Logic and Probability Lecture 3: Beyond Boolean Logic

Transcription

1 Logic and Probability Lecture 3: Beyond Boolean Logic Wesley Holliday & Thomas Icard UC Berkeley & Stanford August 13, 2014 ESSLLI, Tübingen Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 1

2 Overview Last time: Nonmonotonic logic, Bayes nets, graphical models,... Today: Stochastic lambda calculus Probabilistic First-Order Logic Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 2

3 Part I: Stochastic Lambda Calculus Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 3

4 Stochastic Lambda Calculus Probabilistic programming languages give rise to more general formalisms for defining probability distributions, with graphical models as a basic special case. Stochastic Lambda Calculus: a perspicuous, simple formalism. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 4

5 Stochastic Lambda Calculus Review of basic Lambda Calculus The λ-terms are given by a set of variables V and defined inductively: Each variable x V is a λ-term. If M and N are λ-terms, then so is M(N). If x V and M is a λ-term, then λx.m is a λ-term. Main reduction rule, β-conversation: (λx.m)(n) β M[N/x] Let β be the reflexive, transitive closure of β. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 5

6 Stochastic Lambda Calculus Church-Rosser for β M N Q Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 6

7 Stochastic Lambda Calculus Church-Rosser for β M N Q R Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 7

8 Stochastic Lambda Calculus Corollary If a λ-term has a β-normal form, it is unique. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 8

9 Stochastic Lambda Calculus Boolean Logic in Lambda Calculus λx.λy.x λx.λy.y and λa.λb.bab if then else λa.λb.λc.abc not λa. a For M a boolean function of v 1,..., v n, and T 1,..., T n {, }: (λv 1... λv n.m)(t 1 )... (T n ) β { depending on whether the boolean formula is true or false with this input. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 9

10 Stochastic Lambda Calculus Numbers in Lambda Calculus 0 λf.λx.x 1 λf.λx.f (x) 2 λf.λx.f (f (x)). n λf.λx.f n (x) succ λn.λf.λx.f (n(f (x))) And so on... Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 10

11 Stochastic Lambda Calculus Fixed-Point Combinators Θ ( λx.λy.y(xxy) )( λx.λy.y(xxy) ) Turing s combinator Θ has the feature that, for any term M, ΘM β M(ΘM). This allows great flexibility, e.g., in defining recursive functions. But it also means trouble for logic, viz. Curry s Paradox. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 11

12 Stochastic Lambda Calculus Curry s Paradox Suppose we devise a proof system in λ-calculus with the following rules: A B B A ( A (A B) ) (A B) A A = β B B Define M λx.x (x ), and let N ΘM. It is easy to show But then we can evidently prove : N = β N (N ). 1. ( N (N )) (N ) 2. N (N ) 3. N 4. N 5. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 12

13 Stochastic Lambda Calculus Adding Probability Stochastic lambda calculus adds to lambda calculus a new operator: M N with intended interpretation of probabilistic choice between M and N. Write M p N for: M reduces to normal form N with probability p. In particular, then, M N M and M N normal form, and more generally, 1 2 p 1 M N p Stochastic λ-terms are themselves random variables! 1 2 N if M and N are in Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 13

14 Stochastic Lambda Calculus Evidently, this allows us to define term denoting a 50/50 coin flip : flip What about other types of coin flips, i.e., Bernoulli variables? Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 14

15 Stochastic Lambda Calculus Theorem (von Neumann) Suppose p [0, 1] is computable, i.e., there is a λ-term N such that { if the k th digit in the binary expansion of p is 1 N(k) β if the k th digit is 0 Then there is a stochastic λ-term M such that: M p and M 1 p. Proof. Define E to be the following term: ( E λe.λk.flip N(k) e ( succ(k) ) ) ( N(k) e ( succ(k) )). and let M ΘE, which returns with probability p and with 1 p. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 15

16 Stochastic Lambda Calculus Theorem (von Neumann) The same result holds for arbitrary domains, not just and. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 16

17 Stochastic Lambda Calculus Defining Graphical Models For exposition, assume we are working with binary variables, and let T p be a term that returns with probably p, and with probability 1 p. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 17

18 Stochastic Lambda Calculus Defining Graphical Models P(X ) = 0.1 X Y P(Y ) = 0.2 Z P(Z X, Y ) = 0.9 P(Z X, Y ) = 0.8 P(Z X, Y ) = 0.6 P(Z X, Y ) = 0.2 Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 18

19 Stochastic Lambda Calculus Defining Graphical Models X := T 0.1 Y := T 0.2 Z := if (X and Y ) then T 0.9 elif (X and not Y ) then T 0.8 elif (not X and Y ) then T 0.6 elif (not X and not Y ) then T 0.2 Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 19

20 Stochastic Lambda Calculus PCFGs S := NP, VP VP := if T 0.75 then V, NP else V, NP, PP NP := if T 0.7 then det, N else NP, PP PP := P, NP. N := if T 0.02 then cat else... Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 20

21 Stochastic Lambda Calculus Given (meta-)variable names for programs, X 1,..., X n, which we can think of as random variables, we can ask questions such as: P(X i = v)? P(X i = v Y 1 = u 1,..., Y m = u m )?... for an extremely general class of random variables. In principle one could import the same tools on probability-preserving deduction of Suppes, Adams, etc., from Lecture 1 to this setting. But one must be careful, witness Curry s Paradox, etc. (N.B. For more on actual programming languages based on these ideas, viz. Church, etc., see Noah Goodman s class next week.) Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 21

22 Part II: Probabilistic First-Order Logic Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 22

23 Let L be a first-order logical language, given by: a set C of individual constants ; a set V of individual variables, a set P of predicate variables. Terms and formulas of L are defined as usual: ϕ ::= R(t 1,..., t n ) ϕ ϕ ϕ x ϕ x ϕ Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 23

24 Define S L to be the set of sentences of L, i.e., formulas with no free variables, and SL 0 to be the set of quantifier-free sentences of L. As before, a probability on L S L is a function P : L [0, 1], with P(ϕ) = 1, for any first-order tautology ϕ ; P(ϕ ψ) = P(ϕ) + P(ψ), whenever (ϕ ψ). Question: Given a probability P : SL 0 [0, 1], is there a natural extension of P to all of S L, including quantified sentences? (Cf. Markov Logic.) Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 24

25 Question: Given a probability P : SL 0 [0, 1], is there a natural extension of P to all of S L, including quantified sentences? If there are only finitely many constants c such that P ( R(c) ) > 0, then: P ( xr(x) ) = P ( R(c) ) What about in the case where the number of c is infinite? Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 25

26 Example Consider a simple first-order arithmetical language L, with a constant n for each n N +. Let R(x) be a one-place predicate. Define a probability function P : SL 0 [0, 1] on the quantifier-free sentences as follows: P ( R(n) ) = 1/2 n+1, for all n N ; P ( i k R(n i ) ) = i k P ( R(n i ) ). Then, intuitively, P ( xp(x) ) = 1 n=2 2 n = 1 2. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 26

27 Definition (Gaifman s Condition) A probability P : S L [0, 1] satisfies the Gaifman condition if for all formulas with one free variable ϕ(x): or equivalently, P ( x ϕ(x) ) = sup {P ( n ϕ(c i ) ) c 1,..., c n C}, i=1 P ( x ϕ(x) ) = inf {P ( n ϕ(c i ) ) c 1,..., c n C}. i=1 Theorem (Gaifman 1964) Given P : S 0 L [0, 1], there is exactly one extension P of P to all of S L that satisfies the Gaifman condition. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 27

28 Probabilities on Models So far we have been focusing on probability functions P defined on a language L. But we can also start with a probability measure µ over models and induce a probability on formulas. Recall models of first-order languages: M = D, I, γ D is a set of individuals I is a function interpreting predicates, constants, etc. γ is an assignment function for variables. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 28

29 Consider a set of models M = i I {M i }, and σ-algebra E on M. We will require that each set M ϕ := {M M : M = ϕ}, for ϕ S L is measurable, i.e, an element of E. We can define a measure µ : E [0, 1], which induces an obvious probability P µ : S L [0, 1] on our first-order language L: P µ (ϕ) = µ(m ϕ ). It is then easy to show that P µ is a probability in our previous sense. Question: Does P µ always satisfy the Gaifman condition? Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 29

30 Suppose that every element of every domain of every model is named by a constant, according to all interpretation functions. Then: Fact Given µ, the probability P µ satisfies the Gaifman condition. Proof. P µ ( x ϕx) = µ ( {M M : M = x ϕx} ) = µ ( {M M : M = ϕ(c), for some c C} ) = sup {µ{m M : M = } ϕ(c i )} c 1,..., c n C i n ( = sup {P µ ϕ(c i ) ) } c 1,..., c n C i n Note that one may need to use countable additivity in this argument! Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 30

31 First-Order Probability Logic Add probability operators to the first-order language: π(ϕ) and corresponding atomic formulas, for a, b, c,... R, e.g.: aπ(ϕ) 2 + bπ(ψ) = c Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 31

32 We can also express, e.g., certain facts about conditional probability: π(ϕ ψ) 2/3 3π(ϕ ψ) 2π(ψ). Models of this language are tuples: M, E, µ, M M is a set of models with E a σ-algebra over M ; (M, E, µ) is a probability space ; M M is a distinguished model in M. But note that in the background there are also real numbers! Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 32

33 Note on Terminology Beware, we now have three different types of probabilities. We have mostly been using the following terminology: µ(e ) measure of event E, as part of a probability space P(ϕ) meta-language symbol for probability of a sentence ϕ π(ϕ) object language symbol for probability of a sentence ϕ Tomorrow we will add a sentence forming modal operator P r > object language, where P r > ϕ is not a term but a sentence. in the Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 33

34 Language and Interpretation Two sorts of variables: object variables x, y, z,..., and field variables x, y, z,.... Plus field constants 0 and 1 and plus and times. Field terms: t ::= x y 0 1 π(ϕ) t t t + t Then the language is given as follows: ϕ ::= P(x 1,..., x n ) x = y t 1 = t 2 ϕ ϕ ϕ x ϕ x ϕ Interpretation of field terms: [[π(ϕ)]] M,E,µ,M = µ({m M : M = ϕ}) Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 34

35 This is clearly a very expressive language. For instance, all of the probability axioms come out as validities in the object language. If Γ ϕ in this language, that means any probability distribution that satisfies all the probabilistic (and other) requirements in Γ, also satisfies whatever requirement ϕ specifies. One might also like to study maximum entropy inference in this setting. See, e.g., Plaskin (2002). Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 35

36 Complexity In general, the problem of determining validity in this language is undecidable. Indeed, the theory of models in this language is not even axiomatizable. (See Abadi & Halpern 1989.) However, if we assume the domain is finite, it is possible to give an axiomatization, and prove decidability. One can also obtain a complete axiomatization by allowing non-standard reals (see Bacchus 1988). Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 36

37 Axiomatization The axioms and rules of classical first-order logic. The axioms for real closed fields, e.g., that multiplication and addition are commutative and associated, that every odd degree polynomial has a root, etc. (Tarski) ϕ ( π(ϕ) = 1 ), provided the only predicate symbols occurring in ϕ occur only as subformulas of some π(ψ). π(ϕ) 0 π(ϕ) = π(ϕ ψ) + π(ϕ ψ) From ϕ ψ, infer π(ϕ) = π(ψ) x 1... x n y ( y = x 1... y = x n ) Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 37

38 Theorem (Halpern 1990) This logic is sound and complete with respect to probability models of domain size at most n. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 38

39 Another way of curtailing complexity: Still add probability operators to the first-order language: π(ϕ) but corresponding atomic formulas, for a 1,..., a n, b R: a 1 π(ϕ 1 ) a n π(ϕ n ) > b Abbreviations: π(ϕ) π(ψ) > b π(ϕ) + ( 1)π(ψ) > b π(ϕ) > π(ψ) π(ϕ) π(ψ) > 0 π(ϕ) b ( π(ϕ) > b ) π(ϕ) b ( 1)π(ϕ) b ( ) ( ) π(ϕ) = b π(ϕ) b π(ϕ) b Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 39

40 Then the language is given as follows: ϕ ::= P(x 1,..., x n ) a 1 π(ϕ) a n π(ϕ) > b a = b x = y ϕ ϕ ϕ x ϕ x ϕ The crucial truth clause: M, E, µ, M = a 1 π(ϕ 1 ) a n π(ϕ n ) > b iff [[a 1 ]] µ([[ϕ 1 ]]) + + [[a 1 ]] µ([[ϕ n ]]) > b in R Theorem (Halpern 2003) Adding axioms for linear inequalities gives a sound and complete axiomatization for this language. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 40

41 Measures on the Domain This language seem ill-suited for reasoning about what might be called a chance setup, e.g., the probability that a randomly chosen student in Tübingen being German is high. E.g., we should not expect this formula to have high probability: x ( StudentTbg(x) German(x) ) To deal with these kinds of statements, Bacchus (1988) proposed a slightly different language, with operators π x ( ϕ(x) ) > b, meaning, roughly, the probability of a randomly chosen x satisfying ϕ is greater than b. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 41

42 Otherwise the language is just the same as before. Models, however, are simpler: M = D, I, ν D, γ with ν : D [0, 1] a map such that d D ν(d) = 1. The interpretation of a probability subformula π x (ϕ) is given as: [[π x (ϕ)]] M = ν ( {d D : M[d/x] = ϕ} ) The rest of interpretation is as before. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 42

43 Complexity This logic, too, is undecidable and unaxiomatizable (Abadi & Halpern 1989). However, once again, by restricting to models of bounded size, the class becomes axiomatizable. Alternatively, we can restrict the language to linear inequalities and include a truth clause: M = a 1 π x (ϕ 1 ) a n π x (ϕ n ) > b iff [[a 1 ]] [[π x (ϕ 1 )]] + + [[a n ]] [[π x (ϕ n )]] > b in R This system is again axiomatizable. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 43

44 Theorem (Halpern 1990) This class of models of size at most n is axiomatized by the principles and rules of first order logic, those of real closed fields, and: x ϕ ( π x (ϕ) = 1 ) π x (ϕ) 0 π x (ϕ) = π x (ϕ ψ) + π x (ϕ ψ) π x (ϕ) = π y (ϕ[y/x]) From ϕ ψ, infer π x (ϕ) = π x (ψ) x 1... x n y ( y = x 1... y = x n ). Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 44

45 Of course, these two languages (and classes of models) can also be combined, so that one can reason, e.g., about such statements: ( ( ) 1 ) π π x German(x) StudentTbg(x) Combining the two axiomatizations gives completeness for the combined language, again for models of bounded size. See Halpern (1990) for details. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 45

46 Summary and Preview Untyped lambda calculus with a coin flip operation provides a representation language for defining a very rich class of probabilistic processes and models, even if it does not straightforwardly give rise to a logical system as we would typically think of it. There is a canonical way of extending a measure on the quantifier free sentences of a first-order language to all sentences. An important theme is the difference between defining a probability P on a logical language and defining a measure µ on a class of logical models. Today we focused on the latter, including rich first-order languages for talking about these measures. These logics are highly complex. Tomorrow and Friday we will look at much simpler languages for logical reasoning about probability. Wesley Holliday & Thomas Icard: Logic and Probability, Lecture 3: Beyond Boolean Logic 46