# Probabilities on Sentences in an Expressive Logic

@article{Hutter2013ProbabilitiesOS, title={Probabilities on Sentences in an Expressive Logic}, author={Marcus Hutter and John W. Lloyd and Kee Siong Ng and William T. B. Uther}, journal={J. Appl. Log.}, year={2013}, volume={11}, pages={386-420} }

Abstract Automated reasoning about uncertain knowledge has many applications. [...] Key Method We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter)examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed… Expand

#### 24 Citations

A Formal Approach to the Problem of Logical Non-Omniscience

- Computer Science
- TARK
- 2017

How this single criterion implies a number of desirable properties of bounded reasoners is described; for example, logical inductors outpace their underlying deductive process, perform universal empirical induction given enough time to think, and place strong trust in their own reasoning process. Expand

Questions of Reasoning Under Logical Uncertainty

- 2015

A logically uncertain reasoner would be able to reason as if they know both a programming language and a program, without knowing what the program outputs. Most practical reasoning involves some… Expand

Logical Induction

- Computer Science, Mathematics
- Electron. Colloquium Comput. Complex.
- 2016

A computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time, and follows a single logical induction criterion, motivated by a series of stock trading analogies. Expand

Logical Induction (Abridged)

- 2016

We describe a logical induction criterion for algorithms that reason probabilistically about logical facts, such as claims about mathematical conjectures and long-running computer programs. We show… Expand

Logical Induction Abridged version, early draft

- 2016

We present a computable algorithm for assigning probabilities to sentences of logic, such as sentences of the form “this long-running computation outputs 3” or “the twin prime conjecture is true”.… Expand

On Learning to Prove

- Computer Science
- ArXiv
- 2019

This paper introduces a representation of beliefs that assigns probabilities to the exhaustive and mutually exclusive first-order possibilities found in Hintikka's theory of distributive normal forms and suggests an embedding of statements into an associated Hilbert space. Expand

Asymptotic Logical Uncertainty and the Benford Test

- Computer Science, Mathematics
- AGI
- 2016

This work studies the asymptotic properties of beliefs on quickly computable sequences of logical sentences and provides an approach which identifies when such subsequences are indistinguishable from random, and learns their probabilities. Expand

Non-Omniscience, Probabilistic Inference, and Metamathematics

- 2014

We suggest a tractable algorithm for assigning probabilities to sentences of firstorder logic and updating those probabilities on the basis of observations. The core technical difficulty is relaxing… Expand

Dependent Type Networks: A Probabilistic Logic via the Curry-Howard Correspondence in a System of Probabilistic Dependent Types

- 2018

We first introduce a Probabilistic Dependent Type System (PDTS) via a functional language based on a subsystem of intuitionistic type theory including dependent sums and products, which is expanded… Expand

On Nicod's Condition, Rules of Induction and the Raven Paradox

- Computer Science
- ArXiv
- 2013

It is suggested that the informal representation of NC may seem to be intuitively plausible because it can easily be mistaken for reasoning by analogy, and violates a simple kind of inductive inference (namely projectability). Expand

#### References

SHOWING 1-10 OF 57 REFERENCES

Probabilistic Modal Logic

- Computer Science
- AAAI
- 2007

An exact algorithm which takes a probabilistic Kripke stntcture and answers Probabilistic modal queries in polynomial-time in the size of the model is provided and an approximate method for applications in which the authors have very many or infinitely many states is introduced. Expand

Probabilistic Logic

- Computer Science, Mathematics
- Artif. Intell.
- 1986

The method described in the present paper combines logic with probability theory in such a way that probabilistic logical entaihnent reduces to ordinary logical entailment when the probabilities of all sentences are either 0 or 1. Expand

Reasoning about knowledge and probability

- Mathematics, Computer Science
- JACM
- 1994

This work provides a complete axiomatization for reasoning about knowledge and probability, proves a small model property, and obtains decision procedures for adding common knowledge and a probabilistic variant of common knowledge to the language. Expand

Reasoning about uncertainty

- Mathematics, Computer Science
- 2003

This second edition has been updated to reflect Halpern's recent research and includes a consideration of weighted probability measures and how they can be used in decision making. Expand

Stochastic Logic Programs

- 1996

One way to represent a machine learning algorithm's bias over the hypothesis and instance space is as a pair of probability distributions. This approach has been taken both within Bayesian learning… Expand

An Analysis of First-Order Logics of Probability

- Mathematics, Computer Science
- IJCAI
- 1989

This work provides axiom systems that are sound and complete in cases where a complete axiomatization is possible, and shows that they do allow us to capture a great deal of interesting reasoning about probability. Expand

Assigning Probabilities to Logical Formulas

- Mathematics
- 1966

Publisher Summary Probability concepts nowadays are presented in the standard framework of the Kolmogorov axioms. A sample space is given together with an σ-field of subsets, the events, and an… Expand

Probabilistic reasoning in a classical logic

- Computer Science, Mathematics
- J. Appl. Log.
- 2009

The argument rests on the observation that probability densities, being functions, can be represented and reasoned with naturally and directly in (classical) higher-order logic. Expand

Markov logic networks

- Computer Science, Mathematics
- Machine Learning
- 2006

Experiments with a real-world database and knowledge base in a university domain illustrate the promise of this approach to combining first-order logic and probabilistic graphical models in a single representation. Expand

First-Order Probabilistic Languages: Into the Unknown

- Computer Science
- ILP
- 2006

A taxonomy that helps make sense of the profusion of FOPLs that have been proposed over the past fifteen years is provided, including a brief overview of BLOG syntax and semantics, and some of the design decisions that distinguish it from other languages. Expand