Logical Induction (1609.03543v5)

Abstract: We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations, and its own probabilities. We show that our algorithm, an instance of what we call a logical inductor, satisfies a number of intuitive desiderata, including: (1) it learns to predict patterns of truth and falsehood in logical statements, often long before having the resources to evaluate the statements, so long as the patterns can be written down in polynomial time; (2) it learns to use appropriate statistical summaries to predict sequences of statements whose truth values appear pseudorandom; and (3) it learns to have accurate beliefs about its own current beliefs, in a manner that avoids the standard paradoxes of self-reference. For example, if a given computer program only ever produces outputs in a certain range, a logical inductor learns this fact in a timely manner; and if late digits in the decimal expansion of $\pi$ are difficult to predict, then a logical inductor learns to assign $\approx 10\%$ probability to "the $n$th digit of $\pi$ is a 7" for large $n$. Logical inductors also learn to trust their future beliefs more than their current beliefs, and their beliefs are coherent in the limit (whenever $\phi \implies \psi$, $\mathbb{P}\infty(\phi) \le \mathbb{P}\infty(\psi)$, and so on); and logical inductors strictly dominate the universal semimeasure in the limit. These properties and many others all follow from a single logical induction criterion, which is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence $\phi$ is associated with a stock that is worth \$1 per share if [...]

• The paper defines logical inductors as computable algorithms that iteratively adjust probabilities for logical statements amid deductive uncertainty.
• It establishes convergence, timely theorem probability assignment, and calibration against deductive shortcomings.
• The study bridges theoretical logic and practical AI by addressing self-referential paradoxes and refining probabilistic reasoning in complex systems.

Logical Induction: Insights on Deductive Limitations and Probabilistic Reasoning

The paper "Logical Induction" authored by Scott Garrabrant, Tsvi Benson-Tilsen, Andrew Critch, Nate Soares, and Jessica Taylor, presents a novel framework for reasoning under logical uncertainty. It introduces the concept of logical inductors, which are computable algorithms that assign probabilistic values to logical expressions within a formal language and refine these probabilities over time. This work addresses the disconnect between probability theory, which manages empirical uncertainty, and deductive processes required for reasoning about logical facts.

Overview and Main Contributions

The core contribution of the paper is a formal definition of the logical induction criterion, which stipulates that a market (or belief sequence) should not be exploitable by any polynomial-time computable strategy. This is a relaxation of classic "no Dutch book" criteria, focusing on deductively limited reasoners. The authors provide an algorithm—Logical Induction Algorithm (LIA)—ensuring that a given market satisfies this criterion, thus serving as a logical inductor.

Logical inductors offer several desirable properties:

• Convergence and Coherence: The beliefs approach a consistent probability distribution in the limit, fulfilling classic Gaifman definitions of logical coherence.
• Timely Learning: They preemptively allocate correct probabilities to efficiently computable sequences of theorems well before formal proofs are derived.
• Calibration: Logical inductors remain statistically unbiased provided they receive adequate feedback from a deductive process.
• Learning Statistical Patterns: They capture pseudorandom sequences, enabling accurate statistical summaries for inherently unpredictable sequences.

Bold and Contradictory Claims

One key assertion is the ability of logical inductors to refine beliefs in a way that strictly dominates the universal semimeasure. This claim highlights the theoretical superiority of logical inductors in assigning probabilities even to conjunctions of independent logical statements—a task problematic for semimeasures due to convergence to zero probabilities on infinite conjunctions.

Furthermore, logical inductors propose solutions to classical paradoxes of self-reference and diagonalization, providing resistant mechanisms by forcing traders to operate continuously over the market's pricing function. By doing so, they can uphold coherent reasoning in contexts where paradoxical sentences exist.

Practical and Theoretical Implications

Implications of this research are profound both in theoretical and practical dimensions. Theoretically, logical induction provides a groundwork for understanding how deductively limited reasoners might develop rational beliefs and manage self-reference without complete information or infinite deduction capacity. This addresses significant meta-mathematical queries about reasoning systems capable of achieving forms of self-trust and coherence.

On a practical level, ensemble methods inspired by logical induction can aid fields requiring robust inductive mechanisms beyond empirical data. AI systems tasked with modeling self-referential or hypothetically counterfactual scenarios stand to benefit, deriving insights from logical inductors on how to allocate computational resources effectively toward complex logical relationships while maintaining probabilistic integrity.

Future Research Directions

Open questions outlined pertain to extending logical induction to efficiently guide resource allocations in decision-making, refine counterpossible reasoning, and improve the practical efficiency of logical induction-inspired algorithms. Exploring how logical inductors influence design principles in AI and frameworks for counterfactual modeling presents a promising research avenue.

Conclusion

This paper defines a significant frontier in aligning probabilistic reasoning with logical deduction limitations, providing tools for understanding logical uncertainty analogous to Solomonoff's inductive framework for empirical uncertainty. Logical induction not only bridges the gap between logic and probability theory but also sets the stage for novel applications in computational and epistemic domains.