Miodrag Rašković

Some probability logics with new types of probability operators

We introduce new types of probability operators of the form QF , whereF is a recursive rational subset of [0; 1]. A formulaQF is satisfied in a probability model if the measure of the set of worlds that satisfy is in F . The new operators are suitable for describing events in discrete sample spaces. We provide sound and complete axiomatic systems for a number of probability logics augmented with the QF -operators. We show that the new operators are not definable in languages of probability logics that have been used so far. We study decidability of the presented logics. We describe a relation of ‘being more expressive’ between the new probability logics.

Measures of inconsistency and defaults

We introduce a method for measuring inconsistency based on the number of formulas needed for deriving a contradiction. The relationships to previously considered methods based on probability measures are discussed. Those methods are extended to conditional probability and default reasoning.

A Logic with Conditional Probabilities

The paper presents a logic which enriches propositional calculus with three classes of probabilistic operators which are applied to propositional formulas: P ≥ s(α), CP = s(α, β) and CP ≥ s (α, β), with the intended meaning ”the probability of α is at least s”, ”the conditional probability of α given β is s”, and ”the conditional probability of α given β is at least s”, respectively. Possible-world semantics with a probability measure on sets of worlds is defined and the corresponding strong completeness theorem is proved for a rather simple set of axioms. This is achieved at the price of allowing infinitary rules of inference. One of these rules enables us to syntactically define the range of the probability function. This range is chosen to be the unit interval of a recursive nonarchimedean field, making it possible to define another probabilistic operator CP ≈ 1(α, β) with the intended meaning ”probabilities of α ∧ β and β are almost the same”. This last operator may be used to model default reasoning.

A probabilistic extension of intuitionistic logic

We introduce a probabilistic extension of propositional intuitionistic logic. The logic allows making statements such as P≥sα, with the intended meaning “the probability of truthfulness of α is at least s”. We describe the corresponding class of models, which are Kripke models with a naturally arising notion of probability, and give a sound and complete infinitary axiomatic system. We prove that the logic is decidable.

A probabilistic logic with polynomial weight formulas

The paper presents a sound and strongly complete axiomatization of reasoning about polynomial weight formulas. In addition, the PSPACE decision procedure for polynomial weight formulas developed by Fagin, Halpern and Megiddo works for our logic as well. The introduced formalism allows the expression of qualitative probability statements, conditional probability and Bayesian inference.

A Probabilistic Temporal Logic That Can Model Reasoning about Evidence

The aim of the paper is to present a sound, strongly complete and decidable probabilistic temporal logic that can model reasoning about evidence.

Finitely additive probability measures on classical propositional formulas definable by Gödel's t-norm and product t-norm

Suppose that e is any [0,1]-valued evaluation of the set of propositional letters. Then, e can be uniquely extended to finitely additive probability product and Godels measures on the set of classical propositional formulas. Those measures satisfy that the measure of any conjunction of distinct propositional letters is equal to the product of, or to the minimum of the measures of the propositional letters, respectively. Product measures correspond to the one extreme - stochastic or probability independence of elementary events (propositional letters), while Godels measures correspond to the other extreme - logical dependence of elementary events. Any linear convex combination of a product measure and a Godels measure is also a finitely additive probability measure. In that way infinitely many intermediate measures that corresponds to various degrees of dependence of propositional letters can be generated. Such measures give certain truth-functional flavor to probability, enabling applications to preferential problems, in particular classifications according to predefined criteria. Some examples are provided to illustrate this possibility. We present the proof-theoretical and the model-theoretical approaches to a probabilistic logic which allows reasoning about the mentioned types of probabilistic functions. The logical language enables formalization of classification problems with the corresponding criteria expressible as propositional formulas. However, more complex criteria, for example involving arithmetical functions, cannot be represented in that framework. We analyze the well-known problem proposed by Grabisch to illustrate interpretation of such classification problems in fuzzy logic.

A propositional probabilistic logic with discrete linear time for reasoning about evidence

The aim of the paper is to present a sound, strongly complete and decidable probabilistic temporal logic that can model reasoning about evidence. The formal system developed here is actually a solution of a problem proposed by Halpern and Pucella (J Artif Intell Res 26:1–34, 2006).

Qualitative Possibilities and Necessities

Qualitative possibilities and necessities are well known types of confidence relations. They have been extensively studied semantically, as relations on Boolean algebras (or equivalently, relations on algebras of sets). The aim of this paper is to give a syntactical flavor to the subject providing a sound and complete axiomatization of qualitative possibility relations.

Measure Logic

In this paper we investigate logic which is suitable for reasoning about uncertainty in different situations. A possible-world approach is used to provide semantics to formulas. Axiomatic system for our logic is given and the corresponding strong completeness theorem is proved. Relationships to other systems are discussed.