The problem of proof identity, and why computer scientists should care about Hilbert's 24th problem

Abstract

In this short overview article, I will discuss the problem of proof identity and explain how it is related to Hilbert's 24th problem. I will also argue that not knowing when two proofs are ‘the same’ has embarrassing consequences not only for proof theory but also for certain areas of computer science where formal proofs play a fundamental role, in particular, the formal verification of software. Then I will formulate a set of four objectives that a satisfactory notion of proof identity should obey. And finally, I discuss Hughes' combinatorial proofs and argue that they can be seen as a first step towards a possible solution to the problem of proof identity. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

On Combinatorial Proofs for Modal Logic

“…We are not claiming to provide a final answer to this question, but we propose an approach based on combinatorial proofs, introduced by Hughes [15,16] to address the question of proof identity for classical propositional logic and Hilbert's 24th problem [31,30]. Via combinatorial proofs, it is finally possible to ask the question of proof identity also for proofs in different proof formalisms; recent research has investigated this for syntactic proofs in sequent calculus [16,15], calculus of structures [29], resolution calculus, and analytic tableaux [1].…”

Combinatorial Proofs and Decomposition Theorems for First-order Logic

“…To address (ii), we also apply a permutation argument, permuting all instances of ac x up until they either reach the top of the derivation or an instance of m which separates the two atoms in the premise. More precisely, we consider the following inference rule We can therefore assume that all instances of ac x , that contract an atom x ∈ VAR are either at the top of Ψ or below a minstance as in (11). We now lift Ψ to {ac, c ∀ , m, m ∀ , m ∃ , ≡}, proceed by induction on the height of Ψ, beginning at the top, making a case analysis on the topmost rule that is not a ≡.…”

“…This is where combinatorial proofs come in. They were introduced by Hughes [9] for classical propositional logic as a syntax-free notion of proof, and as a potential solution to Hilbert's 24th problem [10] (see also [11]). The basic idea is to abstract away from the syntax of the inference rules used in inductively-generated proofs and consider the proof as a combinatorial object, more precisely as a special kind of graph homomorphism.…”

Discussing Hilbert's 24th problem

“…Lutz Straßburger focuses on the particular problem of proof identity which clearly relates to simplicity. In his paper, The problem of proof identity, and why computer scientists should care about Hilbert's 24th problem [51], '4 objectives that a satisfactory notion of proof identity should obey' are formulated, and the problem is discussed in the context of Hughes's notion of combinatorial proofs [27].…”