This paper presents a sound and strongly complete axiomatization of the reasoning about linear co... more 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.
We show that certain families of sets and functions related to a countable structure A are analyt... more We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alone, that cardinality of every such uncountable set is 2 ℵ 0 .
ABSTRACT Arguably, [0,1]-valued evaluation of formulas is dominant form of representation of unce... more ABSTRACT Arguably, [0,1]-valued evaluation of formulas is dominant form of representation of uncertainty, believes, preferences and so on despite some theoretical issues - most notable one is incompleteness of any unrestricted finitary formalization. We offer an infinitary propositional logic (formulas remain finite strings of symbols, but we use infinitary inference rules with countably many premises, primarily in order to address the incompleteness issue) which is expressible enough to capture finitely additive probabilistic evaluations, some special cases of truth functionality (evaluations in Lukasiewicz, product, Gödel and ŁΠ1/2 logics) and the usual comparison of such evaluations. The main technical result is the proof of completeness theorem (every consistent set of formulas is satisfiable).
The paper offers a probabilistic characterizations of determinacy preservation, fragmented disjun... more 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 ε,μ-relation is constructed using a positive infinitesimal ε and a finitely additive hyperreal valued probability measure μ on the set of propositional formulas.
Annals of Mathematics and Artificial Intelligence, 2012
The aim of the paper is to present a sound, strongly complete and decidable probabilistic tempora... more 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: 2006).
ABSTRACT Spatiotemporal databases can be used to efficiently store and retrieve information about... more ABSTRACT Spatiotemporal databases can be used to efficiently store and retrieve information about objects moving in space and time. Probabilities are added to model the case where the locations are not known with certainty. A few years ago a new formalism was introduced to represent such information in the form of atomic formulas, each of which represents the probability (in the form of an interval because even the probabilities are not known precisely) that a particular object is in a particular location at a particular time. We extend this formalism to obtain several different probabilistic logics by adding logical operators. Furthermore, we axiomatize these logics, provide corresponding semantics, prove that the axiomatizations are sound and complete, and discuss decidability issues. While we relate these logics to previous axiomatizations of probabilistic logics, this article is self-contained: no prior knowledge of probabilistic logics is assumed.
Primarily guided with the idea to express zero-time transitions by means of temporal propositiona... more Primarily guided with the idea to express zero-time transitions by means of temporal propositional language, we have developed a temporal logic where the time flow is isomorphic to ordinal ω 2 (concatenation of ω copies of ω). If we think of ω 2 as lexicographically ordered ω×ω, then any particular zero-time transition can be represented by states whose indices are all elements of some {n} × ω. In order to express noninfinitesimal transitions, we have introduced a new unary temporal operator [ω] (ω-jump), whose effect on the time flow is the same as the effect of α → α + ω in ω 2 . In terms of lexicographically ordered ω × ω, [ω]φ is satisfied in i, j -th time instant iff φ is satisfied in i + 1, 0 -th time instant. Moreover, in order to formally capture the natural semantics of the until operator U, we have introduced a local variant u of the until operator. More precisely, φ uψ is satisfied in i, j -th time instant iff ψ is satisfied in i, j + k -th time instant for some nonnegative integer k, and φ is satisfied in i, j + l -th time instant for all 0 l < k. As in many of our previous publications, the leitmotif is the usage of infinitary inference rules in order to achieve the strong completeness.
ABSTRACT We introduce a Hilbert-style first-order dynamic probability logic and prove the strong ... more ABSTRACT We introduce a Hilbert-style first-order dynamic probability logic and prove the strong completeness theorem for the class of rigid measurable models.
The aim of the paper is to present a sound, strongly complete and decidable probabilistic tempora... more The aim of the paper is to present a sound, strongly complete and decidable probabilistic temporal logic that can model reasoning about evidence.
International Journal of Approximate Reasoning, 2010
We introduce a method for measuring inconsistency based on the number of formulas needed for deri... more 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.
This paper presents a sound and strongly complete axiomatization of the reasoning about linear co... more 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.
We show that certain families of sets and functions related to a countable structure A are analyt... more We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alone, that cardinality of every such uncountable set is 2 ℵ 0 .
ABSTRACT Arguably, [0,1]-valued evaluation of formulas is dominant form of representation of unce... more ABSTRACT Arguably, [0,1]-valued evaluation of formulas is dominant form of representation of uncertainty, believes, preferences and so on despite some theoretical issues - most notable one is incompleteness of any unrestricted finitary formalization. We offer an infinitary propositional logic (formulas remain finite strings of symbols, but we use infinitary inference rules with countably many premises, primarily in order to address the incompleteness issue) which is expressible enough to capture finitely additive probabilistic evaluations, some special cases of truth functionality (evaluations in Lukasiewicz, product, Gödel and ŁΠ1/2 logics) and the usual comparison of such evaluations. The main technical result is the proof of completeness theorem (every consistent set of formulas is satisfiable).
The paper offers a probabilistic characterizations of determinacy preservation, fragmented disjun... more 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 ε,μ-relation is constructed using a positive infinitesimal ε and a finitely additive hyperreal valued probability measure μ on the set of propositional formulas.
Annals of Mathematics and Artificial Intelligence, 2012
The aim of the paper is to present a sound, strongly complete and decidable probabilistic tempora... more 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: 2006).
ABSTRACT Spatiotemporal databases can be used to efficiently store and retrieve information about... more ABSTRACT Spatiotemporal databases can be used to efficiently store and retrieve information about objects moving in space and time. Probabilities are added to model the case where the locations are not known with certainty. A few years ago a new formalism was introduced to represent such information in the form of atomic formulas, each of which represents the probability (in the form of an interval because even the probabilities are not known precisely) that a particular object is in a particular location at a particular time. We extend this formalism to obtain several different probabilistic logics by adding logical operators. Furthermore, we axiomatize these logics, provide corresponding semantics, prove that the axiomatizations are sound and complete, and discuss decidability issues. While we relate these logics to previous axiomatizations of probabilistic logics, this article is self-contained: no prior knowledge of probabilistic logics is assumed.
Primarily guided with the idea to express zero-time transitions by means of temporal propositiona... more Primarily guided with the idea to express zero-time transitions by means of temporal propositional language, we have developed a temporal logic where the time flow is isomorphic to ordinal ω 2 (concatenation of ω copies of ω). If we think of ω 2 as lexicographically ordered ω×ω, then any particular zero-time transition can be represented by states whose indices are all elements of some {n} × ω. In order to express noninfinitesimal transitions, we have introduced a new unary temporal operator [ω] (ω-jump), whose effect on the time flow is the same as the effect of α → α + ω in ω 2 . In terms of lexicographically ordered ω × ω, [ω]φ is satisfied in i, j -th time instant iff φ is satisfied in i + 1, 0 -th time instant. Moreover, in order to formally capture the natural semantics of the until operator U, we have introduced a local variant u of the until operator. More precisely, φ uψ is satisfied in i, j -th time instant iff ψ is satisfied in i, j + k -th time instant for some nonnegative integer k, and φ is satisfied in i, j + l -th time instant for all 0 l < k. As in many of our previous publications, the leitmotif is the usage of infinitary inference rules in order to achieve the strong completeness.
ABSTRACT We introduce a Hilbert-style first-order dynamic probability logic and prove the strong ... more ABSTRACT We introduce a Hilbert-style first-order dynamic probability logic and prove the strong completeness theorem for the class of rigid measurable models.
The aim of the paper is to present a sound, strongly complete and decidable probabilistic tempora... more The aim of the paper is to present a sound, strongly complete and decidable probabilistic temporal logic that can model reasoning about evidence.
International Journal of Approximate Reasoning, 2010
We introduce a method for measuring inconsistency based on the number of formulas needed for deri... more 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.
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Papers by Dragan Doder