Zoran Marković

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.

On the structure of kripke models of heyting arithmetic

Since in Heyting Arithmetic (HA) all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic (PA)? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some partial answers to these questions were obtained in [6], [3], [1] and [2]. Here we present some results in the same direction, announced in [7]. In particular, it is proved that the classical structures at the nodes of a Kripke model of HA must be models of IΔ1 (PA- with induction for provably Δ1 formulas) and that the relation between these classical structures must be that of a Δ1-elementary submodel. MSC: 03F30, 03F55.

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.

How to Restore Compactness into Probabilistic Logics

The paper offers a proof of the compactness theorem for the