Aleksandar Perović

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.

Some computational aspects of the brain computer interfaces based on inner music

We discuss the BCI based on inner tones and inner music. We had some success in the detection of inner tones, the imagined tones which are not sung aloud. Rather easily imagined and controlled, they offer a set of states usable for BCI, with high information capacity and high transfer rates. Imagination of sounds or musical tunes could provide a multicommand language for BCI, as if using the natural language. Moreover, this approach could be used to test musical abilities. Such BCI interface could be superior when there is a need for a broader command language. Some computational estimates and unresolved difficulties are presented.

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 p-adic probability logic

In this article we present a p-adic valued probabilistic logic which is a complete and decidable extension of classical propositional logic. The key feature of lies in ability to formally express boundaries of probability values of classical formulas in the field of p-adic numbers via classical connectives and modal-like operators of the form Kr, ρ. Namely, is designed in such a way that the elementary probability sentences Kr, ρα actually do have their intended meaning—the probability of propositional formula α is in the -ball with the center r and the radius ρ. Due to modal nature of the operators Kr, ρ, it was natural to use the probability Kripke like models as -structures, provided that probability functions range over instead of or .

Detection of Structural Features in Biological Signals

In this article structures in biological signals are treated. The simpler—directly visible in the signals, which still demand serious methods and algorithms in the feature detection, similarity investigation and classification. The major actions in this domain are of geometric, thus simpler sort, though there are still hard problems related to simple situations. The other large class of less simple signals unsuitable for direct geometric or statistic approach, are signals with interesting frequency components and behavior, those suitable for spectroscopic analysis. Semantics of spectroscopy, spectroscopic structures and research demanded operations and transformations on spectra and time spectra are presented. The both classes of structures and related analysis methods and tools share a large common set of algorithms, all of which aiming to the full automatization. Some of the signal features present in the brain signal patterns are demonstrated, with the contexts relevant in BCI, brain computer interfaces. Mathematical representations, invariants and complete characterization of structures in broad variety of biological signals are in the central focus.

A logic with conditional probability operators

This paper presents a sound and strongly complete axiomatization of the reasoning about linear combinations of conditional probabilities, including compar- ative statements. The developed logic is decidable, with a PSPACE containment for the decision procedure.

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.

Probabilistic approach to nonmonotonic consequence relations

The paper offers a probabilistic characterizations of determinacy preservation, fragmented disjunction and conditional excluding middle for preferential relations. The paper also presents a preferential relation that is above Disjunctive rationality and strictly below Rational monotonicity. This so called ɛ, µ-relation is constructed using a positive infinitesimal e and a finitely additive hyperreal valued probability measure µ on the set of propositional formulas.