It has traditionally been argued that the value of information, where known in advance, can never be less than zero, because decision-making can always ignore extraneous information, proceeding as if such information were not available. In this paper we follow a formal approach that suggests that inquiring after or figuring out (as opposed to being given in advance) the value of information with a view to shortening (and therefore speeding-up) the length of mathematical proofs (in propositional logic), may be as hard as solving the problem at hand. This indicates that there may be a fundamental tradeoff between information value and computational (e.g. decision) time in general. Intuitively, this seems to make more sense than the usual definition of the value of information, which assumes that information can easily be ignored when identified as useless. In this paper we show that this is not so in a specific formal system, and that this case may exemplify a generalized phenomenon of a more fundamental character that has hitherto been overlooked.

**Rare Speed-up in Automatic Theorem Proving Reveals Tradeoff Between Computational Time and Information Value**

Santiago Hernández-Orozco, Francisco Hernández-Quiroz, Hector Zenil, Wilfried Sieg

We show that strategies implemented in automatic theorem proving involve an interesting tradeoff between execution speed, proving speedup/computational time and usefulness of information. We advance formal definitions for these concepts by way of a notion of normality related to an expected (optimal) theoretical speedup when adding useful information (other theorems as axioms), as compared with actual strategies that can be effectively and efficiently implemented. We propose the existence of an ineluctable tradeoff between this normality and computational time complexity. The argument quantifies the usefulness of information in terms of (positive) speed-up. The results disclose a kind of no-free-lunch scenario and a tradeoff of a fundamental nature. The main theorem in this paper together with the numerical experiment—undertaken using two different automatic theorem provers AProS and Prover9 on random theorems of propositional logic—provide strong theoretical and empirical arguments for the fact that finding new useful information for solving a specific problem (theorem) is, in general, as hard as the problem (theorem) itself.

The paper is available here.