Logics with Probability Operators

Transcription

1 Logics with Probability Operators Zoran Ognjanovi and Dragan Doder Mathematical Institute SANU and IRIT, Universite Paul Sabatier August 6, 2018 Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

2 About the course Goal of the course Probability is one of the most popular tools for reasoning about uncertainty in a quantitative way Modal logic is a powerful tool for reasoning about computation and (qualitative) uncertainty Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

3 About the course Goal of the course Probability is one of the most popular tools for reasoning about uncertainty in a quantitative way Modal logic is a powerful tool for reasoning about computation and (qualitative) uncertainty Combining probability and modal approach: Philosophy Articial Intelligence Computer Science Economics Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

4 About the course Goal of the course Probability is one of the most popular tools for reasoning about uncertainty in a quantitative way Modal logic is a powerful tool for reasoning about computation and (qualitative) uncertainty Combining probability and modal approach: Philosophy Articial Intelligence Computer Science Economics The goal of this course is to show how to combine those two approaches into a powerful formalism for reasoning about uncertainty. Syntax, semantics, axiomatic systems, completeness theorem, decidability... Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

5 About the course Outline of the course (tentative) Lecture 1: Historical roots and basic concepts: The ideas of Leibniz and Boole, work of Keisler and Nilsson. The key concepts of probability logics (syntax, semantics and satisability relation). Lecture 2: Basic probability logics: Propositional, without iterations of probability operators. Logic with linear weight formulas (FHM), the fuzzy approach (HGE), ω-rules. The axiomatization techniques, decidability. Lecture 3: Probabilistic modal logic: Reasoning about higher-order probabilities. First order case. Lecture 4: Extensions of probability logics: new types of operators (conditional probability, imprecise probabilities, qualitative probability, independence); alternative ranges of probability functions. Lecture 5: Probability and other modalities: Temporal probability logics, probabilistic common knowledge. Dealing with transitive closure operators. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

6 About the course What is not the topic of the course Probability valued logics Fuzzy logics non-monotonic reasoning Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

7 About the course Based on the book Ognjanovic et.al. Probability Logics Probability-Based Formalization of Uncertain Reasoning Springer Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

8 Roots of Probability logic Mathematical logic and Probability For a long time, logic and probability are developed in parallel: Gotfried W. Leibnitz ( ), Jacobus Bernoulli ( ), Johann Bernoulli ( ), Thomas Bayes ( ), Johann Heinrich Lambert ( ), Bernard Bolzano ( ), Pierre Simon de Laplace ( ) Augustus De Morgan ( ), John Venn ( ), Hugh MacColl ( ), Charles S. Pierce ( ) George Boole, An Investigation into the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities, Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

9 Roots of Probability logic Mathematical logic and Probability For a long time, logic and probability are developed in parallel: Gotfried W. Leibnitz ( ), Jacobus Bernoulli ( ), Johann Bernoulli ( ), Thomas Bayes ( ), Johann Heinrich Lambert ( ), Bernard Bolzano ( ), Pierre Simon de Laplace ( ) Augustus De Morgan ( ), John Venn ( ), Hugh MacColl ( ), Charles S. Pierce ( ) George Boole, An Investigation into the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities, The end of XIX and in XX century big progress in the development in both of the elds (Gottlob Frege, Bertrand Russell, Kurt Goedel classical rst-order logic, Andrey Kolmogorov probability theory). The independent developments of the elds begin Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

10 Roots of Probability logic Mathematical logic and Probability For a long time, logic and probability are developed in parallel: Gotfried W. Leibnitz ( ), Jacobus Bernoulli ( ), Johann Bernoulli ( ), Thomas Bayes ( ), Johann Heinrich Lambert ( ), Bernard Bolzano ( ), Pierre Simon de Laplace ( ) Augustus De Morgan ( ), John Venn ( ), Hugh MacColl ( ), Charles S. Pierce ( ) George Boole, An Investigation into the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities, The end of XIX and in XX century big progress in the development in both of the elds (Gottlob Frege, Bertrand Russell, Kurt Goedel classical rst-order logic, Andrey Kolmogorov probability theory). The independent developments of the elds begin In the 70's and 80's a form or reunion through probability quantiers and operators Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

11 Roots of Probability logic Leibnitz Figure: The blogger of his age: Gottfried Leibniz ( ) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

12 Roots of Probability logic Leibnitz Leibnitz (1) Dissertation (1665): he used numbers from [0, 1] to represent legal conditional rights 0 and 1 denote non-existence of rights and absolute rights, respectively; fractions stand for dierent degrees of certainty Leibnitz gave a denition of probability, relaying on equally possible cases, as the ratio of favorable cases to the total number of cases. Moral certainty something which is innitely probable. Probability, or better said probability logic, had the central role in his attempts to create a powerful universal calculus. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

13 Roots of Probability logic Leibnitz Leibnitz (2) Leibnitz is particularly focused on analogies between the processes of thinking and computation. Thomas Hobbes, everywhere a profound examiner of principles, rightly stated that everything done by our mind is a computation Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

14 Roots of Probability logic Leibnitz Leibnitz (2) Leibnitz is particularly focused on analogies between the processes of thinking and computation. Thomas Hobbes, everywhere a profound examiner of principles, rightly stated that everything done by our mind is a computation Starting from his thesis, Leibnitz publicized ideas to develop a doctrine - or a new kind of logic - of de gradibus probabilitatis (degrees of probability) He used the new ideas in a debate about inheritance of throne of Poland in Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

15 Roots of Probability logic Leibnitz Leibnitz (2) New Essays on Human Understanding, nished in 1704 (translations): (66) As for the inevitability of the result, and degrees of probability, we don't yet possess the branch of logic that would let them be estimated. (372) Perhaps opinion based on likelihood also deserves the name of knowledge; otherwise nearly all historical knowledge will collapse, and a good deal more... probability or likelihood is broader: it must be drawn from the nature of things; and the opinion of weighty authorities is one of the things which can contribute to the likelihood of an opinion, but it does not produce the entire likelihood by itself. (464) The entire form of judicial procedures is, in fact, nothing but a kind of logic, applied to legal questions. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

16 Roots of Probability logic Leibnitz Leibnitz (3) (466) I have said more than once that we need a new kind of logic, concerned with degrees of probability, since Aristotle in his Topics couldn't have been further from it. He is satised with arranging a few familiar rules according to common patterns; these could serve on the occasion when one is concerned with amplifying a discourse so as to give it some likelihood. No eort is made to provide a balance necessary to weight the likelihoods in order to obtain a rm judgment. Anyone wanting to deal with this question would do well to pursue the investigation of games of chance. In general, I wish that some skillful mathematician were interested in producing a detailed study of all kinds of games, carefully reasoned and with full particulars. This would be of great value in improving discovery-techniques, since the human mind appears to better advantage in games than in the most serious pursuits. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

17 Roots of Probability logic Leibnitz Leibnitz (4) Once realized, all these ideas would lead to a formal system - universal language and a powerful logical calculus - which could be the basis for all sciences and replace arguments by formal computation: If this is done, whenever controversies arise, there will be no more need for arguing among two philosophers than among two mathematicians. For it will suce to take pens into the hand and to sit down by the abacus, saying to each other (and if they wish also to a friend called for help): Let us calculate! Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

18 Roots of Probability logic Bolzano Bernhard Placidus Johann Nepomuk Bolzano ( ) Bernard Bolzano, Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik, Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

19 Roots of Probability logic Bolzano Wissenschaftslehre: describes probability as a part of logic Bolzano relates deductive inference and conditional probability. Deductive consequence and incompatibility are two extremes of conditional probability. The notion of validity: degree of validity of A(x) (its absolute probability) the usual ratio of favorable cases to the total number of cases x U : A(x) U Relative validity of a proposition M with respect to a set of propositions A, B, C,... was actually the conditional probability P(M A B C...) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

20 Roots of Probability logic De Morgan Augustus De Morgan ( ) A. De Morgan, Formal logic, or, The calculus of inference, necessary and probable, Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

21 Roots of Probability logic De Morgan De Morgan considered probabilistic inference as a part of logic The old doctrine of modals is made to give place to the numerical theory of probability. Many will object to this theory as extralogical... I cannot understand why the study of the eect which partial belief of the premises produces with respect to the conclusion, should be separated from that of the consequences of supposing the former to be absolutely true Subjective, epistemic approach to probability. By degree of probability we really mean, or ought to mean, degree of belief... I will take it then that all the grades of knowledge, from knowledge of impossibility to knowledge of necessity, are capable of being quantitatively conceived He discussed uncertain reasoning with necessary valid inferences (arguments) and probable premises (testimonies) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

22 Roots of Probability logic Boole George Boole ( ) G. Boole, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

23 Roots of Probability logic Boole An investigation of the laws of thought almost half of the text was devoted to the relationship between logic and probability. The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, nally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind... The general doctrine and method of Logic above explained form also the basis of a theory and corresponding method of Probabilities... Hence the subject of Probabilities belongs equally to the science of Number and to that of Logic. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

24 Roots of Probability logic Boole Boole's understanding of probability was that it is founded upon partial knowledge about the relative frequency of occurrences of events. Instead of considering the numerical probability of the occurrence of an event, Boole often expressed it as the probability of the truth of the proposition declaring that the event will occur. The goal of probability theory can be dened as: for a given probability of an event, nd the probability of another, related event. Hailperin: Boole developed probabilistic inference, from probabilistic assumptions derive probability of the conclusion. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

25 Roots of Probability logic Boole Principles Boole's principles (mostly taken from Laplace) the basis for the above mentioned doctrine. They corresponded to additivity of probabilities, probabilities of (in)dependent events, Bayes' theorem for a priori equally probable causes, etc. Conditional probability: 4th. The probability that if an event, E, take place, an event, F, will also take place, is equal to the probability of the concurrence of the events E and F, divided by the probability of the occurrence of E. By combining these principles one can calculate the probability of a compound event which depends on a set of independent events with known probabilities. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

26 Roots of Probability logic Probability logic in the 20th century 20th century Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

27 Roots of Probability logic Probability logic in the 20th century 20th century Gaifman, Scott, and Krauss: probabilities on rst-order sentences Keisler: probability quantiers Probabilities in the framework of modal logics Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

28 Roots of Probability logic Probability logic in the 20th century Jerome Keisler Px 1 2 a probabilistic quantier M, µ = (Px r)φ(x) i µ({a : M = φ(a)}) r Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

29 Roots of Probability logic Probability logic in the 20th century Jerome Keisler Px 1 2 a probabilistic quantier M, µ = (Px r)φ(x) i µ({a : M = φ(a)}) r nitary, innitary languages, axiomatizations, completeness Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

30 Roots of Probability logic Probability logic in the 20th century Jerome Keisler Px 1 2 a probabilistic quantier M, µ = (Px r)φ(x) i µ({a : M = φ(a)}) r nitary, innitary languages, axiomatizations, completeness H. J. Keisler. Hypernite model theory. In: R. O. Gandy, J. M. E. Hyland (editors), Logic Colloquim 76, North-Holland H. J. Keisler. Probability quantiers. In: J. Barwise and S. Feferman (editors), Model Theoretic Logics, Springer-Verlag, Berlin Douglas N. Hoover. Probability logic. Annals of mathematical logic, 14, Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

31 Roots of Probability logic Probability logic in the 20th century N. Nilsson, Probabilistic logic, Articial intelligence, Nilsson asked for a generalization of classical logic to deal with uncertain knowledge. Knowledge base: if A 1 then B 1 if A 2 then B 2 if A 3 then B 3... Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

32 Roots of Probability logic Probability logic in the 20th century N. Nilsson, Probabilistic logic, Articial intelligence, Nilsson asked for a generalization of classical logic to deal with uncertain knowledge. Knowledge base: if A 1 then B 1 (cf c 1 ) if A 2 then B 2 (cf c 2 ) if A 3 then B 3 (cf c 3 )... Uncertain knowledge: from statistics, our experiences and beliefs, etc. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

33 Roots of Probability logic Probability logic in the 20th century N. Nilsson, Probabilistic logic, Articial intelligence, Nilsson asked for a generalization of classical logic to deal with uncertain knowledge. Knowledge base: if A 1 then B 1 (cf c 1 ) if A 2 then B 2 (cf c 2 ) if A 3 then B 3 (cf c 3 )... Uncertain knowledge: from statistics, our experiences and beliefs, etc. To check consistency of (nite) sets of sentences. To deduce probabilities of conclusions from uncertain premises. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

34 Roots of Probability logic Probability logic in the 20th century Probability operators The probability logics allow strict reasoning about probabilities using well-dened syntax and semantics. Example: the probability that Tweety ies is at least 0.75 P 0.75 flies Tweety Formulas in these logics remain either true or false. Formulas do not have probabilistic (numerical) truth values. Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

35 Roots of Probability logic Probability logic in the 20th century Probability quantiers vs. probability operators the probability that birds y is 0.75 the probability that A ies is 0.75 (Px = 0.75)Fly(x) P =0.75 flies A M = (Px = 0.75)Fly(x) i µ({d : M = F (d)}) = 0.75 M = P =0.75 flies A i µ({w : w = flies A }) = 0.75 Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

36 Roots of Probability logic Probability logic in the 20th century Early papers H. Gaifman. A Theory of Higher Order Probabilities. In: Proceedings of the Theoretical Aspects of Reasoning about Knowledge (edts. J.Y. Halpern), Morgan-Kaufmann, San Mateo, California, M. Fattorosi-Barnaba and G. Amati. Modal operators with probabilistic interpretations I. Studia Logica 46(4), R. Fagin, J. Halpern and N. Megiddo. A logic for reasoning about probabilities. Information and Computation 87(1-2): M. Ra²kovi. Classical logic with some probability operators. Publications de l'institut Mathématique, n.s. 53(67), R. Fagin and J. Halpern. Reasoning about knowledge and probability. Journal of the ACM, 41(2):340367, A. Frish and P. Haddawy. Anytime deduction for probabilistic logic. Articial Intelligence 69, Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

37 Logics with probability operators Logics with probability operators Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

38 Logics with probability operators Logic Logical language, formulas axiom system (axioms, rules) semantics (models, satisability) proofs consequence relation Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

39 Logics with probability operators Syntax and Semantics Language of propositional logic Unary operators: P s, s Q [0, 1] α - a classical propositional formula Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

40 Logics with probability operators Syntax and Semantics Language of propositional logic Unary operators: P s, s Q [0, 1] α - a classical propositional formula Basic probabilistic formula: P s α Probabilistic formula = a Boolean combination of basic probabilistic formulas P <s α means P s α,... Example: P <r α P s β P <t (α β), p P s P =t (p q), Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

41 Logics with probability operators Syntax and Semantics Language of propositional logic Unary operators: P s, s Q [0, 1] α - a classical propositional formula Basic probabilistic formula: P s α Probabilistic formula = a Boolean combination of basic probabilistic formulas P <s α means P s α,... Example: P <r α P s β P <t (α β), p P s P =t (p q), Extensions: Richer language: linear / polynomial combinations of probabilities FO logic iterations of operators changing the background logic changing the co-domain of probabilities Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

42 Logics with probability operators Syntax and Semantics Semantics (1) Modal logic: α necessarily α α possibly α P s α somewhere between Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

43 Logics with probability operators Syntax and Semantics Semantics (1) Modal logic: α necessarily α α possibly α P s α somewhere between Possible world semantics M = W, R, v : W a set worlds, R W W accessibility relation v a valuation Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

44 Logics with probability operators Syntax and Semantics Semantics (1) Modal logic: α necessarily α α possibly α P s α somewhere between Possible world semantics M = W, R, v : W a set worlds, R W W accessibility relation v a valuation Satisability: w = α i u = α for every u s.t. wru w = α i u = α for some u s.t. wru w = P s α if µ({u wru}) s Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

45 Logics with probability operators Syntax and Semantics Semantics (2) A probability space M = S, H, µ : S is a nonempty set H P(S) is a σ-algebra: H and closed under complements and countable unions. (closeness under nite unions an algebra) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

46 Logics with probability operators Syntax and Semantics Semantics (2) A probability space M = S, H, µ : S is a nonempty set H P(S) is a σ-algebra: H and closed under complements and countable unions. (closeness under nite unions an algebra) µ : H [0, 1] is a probability measure: µ(s) = 1 µ( + n=1 An) = + n=1 µ(an), whenever {An}+ n=1 is a collection of pairwise disjoint sets. (countable additivity) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

47 Logics with probability operators Syntax and Semantics Semantics (2) A probability space M = S, H, µ : S is a nonempty set H P(S) is a σ-algebra: H and closed under complements and countable unions. (closeness under nite unions an algebra) µ : H [0, 1] is a probability measure: µ(s) = 1 µ( + n=1 An) = + n=1 µ(an), whenever {An}+ n=1 is a collection of pairwise disjoint sets. (countable additivity) µ(a B) = µ(a) + µ(b), whenever A and B are disjoint sets. additivity) (nite Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

48 Logics with probability operators Syntax and Semantics Semantics (2) A probability space M = S, H, µ : S is a nonempty set H P(S) is a σ-algebra: H and closed under complements and countable unions. (closeness under nite unions an algebra) µ : H [0, 1] is a probability measure: µ(s) = 1 µ( + n=1 An) = + n=1 µ(an), whenever {An}+ n=1 is a collection of pairwise disjoint sets. (countable additivity) µ(a B) = µ(a) + µ(b), whenever A and B are disjoint sets. additivity) (nite Finitary language focus on nite additivity! P( + n=1 α n) = + n=1 P(α n) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

49 Logics with probability operators Syntax and Semantics Semantics (3) A probability model M = W, H, µ, v : W is a nonempty set of elements called worlds, H is an algebra of subsets of W, µ : H [0, 1] is a nitely additive probability measure, and v : W Var {, } is a valuation Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

50 Logics with probability operators Syntax and Semantics Semantics (3) A probability model M = W, H, µ, v : W is a nonempty set of elements called worlds, H is an algebra of subsets of W, µ : H [0, 1] is a nitely additive probability measure, and v : W Var {, } is a valuation Measurable models α For C [α] = {w W : w = α} [α] H Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

51 Logics with probability operators Syntax and Semantics Satisability M = P s α if µ([α] M ) s, M = α if M = α, M = α β if M = α and M = β. A set of formulas F = {α 1, α 2,...} is satisable if there is a model M: M = α i for i = 1, 2,... Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

52 Logics with probability operators Syntax and Semantics Satisability M = P s α if µ([α] M ) s, M = α if M = α, M = α β if M = α and M = β. A set of formulas F = {α 1, α 2,...} is satisable if there is a model M: M = α i for i = 1, 2,... Remark: Iterations of probability operators every world is equipped with a probability measure! Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

53 Logics with probability operators Axiomatization Logic Logical language, formulas axiom system (axioms, rules) semantics (models, satisability) proofs consequence relation Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

54 Logics with probability operators Axiomatization Logic Logical language, formulas axiom system (axioms, rules) semantics (models, satisability) proofs consequence relation T α vs T = α Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

55 Logics with probability operators Axiomatization Logical issues (1) Providing a sound and complete axiom system simple completeness (every consistent formula is satisable, = A i A) extended completeness (every consistent set of formulas is satisable) Decidability (there is a procedure which decides if an arbitrary formula formula is valid) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

56 Logics with probability operators Axiomatization Logical issues (1) Providing a sound and complete axiom system simple completeness (every consistent formula is satisable, = A i A) extended completeness (every consistent set of formulas is satisable) Decidability (there is a procedure which decides if an arbitrary formula formula is valid) Compactness (a set of formulas is satisable i every nite subset is satisable) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

57 Logics with probability operators Axiomatization Logical issues (2) Inherent non-compactness: F = { P =0 p} {P <1/n p : n is a positive integer} Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

58 Logics with probability operators Axiomatization Logical issues (2) Inherent non-compactness: F = { P =0 p} {P <1/n p : n is a positive integer} F k = { P =0 p, P <1/1 p, P <1/2 p,..., P <1/k p} c: 0 < c < 1 k, µ[p] = c M satises F k, but does not satisfy F Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

59 Logics with probability operators Axiomatization Logical issues (2) Inherent non-compactness: F = { P =0 p} {P <1/n p : n is a positive integer} F k = { P =0 p, P <1/1 p, P <1/2 p,..., P <1/k p} c: 0 < c < 1 k, µ[p] = c M satises F k, but does not satisfy F nitary axiomatization + extended completeness compactness nitary axiomatization for real valued probability logics: there are consistent sets that are not satisable Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

60 Logics with probability operators Axiomatization Choices (1) Finitary axiomatization [simple completeness] Restrictions on ranges of probabilities: {0, 1 n, 2 n,..., n 1 n, 1} Innitary axiomatization Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

61 Logics with probability operators Axiomatization Choices (2) Describing nite additivity: With a formula Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

62 Logics with probability operators Axiomatization Choices (2) Describing nite additivity: With a formula An extension of the language (linear weight formula) If (α β) is a propositional tautology, then w(α β) = w(α) + w(β) Fagin, Halpern, Megiddo. A logic for reasoning about probabilities. Information and Computation Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

63 Logics with probability operators Axiomatization Choices (2) Describing nite additivity: With a formula An extension of the language (linear weight formula) If (α β) is a propositional tautology, then w(α β) = w(α) + w(β) Fagin, Halpern, Megiddo. A logic for reasoning about probabilities. Information and Computation With a set of formulas If (α β) is a propositional tautology, then (P =r α P =s β) P =r+s (α β) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

64 Logics with probability operators Extensions Extensions Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

65 Logics with probability operators Extensions Extensions Iterations of probability operators P s P t α, β P s α Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

66 Logics with probability operators Extensions Extensions Iterations of probability operators P s P t α, β P s α First order case ( x)p 1/2 α(x) Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

67 Logics with probability operators Extensions Extensions Iterations of probability operators P s P t α, β P s α First order case ( x)p 1/2 α(x) New probability operators α β CP s (β, α) α β L s α, U s α Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

68 Logics with probability operators Extensions Extensions Iterations of probability operators P s P t α, β P s α First order case ( x)p 1/2 α(x) New probability operators α β CP s (β, α) α β L s α, U s α Combining probability and other modalities Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40

69 Logics with probability operators Extensions Extensions Iterations of probability operators P s P t α, β P s α First order case ( x)p 1/2 α(x) New probability operators α β CP s (β, α) α β L s α, U s α Combining probability and other modalities Fuzzy probability operator Ognjanovi, Doder (SANU, IRIT) Logics with probability operators 6/8/ / 40