Quantifying over events in probability logic: expressibility vs. - PowerPoint PPT Presentation

Probabilistic elementary analysis Main results, and references Quantifying over events in probability logic: expressibility vs. computability Stanislav O. Speranski Sobolev Institute of Mathematics Novosibirsk State University Munich 2013 S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics Let A and B be sets of natural numbers. Say that A is m -reducible to B (denoted A � m B ) iff there exists a computable function f : N → N satisfying n ∈ A ⇐ ⇒ f ( n ) ∈ B ; A and B are called m -equivalent (denoted A ≡ m B ) iff A � m B and B � m A . Now we define [ A ] := { B | A ≡ m B } . Further, identify each problem specified by a question of the type Whether a given input has the desired property? with the set of inputs for which the answer is affirmative, and view, in turn, this set as a collection of natural numbers. S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics Take P n (respectively S n ) to be the set of Π 1 n (Σ 1 n )-sentences of second-order arithmetic true in the standard model N , and P ∞ to be the full second-order theory of N . The analytical hierarchy includes the following milestones: Π 1 n := [ P n ] and Σ 1 n := [ S n ] for all n ∈ N . Define Π 1 ∞ := [ P ∞ ]. A portion of the related terminology: for λ ∈ N ∪ {∞} , A is Π 1 λ -hard iff P λ � m A , A is Π 1 λ -bounded iff A � m P λ , A is Π 1 λ -complete iff P λ ≡ m A ; and similarly for Σ 1 λ (in place of Π 1 λ ) with λ ∈ N . S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics We present a bunch of quantified probability logics each of which has the complexity of P 1 ∞ and, in addition, obeys the conditions: the validity problem for its quantifier-free fragment is decidable; only two quantifiers, ∀ and ∃ , are available in the logic, both ranging over the unique sort of objects; no quantifiers may occur in the scope of the probability symbol, i. e., the formulas cannot contain µ ( . . . ∀ . . . ) or µ ( . . . ∃ . . . ); the quantification employed must be intuitively attractive from the viewpoint of probability theory, and the syntax/semantics of the logic should be easily describable. S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics Let X = { x i | i ∈ N } and C = { c i | i ∈ I } , where I is a non-empty computable subset of N . The collection of e -terms is the smallest set containing X ∪ C , and s. t. if t 1 and t 2 are e -terms, then t 1 and t 1 ∩ t 2 are also e -terms. Definition By a QPL C -atom we mean an expression of the sort f ( µ ( t 1 ) , . . . , µ ( t n )) � g ( µ ( t n +1 ) , . . . , µ ( t n + m )) , where f and g are polynomials with coefficiants in Q , µ is a fixed special symbol, and t 1 , . . . , t n + m are e -terms. The QPL C -formulas are obtained from the QPL C -atoms by closing under ¬ , ∧ and the applications of ∀ x , with x ∈ X . As usual, ∃ x Φ := ¬∀ x ¬ Φ. S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics A QPL C -formula belongs to Π n (Σ n ) iff it has the form ∀ x 1 ∃ x 2 . . . Ψ ( ∃ x 1 ∀ x 2 . . . Ψ) � �� � � �� � n − 1 alternations n − 1 alternations with { x 1 , x 2 , . . . } a set of tuples from X and Ψ quantifier-free. A QPL C -structure is a discrete probability space � Ω , A , P � augmeted by a valuation v : X ∪ C → A . So Ω is an at most countable set, A = { S | S ⊆ Ω } , and P is a discrete probability measure on A determined by a distribution p : Ω → [0 , 1] s. t. � � p ( ω ) = 1 , and P ( S ) = p ( ω ) for all S ⊆ Ω . ω ∈ Ω ω ∈ S S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics M = ( � Ω , A , P � , v : X ∪ C → A ) Let’s expand v from X ∪ C to the e -terms by interpreting t 1 as the complement of t 1 , t 1 ∩ t 2 as the intersection of t 1 and t 2 . And for every quantifier-free formula Φ of QPL C , naturally define M � Φ ⇐ ⇒ the result of replacing each µ ( t ) in Φ with P ( v ( t )) is true in R (which is, essentially, a variation on the quantifier-free probability logic of Fagin–Halpern–Megiddo). We extend � to all QPL C -formulas by: treating the connectives ¬ and ∧ clasically ; viewing the quantifier ∃ as ranging over all events of A . S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics Call a QPL C -sentence valid if it holds in any QPL C -structure. Along with the problem of testing validity for all QPL C -sentences comes the hierarchy of validity problems for QPL C containing Π n - Val C := the set of valid Π n -QPL C -sentences , Σ n - Val C := the set of valid Σ n -QPL C -sentences — hence we have Π n - Val C � m Π n +1 - Val C , Σ n +1 - Val C and Σ n - Val C � m Σ n +1 - Val C , Π n +1 - Val C . Such a hierarchy collapses if there exists n fulfilling the condition: for each k � n , Π k - Val C ≡ m Π n - Val C (clearly, one may switch from Π to Σ here). S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Preliminaries Main results, and references Syntax and semantics Before proceeding, it is helpful to list some observations. Since every event is uniquely specified by its characteristic function, quantifiers over events correspond to quantifiers over Bernoulli random variables — so the quantification employed in QPL C is very appealing from the viewpoint of probability theory. In addition, the logics QPL C are closely related to the logic with quantifiers over propositions, and hence are indirectly connected with formalisms introduced by H. J. Keisler, J. B. Paris, etc. The validity problem for quantifier-free QPL C -sentences is easily shown to be decidable by an argument of Fagin–Halpern–Megiddo, via m -reduction to determining membership in Th ( � R , + , × , � � ) along with implementation of the Tarski’s decision procedure. S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Issues of Computability Main results, and references Further reading Each logic QPL C has the same complexity as elementary analysis: Theorem The validity problem for QPL C is Π 1 ∞ -complete. And there are infinitely many pairwise non- m -equivalent elements of the nondecreasing sequence Π 0 - Val C � m Π 1 - Val C � m Π 2 - Val C � m . . . Theorem The hierarchy of validity problems for QPL C does not collapse. Notice: both proofs exploit some technique of monadic second-order definability in � N , + � (that generalises the result of Halpern about the Π 1 1 -completeness of the theory of � N , + � with a free unary predicate). S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Issues of Computability Main results, and references Further reading We turn to the investigation of the decision problems in the context of QPL C , viz. to the characterisation of all maximum prefix fragments of QPL C among those for which the validity problem is decidable. Theorem The validity problem for Π 2 - QPL C -sentences is decidable, while the vali- dity problem for Σ 2 - QPL C -sentences is undecidable. Notice: the proof employs the method of first-order elementarily definability and some related results. (And one may see the parallel with formulating the Skolem–Bernays–Sh¨ onfinkel classification of decision problems for pure first-order predicate logic.) S. O. Speranski Quantifying over events in probability logic

Probabilistic elementary analysis Issues of Computability Main results, and references Further reading Let’s present the probability logic QPL ◦ with quantifiers over propositions in the following way: the QPL ◦ -formulas and the QPL ◦ -structures are the same as for QPL C with C = { c i } i ∈ N ; in QPL ◦ the atoms, ∧ and ¬ are viewed semantically as in QPL C ; the significant distinction concerns the treatment of quantifiers in QPL ◦ : for every M = ( � Ω , A , P � , v : X ∪ C → A ), ∃ ranges over all events of A definable via ground e-terms . Even though the maximum decidable prefix fragments of QPL ◦ turn out to be the same as of QPL C , the two logics differ strikingly from the perspective of expressibility. S. O. Speranski Quantifying over events in probability logic