Elsevier

Artificial Intelligence

Volume 299, October 2021, 103524
Artificial Intelligence

Deciding Koopman's qualitative probability

To Petr Hájek, in memoriam
https://doi.org/10.1016/j.artint.2021.103524Get rights and content

Abstract

In their recent paper in this journal, Delgrande, Renne and Sack study qualitative probability and discuss its role in Artificial Intelligence. Building on related work by de Finetti, Scott, Segerberg and others, the authors provide general axioms for qualitative probability. In this paper we investigate Koopman's conditional qualitative probability from the computational viewpoint. An important part of his work, published in the Annals of Mathematics in 1940-1941, deals with finite conjunctions K of statements of the form “the probability of a given h does not exceed the probability of b given k”, with a,b,h,k elements of a boolean algebra. Upon coding these elements by boolean formulas, we provide a decision procedure to check the consistency of any such K. As an immediate consequence, also inferences in Koopman's probability theory are shown to be computable. These problems of qualitative probability theory significantly generalize Boole's (typically quantitative) problem of estimating the possible probabilities of a new event given the probabilities of other events. Boole's classical problem today is known as the optimization version of the probabilistic satisfiability problem PSAT. In 1986 Nilsson published an influential paper on this subject in this journal. The scope of our results is much larger than that of PSAT, because Koopman's conjunctions K also formalize the key notion of independence. Some familiarity with boolean logic and finite boolean algebras is the only prerequisite for this paper.

Section snippets

Introduction: conditional qualitative probability

In their 2019 paper [14] in this journal, the authors investigate a theory of qualitative probability, generalizing the traditional approach where formulas α,β are equipped with a binary relation ≼ stating that α is no more probable than β.

In [14] one also finds a discussion of the role of qualitative probability in Artificial Intelligence. This will not be paraphrased here. Rather, we will only quote the first lines of the abstract of Nilsson's paper [36] published in this journal in 1986:

Koopman's conditional probability, [31–33]

The fundamental viewpoint of the present work is that the primal intuition of probability expresses itself in a (partial) ordering of eventualities: A certain individual at a certain moment considers the propositions a,b,h,k, the meanings of which he apprehends, and which he considers to be determinate (i.e., either true or false) without (in general) knowing whether they are true or false (for the sharpening of these ideas, see §2). Then the phrase

“a on the presumption that h is true is

The bare minimum on boolean algebras

Boolean algebras and their ideals were extensively used by Koopman to frame his qualitative comparative probability. To help the reader, this section is devoted to a succinct exposition of finite boolean algebras, which is all we need in this paper. The relationship between the basic vocabularies of boolean logic and algebra is illustrated in Table 1. Free boolean algebras are just boolean formulas up to logical equivalence. Koopman's terse elementary algebraic setup dispenses with all

The syntax of Koopman's comparative probability

Koopman's universe of discourse is a “boolean ring determined by all the experimental propositions in a given discussion”, [32, p.766]. Since algorithmic issues are not considered in his papers [31], [32], [33], there is no need for Koopman to present his universe of discourse as the Lindenbaum-Tarski algebra determined by pre-conditions ϕ1,,ϕu.

On the contrary, for our decidability results to make sense, all elements of boolean algebras (≅ boolean rings) considered in this paper must be coded

Semantics: consistency and inference in probability

The consistency problem in Koopman's probability has the following formal definition:

INSTANCE: Boolean formulas ϕ1,,ϕu coding the pre-conditions c1,,cu, with 0c1cu. A system K={E1,,Et} of inequalities, with each El of the formp(αlηl)p(κl)lp(βlκl)p(ηl),l{,,>,<},(l=1,,t). It is assumed that all ϕi and αl,ηl,βl,κl are boolean formulas in the variables X1,,Xn for some fixed n=1,2,.

QUESTION: With =c the filter of Fn generated by c=c1cn, does the Lindenbaum-Tarski algebra Fn/

De Finetti's semantics, Boole, and PSAT

Theorem 5.3 shows that Koopman's qualitative probability theory allows the formalization of quantitative probabilistic inferences concerning conditionals and independence. In this section we will discuss the equivalent quantitative operational semantics for Koopman's comparative probability theory provided by de Finetti.

Examples

In this section some examples are given of consistency and inference problems in Koopman's probability. Building on the syntactic/semantic setup of Sections 3 and 4, detailed solutions are provided paralleling the steps in the proof of Theorem 5.1, Theorem 5.3. De Finetti's consistency notion (6.1-6.2) will have a key role in Example 7.3.

To better appreciate the relative simplicity of Koopman's probability, the reader is encouraged to work out these exercises in the framework of other more

Concluding remarks

A comprehensive account of comparative conditional probability would be beyond the scope of this paper. The interested reader is referred to [15], [18], [26] for detailed information. In this final section we will touch upon a few selected topics directly related to Koopman's qualitative probability theory and their relations with de Finetti's work on probability.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The author thanks the three referees and the associate editor for their insightful comments and valuable suggestions for improvement.

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