The Faulting Problem

by Robert Valentine (A.K.A. Robert Chubb)

Faulting Problem Conjecture:

For any software system $𝑆$, there exists at least one input $𝐼$ such that $𝑆$ exhibits unexpected or unintended behavior, implying the presence of at least one bug or vulnerability.

Assume there exists a software system $𝑆$ that is completely bug-free and devoid of vulnerabilities.

Let $𝐵$ represent the set of all possible bugs and vulnerabilities that could potentially exist in $𝑆$.

Let $𝑆$ be represented by a function $𝐹$ that maps inputs $𝐼$ to outputs $𝑂$. Mathematically, $𝐹:𝐼→𝑂$.

If $𝑆$ is truly bug-free and devoid of vulnerabilities, then $𝐵$ is an empty set, implying $𝐵=∅$.

However, due to the complexity of software and the vast space of possible inputs, the space of $𝐼$ is often infinite or exceedingly large.

According to the Halting Problem, there exist inputs $𝐼$ for which it is undecidable whether $𝐹$ will halt or not.

This implies that for some inputs, $𝐹$ may enter an unexpected or unintended state, even in a bug-free system.

Therefore, there must exist at least one input 𝐼 for which $𝐹$ exhibits unexpected behavior, contradicting the assumption that $𝑆$ is completely bug-free.

Hence, $𝑆$ cannot be completely bug-free, and there must exist at least one bug or vulnerability present in $𝑆$ for it to be functional across all possible inputs.

Mathematically, this can be expressed as: $∀𝑆,∃𝐼$ such that $𝐵(𝑆,𝐼)$, where $𝐵(𝑆,𝐼)$ indicates that $𝐼$ reveals a bug in $𝑆$.

Coq Implementation:

From Coq Require Import Classical.

(* Define the type of software *)
Parameter Software : Type.

(* Define the type of inputs *)
Parameter Input : Type.

(* Predicate representing the presence of bugs *)
Parameter B : Software -> Input -> Prop.

(* Predicate for deciding if software has bugs *)
Parameter decides_bug : Software -> Prop.

(* Bug existence problem definitions *)
Definition bug_existence_problem : Prop :=
  (exists s, decides_bug s) -> False.

Definition bug_existence_problem' : Prop :=
  forall s, decides_bug s -> False.

(* Equivalence proof between two formulations of the bug existence problem *)
Theorem statements_equivalent_bug_existence :
  bug_existence_problem <-> bug_existence_problem'.
Proof.
  unfold bug_existence_problem, bug_existence_problem'; split; intros.
  - exact (H (ex_intro decides_bug s H0)).
  - destruct H0.
    exact (H x H0).
Qed.

(* Definition of a bug-free software system *)
Definition bug_free (s : Software) : Prop :=
  forall i, ~ B s i.

(* Main theorem stating that if a software system is not bug-free,
        there exists an input that reveals a bug *)
Theorem bug_existence_theorem :
  forall s : Software,
    ~ bug_free s -> exists i : Input, B s i.
Proof.
  intros s H.
  unfold bug_free in H.
  apply NNPP.
  intro contra.
  apply H.
  intros i.
  intro Hbug.
  apply contra.
  exists i.
  exact Hbug.
Qed.

(* Combined proof that integrates the two results *)
Theorem combined_bug_existence_proof :
  (forall s : Software, ~ bug_free s -> exists i : Input, B s i) /\
  (bug_existence_problem <-> bug_existence_problem').
Proof.
  split.
  - (* Part 1: Prove bug_existence_theorem *)
    intros s H.
    apply bug_existence_theorem.
    exact H.
  - (* Part 2: Prove statements_equivalent_bug_existence *)
    apply statements_equivalent_bug_existence.
Qed.

Conclusion

The Faulting Problem, akin to the classic Halting Problem, emphasizes the inherent complexity in determining whether a given software system is devoid of faults. Analogous to the Halting Problem's focus on ascertaining if a Turing machine will halt or run indefinitely on a given input, the Faulting Problem centers on the infeasibility of devising an algorithm capable of accurately detecting all faults within a software system. Despite advancements in software engineering methodologies, such as formal verification and rigorous testing, the Faulting Problem remains a quintessential challenge. It highlights the intrinsic limitations of computational systems and underscores the need for robust techniques to manage and mitigate software defects. This perspective underscores the critical importance of acknowledging and managing software defects rather than aiming for an unattainable ideal of bug-free software. By embracing this reality, software developers and engineers can adopt more pragmatic strategies for ensuring the reliability, security, and resilience of software systems in the face of inevitable bugs and errors.

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However the whitepaper and Coq implementation are provided here are under the license given.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International License.