captured or derived within the system. The proof in a ZKP remains inaccessible (or "undefined") to the verifier, preserving its zero-knowledge nature, akin to how some true mathematical statements remain unprovable within their own systems.

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3. Reductio ad Absurdum

is a classical method of argumentation used in logic and mathematics, which demonstrates that a proposition is true by showing that a false, contradictory, or absurd result follows from its denial. Essentially, it involves the following steps:1. Assume the Negation: Start by assuming that the proposition you want to prove is false. 2. Derive a Contradiction: Show that this assumption leads to a contradiction or an absurdity, something that is clearly false or logically inconsistent. 3. Conclude the Proposition: Since the assumption of the negation leads to a contradiction, conclude that the original proposition must be true.

Formal Structure

In formal terms, if you want to prove a proposition $P$:

1. Assume $eg P$ (the negation of $P$).

2. Derive a contradiction $C$ (something that is obviously false, e.g., $Q \land \neg Q$).

3. Conclude $P$ since $eg P$ leads to a contradiction.

Example

Let's apply reductio ad absurdum to prove that $\sqrt{2}$ is irrational:

1. Assume the Negation: Assume $\sqrt{2}$ is rational, meaning it can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers with no common factors.

2. Derive a Contradiction:

   • $\sqrt{2} = \frac{a}{b}$ implies $2 = \frac{a^2}{b^2}$, so $a^2 = 2b^2$.

   • This means $a^2$ is even (since it is two times some integer), which implies $a$ must be even (since the square of an odd number is odd).

   • Let $a = 2k$ for some integer $k$. Then $a^2 = (2k)^2 = 4k^2$.

   • Substituting back, we get $4k^2 = 2b^2$ or $2k^2 = b^2$. This means $b^2$ is also even, so $b$ must be even.

   • Since both $a$ and $b$ are even, they have a common factor of 2, contradicting our original assumption that $\frac{a}{b}$ is in its simplest form.

3. Conclude the Proposition: Since assuming $\sqrt{2}$ is rational leads to a contradiction, $\sqrt{2}$ must be irrational.

Reductio ad absurdum is a powerful tool in logic and mathematics, used to prove the validity of statements by demonstrating that their negation leads to untenable or contradictory conclusions.

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4. ZKPs are Incomplete Gödel Statements

To define the incompleteness attribute of zero-knowledge proofs (ZKP) as an implementation of Gödel's Incompleteness, we illustrate how a statement can be true and accepted as true by a verifier without revealing the proof within the formal system.

Here's a concise primitive equation:

$$\forall S\left(\text { if } S \equiv S \text { then } P \rightarrow{ }^{Z K P} V: S \vDash Q \text { without } S \vdash Q\right)$$

Where:

$S$ is a statement.

$\mathbf{Q}$ is the formal system.

$$S \vDash Q$$

means $S$ is true within the system $Q$

$$S \vdash Q$$

means $S$ is provable within the system $Q$

$P$ is the prover.

$V$ is the verifier.

$$P \rightarrow{ }^{Z K P} V$$

indicates that the prover convinces the verifier using a zero-knowledge proof.

The equation states that if a statement $S$ is true in the system $Q$, then the prover can convince the verifier of $S$ 's truth through a zero-knowledge proof, without $S$ being formally provable within $Q$

5. Completeness

Let $S$ be a statement, $P$ be the prover, and $V$ be the verifier.

If $S$ is true, then an honest prover $P$ can convince the verifier $V$ of this fact.

If $S$ is true $ightarrow P \rightarrow{ }^{Z K P} V$ is convinced

6. Soundness

that $S$ is true, except with some small probability $\epsilon$

$$\text { If } S \text { is false } \rightarrow P^{\prime} \rightarrow{ }^{Z K P} V \text { is convinced with probability } \epsilon$$

7. Zero-Knowledge

{If $S$} is true, the verifier $V$ learns nothing other than the fact that $S$ is true.

If $S$ is true $ightarrow V$ learns $S$ is true and nothing else

8. A Scenario of ZKP/Gödel's Incompleteness

Consider a scenario where a prover wants to prove they know a solution to a complex mathematical problem (e.g., a Hamiltonian cycle in a large graph) without revealing the solution itself:

• Gödel's Perspective: There might be a true statement about the graph (e.g., it has a Hamiltonian cycle) that cannot be proven within the formal system defining the graph's properties. This is analogous to the fact that the prover knows the Hamiltonian cycle exists (a true statement) but cannot derive (reveal) it within the zero-knowledge protocol.

• Zero-Knowledge Proof: The prover uses a ZKP to convince the verifier that they know a Hamiltonian cycle without disclosing the cycle itself. The verifier becomes convinced of the truth (the graph has a Hamiltonian cycle) without learning the actual cycle (the proof remains undefined to the verifier).

Gödel's Incompleteness Theorems highlight the existence of true but unprovable statements within formal systems, paralleling the way zero-knowledge proofs allow one to demonstrate the truth of a statement without revealing the proof itself. This interplay underscores the philosophical and practical connections between mathematical logic and cryptographic protocols, showcasing how concepts of provability and secrecy intertwine in both fields.

To demonstrate how zero-knowledge proofs (ZKPs) can be considered incomplete in the context of Gödel's Incompleteness Theorems, we can represent the principles of ZKPs using mathematical formulations:

9. Incompleteness Example

Consider the scenario of proving the existence of a Hamiltonian cycle in a graph $G$ :

Let $G$ be a large graph, and let $H$ represent the Hamiltonian cycle within $G$

The prover $P$

knows $H$

exists and wants to convince the verifier $V$

without revealing $H$

10. Gödel's Perspective Applied to ZKPs

In this context, Gödel's Incompleteness Theorems suggest the following:

True but Unprovable Statements: There exists a true statement about $G$ (e.g., it has a Hamiltonian cycle) that cannot be proven within the formal system of graph theory:

$\exists G$ : Hamiltonian (G) is true but not provable

Proof Beyond Formal Systems: The prover $P$

can convince the verifier $V$

of the truth of Hamiltonian $(G)$

without revealing $H: P \rightarrow{ }^{Z K P} V$ : Hamiltonian $(G)$ is true

11. Mathematical Representation of ZKP Incompleteness

Hamiltonian Cycle Existence: Let Hamiltonian $(G)$ denote the existence of a Hamiltonian cycle in graph $G$

If Hamiltonian $(G)$ is true $ightarrow$ Prover $P \rightarrow{ }^{Z K P}$ Verifier V is convinced

Gödel Sentence Analogy: There exists a Gödel sentence $G_{S}$ stating, "This statement is not provable in the system." $G_{S}$ : This statement is not provable ZKP Incompleteness: The prover $P$ knows the Hamiltonian cycle exists (true statement) but cannot provide a formal proof within the system (hidden proof):

$$P \rightarrow{ }^{Z K P} V: \text { Hamiltonian }(G) \text { is true without revealing the proof (cycle) itself }$$

Gödel's Incompleteness Theorems and zero-knowledge proofs share the idea that certain truths (or proofs) can exist and be acknowledged without being formally derived or revealed. The equations above capture this relationship, illustrating how ZKPs allow the verification of truths that remain hidden, resonating with the concept that some truths in formal systems are inherently unprovable.

12. Explicit Description

Example of a zero-knowledge proof (ZKP) corollary where Gödel's Incompleteness Theorems and zero-knowledge proofs is illustrated.

Here is a concise extraction with mathematical focus

Undecidable Statements and Gödel's Incompleteness: Gödel's theorems imply there exist true statements $G$ within a formal system $S$ that cannot be proven within $S$. Let $S$ be a formal system capable of expressing elementary arithmetic.

There exists a statement $G_{S}$ such that $G_{S}$ is true, but $S \nvdash G_{S}$ (i.e., $S$ cannot prove $G_{S}$ ). Zero-Knowledge Proofs (ZKPs) and Incompleteness:

In a ZKP, the prover $P$ aims to convince the verifier $V$ of the truth of a statement $\phi$ without revealing the proof.

Completeness: If $\phi$ is true, $V$ will be convinced by $P$.

{$\forall \phi$ (if $\phi$ is true, then $V$ is convinced by $P$ )}

Soundness: If $\phi$ is false, no cheating $P$ can convince $V$ that $\phi$ is true, except with some small probability $\epsilon$

$$\forall \phi( \ if \phi is false, \ then (P) cannot \ convince (V) with \ probability >\epsilon)$$

Zero-Knowledge: If $\phi$ is true, $V$ learns nothing other than the fact that $\phi$ is true.

Example: Hamiltonian Cycle in Graph Theory:

Let $G=(V, E)$ be a graph, and let $H$ be the statement "Graph $G$ has a Hamiltonian cycle."

Gödel's Perspective: $H$ may represent a true statement about the graph $G$ that cannot be proven within the formal system defining the properties of $G . S \nvdash H$, but $H$ is true.

Zero-Knowledge Proof: The prover $P$ knows a Hamiltonian cycle $C$ in $G \times P$ uses a ZKP to convince $V$

that $G$ has a Hamiltonian cycle without revealing $C$.

$V$ becomes convinced that $H$ is true without learning $C$

13. Conclusion

Gödel's Incompleteness Theorems highlight the existence of true but unprovable statements within formal systems, paralleling the way zero-knowledge proofs allow one to demonstrate the truth of a statement without revealing the proof itself. The equations and concepts above illustrate the mathematical and logical connections between these two areas, showing how provability and secrecy are intertwined.