## A Logic for Reasoning about Probabilities (1990)

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Venue: | Information and Computation |

Citations: | 213 - 20 self |

### BibTeX

@ARTICLE{Fagin90alogic,

author = {Ronald Fagin and Joseph Y. Halpern and Nimrod Meciddo},

title = {A Logic for Reasoning about Probabilities},

journal = {Information and Computation},

year = {1990},

volume = {87},

pages = {78--128}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the proposi-tional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatiza-tion and show that the problem of deciding satistiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatiza-tions, we give a complete axiomatization for reasoning about Boolean combina-tions of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.

### Citations

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Citation Context ... 41 However, in some special cases, the logic is decidable; complete axiomatizations for these cases are provided in [Hal90]. 7 Dempster-Shafer belief functions The Dempster-Shafer theory of evidence =-=[Sha76]-=- provides one approach to attaching likelihoods to events. This theory starts out with a belief function (sometimes called a support function). For every event A, the belief in A, denoted Bel(A), is a... |

2045 | An introduction to probability theory and its applications, Volume I - Feller - 1957 |

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A Decision Method for Elementary Algebra and Geometry
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Citation Context ...d constants 0; 1; \Gamma1). That is, any first-order formula that involves only +; \Delta; !; 0; 1; \Gamma1 is true about the real numbers if and only if it is true of every real closed field. Tarski =-=[Tar51]-=- showed that the decision problem for this theory is decidable. BenOr, Kozen, and Reif [BKR86] have shown that the decision problem is decidable in exponential space. Fischer and Rabin [FR74] prove a ... |

417 |
Probabilistic logic
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- 1986
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Citation Context ... There is a fairly extensive literature on reasoning about probability (see for example [Bac90, Car50, Gai64, GKP88, GF87, HF87, Hoo78, Kav89, Kei85, Luk70, Nil86, Nut87, Sha76] and the references in =-=[Nil86]-=-), but remarkably few attempts at constructing a logic to reason explicitly about probabilities. We start by considering a language that allows linear inequalities involving probabilities. Thus, typic... |

319 | Real Analysis - Royden - 1988 |

315 | A generalization of Bayesian inference - Dempster - 1968 |

306 | Measure Theory - Halmos - 1950 |

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Introduction to Mathematical Logic
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Citation Context ...ight formula, and all of our axioms are, of course, weight formulas. We remark that we could replace Taut by a simple collection of axioms that characterize propositional tautologies (see for example =-=[Men64]). We have-=- not done so here because we want to focus here on the axioms for probability. The axiom Ineq includes "all valid formulas about linear inequalities." To make this precise, assume that we st... |

271 | An analysis of first-order logics of probability
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- 1990
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Citation Context ...when probability is interpreted in the more standard way and, as Bacchus shows, they do enable us to prove many facts of interest regarding the probability of first-order sentences. More recently, in =-=[Hal90]-=-, two first-order logics of probability are presented. One, in the spirit of Bacchus, puts probability on the domain while the other, more in the spririt of our approach here, puts probability on the ... |

264 | Logical Foundations of Probability, The - Carnap - 1950 |

252 |
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Citation Context ..., for each n-region ae. Let ae be a size r region. Then y ae = r X t=0 X ae 0 a size t subregion of ae (\Gamma1) r\Gammat x ae 0 : Proof: This proposition is simply a special case of Mobius inversion =-=[Rot64]-=- (see [Hal67, pp. 14--18]). Since the proof of Proposition 3.2 is fairly short, we now give it. Replace each x ae 0 in the right-hand side of the equality in the statement of the proposition by P ae 0... |

250 |
Representing and reasoning with probabilistic knowledge
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- 1990
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Citation Context ...s to allow linear combinations of terms. 2 language of our first logic. The measurable case of our richer logic bears some similarities to the first-order logic of probabilities considered by Bacchus =-=[Bac90]-=-. There are also some significant technical differences; we compare our work with that of Bacchus and the more recent results on first-order logics of probability in [AH94, Hal90] in more detail in Se... |

216 |
Mathematical Logic
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Citation Context ...om linear programming to get decision procedures and axiomatizations. However, the decision problem for the resulting logic can be reduced to the decision problem for the theory of real closed fields =-=[Sho67]-=-. By combining a recent result of Canny [Can88] with some of the techniques we develop in the linear case, we can obtain a polynomial space decision procedure for both the measurable case and the gene... |

154 | Reasoning about knowledge and probability
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(Show Context)
Citation Context ...wledge and probability. In [FH91] we consider the issue of appropriate models for reasoning about uncertainty in more detail, and compare the probabilistic approach to the DempsterShafer approach. In =-=[FH94]-=-, we consider a logic of knowledge and probability that allows arbitrary nesting of knowledge and probability operators. In particular, we allow higherorder weight formulas such as w(w(')s1=2)s1=3. (S... |

125 |
Some algebraic and geometric computations in PSPACE
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Citation Context ...s and axiomatizations. However, the decision problem for the resulting logic can be reduced to the decision problem for the theory of real closed fields [Sho67]. By combining a recent result of Canny =-=[Can88]-=- with some of the techniques we develop in the linear case, we can obtain a polynomial space decision procedure for both the measurable case and the general case of the logic. We can further extend th... |

118 |
The complexity of elementary algebra and geometry
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Citation Context ...ve complete axiomatizations and decision procedures for the extended language, for both the measurable and general case. In this case, combining our techniques with results of Ben-Or, Kozen, and Reif =-=[BKR86]-=-, we get an exponential space decision procedure. Thus, allowing nonlinear terms in the logic seems to have a high cost in terms of complexity, and further allowing quantifiers an even higher cost. Th... |

96 |
A probabilistic PDL
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- 1983
(Show Context)
Citation Context ...ision procedure for his logic, also by reduction to the decision problem for the theory of real closed fields. (The extra complexity in his logic arises from the presence of program operators.) Kozen =-=[Koz85]-=- too considers a probabilistic propositional dynamic logic (which is a fragment of Feldman's logic) for which he shows that the decision problem is PSPACE-complete. While a formula such as w(')s2w(/) ... |

95 | Super-exponential complexity of Presburger arithmetic
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- 1974
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Citation Context ... Tarski [Tar51] showed that the decision problem for this theory is decidable. BenOr, Kozen, and Reif [BKR86] have shown that the decision problem is decidable in exponential space. Fischer and Rabin =-=[FR74]-=- prove a nondeterministic exponential time lower bound for the complexity of the decision problem. In fact, Fischer and Rabin's lower bound holds even if the only nonlogical symbol is + (plus). Berman... |

71 | Knowledge, probability, and adversaries
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(Show Context)
Citation Context ...arify the role of probability in the analysis. It is all too easy to lose track of precisely which events it is that are being assigned a probability, and how that probability should be assigned (see =-=[HT93]-=- for a discussion of the situation in the context of distributed systems). There is a fairly extensive literature on reasoning about probability (see for example [Bac90, Car50, Gai64, GKP88, GF87, HF8... |

62 | Anytime Deduction for Probabilistic Logic
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- 1994
(Show Context)
Citation Context ...ed by Nilsson in [Nil86]. 1 The question of providing a complete axiomatization and decision procedure for Nilsson 's logic has attracted the attention of other researchers before. Haddawy and Frisch =-=[HF87]-=- provide some sound axioms (which they observe are not complete), and show how interesting consequences can be deduced using their axioms. Georgakopoulos, Kavvadias, and Papadimitriou [GKP88] show tha... |

49 | Concerning measures in first order calculi - Gaifman - 1964 |

45 | Uncertainty, belief and probability - Fagin, Halpern - 1989 |

40 | Decidability and expressiveness for first-order logics of probability
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- 1994
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Citation Context ...d to allow a logic where we can reason simultaneously about probabilities on the domain and on possible worlds. In all cases, the probabilities are countably additive and take values in the reals. In =-=[AH94]-=- it is shown that in general the decision problem for these logics is wildly undecidable (technically, it is \Pi 2 1 -complete). 41 However, in some special cases, the logic is decidable; complete axi... |

39 |
The complexity of logical theories
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- 1980
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Citation Context ...prove a nondeterministic exponential time lower bound for the complexity of the decision problem. In fact, Fischer and Rabin's lower bound holds even if the only nonlogical symbol is + (plus). Berman =-=[Ber80]-=- gives a slightly sharper lower bound in terms of alternation. Canny [Can88] has recently shown that the quantifier-free fragment is decidable in polynomial space. We do not know a sound and complete ... |

38 |
Probabilistic satisfiability
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(Show Context)
Citation Context ...nd Frisch [HF87] provide some sound axioms (which they observe are not complete), and show how interesting consequences can be deduced using their axioms. Georgakopoulos, Kavvadias, and Papadimitriou =-=[GKP88]-=- show that a less expressive logic than ours (where formulas have the form (w(' 1 ) = c 1 )s: : :s(w('m ) = c m ), and each ' i is a disjunction of primitive propositions and their negations) is also ... |

32 |
A logic to reason about likelihood
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- 1987
(Show Context)
Citation Context ...[LS82, HR87]) consider modal logics that allow more qualitative reasoning. In [LS82] there are modal operators that allow one to say "with probability one" or "with probability greater =-=than zero"; in [HR87] there is -=-a modal operator which says "it is likely that". Decision procedures and complete axiomatizations are obtained for these logics. However, neither of them allows explicit manipulation of prob... |

31 | A theory of higher order probabilities
- Gaifman
- 1988
(Show Context)
Citation Context ...sider a logic of knowledge and probability that allows arbitrary nesting of knowledge and probability operators. In particular, we allow higherorder weight formulas such as w(w(')s1=2)s1=3. (See also =-=[Gai86]-=- for discussion and further references on the subject of higher-order probabilities.) We are again able to prove technical results about complete axiomatizations and decision procedures for the result... |

25 | Some algebraic and geometric computations - Canny - 1988 |

23 |
Solution d’une question particulière du calcul des inegalités. Nouveau Bulletin des Sciences par la Scociété Philomatique de Paris
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(Show Context)
Citation Context ...usion that f is inconsistent, a contradiction. We now consider the case where s ? 0. Farkas' lemma does not apply, but a variant of it, called Motzkin's transposition theorem, which is due to Fourier =-=[Fou26]-=-, Kuhn [Kuh56], and Motzkin [Mot56] (see [Sch86, page 94]), does. A and A 0 are matrices, b and b 0 are column vectors, and x is a column vector of distinct variables. Lemma 4.5: If the system Axsb; A... |

22 |
Probability quantifiers
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(Show Context)
Citation Context ...ozen's logic is not closed under Boolean combination). Kozen also does not allow nonlinear combinations. None of the papers mentioned above consider the nonmeasurable case. Hoover [Hoo78] and Keisler =-=[Kei85]-=- provide complete axiomatizations for their logics (their language is quite different from ours, in that they allow infinite conjunctions, and do not allow sums of probabilities). Other papers (for ex... |

16 |
Solvability and consistency for linear equations and inequalities
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Citation Context ...s inconsistent, a contradiction. We now consider the case where s ? 0. Farkas' lemma does not apply, but a variant of it, called Motzkin's transposition theorem, which is due to Fourier [Fou26], Kuhn =-=[Kuh56]-=-, and Motzkin [Mot56] (see [Sch86, page 94]), does. A and A 0 are matrices, b and b 0 are column vectors, and x is a column vector of distinct variables. Lemma 4.5: If the system Axsb; A 0 x ? b 0 is ... |

13 | Logical Foundations of Probability. Univ - Carnap - 1950 |

12 | Reasoning about knowledge and probability: preliminary report - Fagin, Halpern - 1988 |

10 |
Reasoning about time and chance
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(Show Context)
Citation Context ...t from ours, in that they allow infinite conjunctions, and do not allow sums of probabilities). Other papers (for example [LS82, HR87]) consider modal logics that allow more qualitative reasoning. In =-=[LS82] there are modal operators tha-=-t allow one to say "with probability one" or "with probability greater than zero"; in [HR87] there is a modal operator which says "it is likely that". Decision procedures... |

6 |
A decidable propositional probabilistic dynamic logic
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Citation Context ...obability in [AH94, Hal90] in more detail in Section 6. The measurable case of the richer logic can also be viewed as a fragment of the probabilistic propositional dynamic logic considered by Feldman =-=[Fel84]-=-. Feldman provides a double-exponential space decision procedure for his logic, also by reduction to the decision problem for the theory of real closed fields. (The extra complexity in his logic arise... |

6 | Logical foundations of probability theory - Lukasiewicz |

6 |
The assignment problem
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Citation Context ...tradiction. We now consider the case where s ? 0. Farkas' lemma does not apply, but a variant of it, called Motzkin's transposition theorem, which is due to Fourier [Fou26], Kuhn [Kuh56], and Motzkin =-=[Mot56]-=- (see [Sch86, page 94]), does. A and A 0 are matrices, b and b 0 are column vectors, and x is a column vector of distinct variables. Lemma 4.5: If the system Axsb; A 0 x ? b 0 is unsatisfiable, then t... |

6 | Uncertainty and probability - Nutter - 1987 |

4 |
Theorie der enfachen ungleichungen
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Citation Context ...he system above or, equivalently, the number of negations of basic inequality formulas in f) is zero or greater than zero. We first assume s = 0. We make use of the following variant of Farkas' lemma =-=[Far02]-=- (see [Sch86, page 89]) from linear programming, where A is a matrix, b is a column vector, and x is a column vector of distinct variables: Lemma 4.4: If Axsb is unsatisfiable, then there exists a row... |

4 | Foundations of Probabilistic Logic - Guggenheimer, Freedman - 1987 |

4 | Linear Programming - CHVTAL - 1983 |

3 |
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Citation Context ...e so viewed (since Kozen's logic is not closed under Boolean combination). Kozen also does not allow nonlinear combinations. None of the papers mentioned above consider the nonmeasurable case. Hoover =-=[Hoo78]-=- and Keisler [Kei85] provide complete axiomatizations for their logics (their language is quite different from ours, in that they allow infinite conjunctions, and do not allow sums of probabilities). ... |

2 | Concerning measures in first order calculi - unknown authors - 1964 |

2 | Probability quantifiers', in Model-Theoretic Logics - Keisler - 1985 |

2 | Theory of Linear and Integer Programming - SCHRUVER - 1986 |

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1 | Representing and Reasoning with Probabilistic Knowledge - BACHS - 1988 |