Complexity for probability logic with quantifiers ... - Semantic Scholar
Short introduction Syntax and semantics Complexity results
Complexity for probability logic with quanti ers over propositions Stanislav O. Speranski
Novosibirsk State University Mechanics and Mathematics Department Maltsev Meeting 2011
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Abstract Underlying idea
We describe the generalization of the language proposed in [FHM] R. Fagin, J. Y. Halpern, N. Megiddo, `A logic for reasoning about probabilities', Information and Computation, 87:1,2 (1990), 78128. by allowing meta-variables in propositions under the probability sign together with quanti ers bounding these variables. After the presentation of its syntax and semantics, we proceed by investigating the complexity issues.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Abstract Underlying idea
The characteristic feature of the semantic constructions in [FHM]: for a xed probability structure M , a measurable [[ψ]] is associated with each ψ . Intuitively, the meaning is that
ψ is true exactly at the worlds from [[ψ]] . Thus, for every ψ , one may consider the Bernulli random variable ξψ given by the property
ξψ (ω) = 1
S. O. Speranski
⇐⇒
ω ∈ [[ψ]] .
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Abstract Underlying idea
The characteristic feature of the semantic constructions in [FHM]: for a xed probability structure M , a measurable [[ψ]] is associated with each ψ . Intuitively, the meaning is that
ψ is true exactly at the worlds from [[ψ]] . Thus, for every ψ , one may consider the Bernulli random variable ξψ given by the property
ξψ (ω) = 1
S. O. Speranski
⇐⇒
ω ∈ [[ψ]] .
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
Let Prop and X be (in nite) computable collections of propositional symbols and variables, respectively; the logical connectives are ∧, ¬. Propositional formulae are introduced in the usual way. Analogously parametric propositional formulae are built up from Prop ∪ X using logical connectives. De nition The notion of QPL-term is given by induction: 1
2
for a parametric propositional formula ϕ, the expression µ (ϕ) is an (atomic) QPL-term; if f (x1 , . . . , xn ) is a polynomial with rational coe cients, and t1 , . . . , tn are QPL-terms, then the expression f (t1 , . . . , tn ) is a QPL-term.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
Let Prop and X be (in nite) computable collections of propositional symbols and variables, respectively; the logical connectives are ∧, ¬. Propositional formulae are introduced in the usual way. Analogously parametric propositional formulae are built up from Prop ∪ X using logical connectives. De nition The notion of QPL-term is given by induction: 1
2
for a parametric propositional formula ϕ, the expression µ (ϕ) is an (atomic) QPL-term; if f (x1 , . . . , xn ) is a polynomial with rational coe cients, and t1 , . . . , tn are QPL-terms, then the expression f (t1 , . . . , tn ) is a QPL-term.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
De nition For any QPL-terms t1 and t2 , the expression t1 6 t2 is a QPL-atom. The set of QPL-formulas is de ned inductively as the smallest class containing all QPL-atoms, subject to the constraint: if Φ1 and Φ2 are QPL-formulas , α is a variable from X , then ¬Φ1 , (Φ1 ∧ Φ2 ) , ∀α Φ1 and ∃α Φ1 are also QPL-formulas . The related notions (e.g., ∀-formulas, ∃-formulas) are introduced by analogy with the well-known de nitions for the rst-order logic.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
According to [FHM], a (measurable) probability structure is a tuple
M = hΩ, F , P, [[·]]i where hΩ, F , Pi is a probability space, and [[·]] is a mapping of Prop into F which is expanded to all propositional formulae by putting
[[ψ1 ∧ ψ2 ]] := [[ψ1 ]] ∩ [[ψ2 ]]
and
[[¬ψ1 ]] := Ω \ [[ψ1 ]] .
The semantics of quanti er-free QPL-sentences is fairly standard and provided in [FHM] (where they are called `polynomial weight formulas'): the idea is to replace all occurrences of the sort µ (ψ) with P ([[ψ]]) and then verify the resulting expression in the ordered eld R.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
According to [FHM], a (measurable) probability structure is a tuple
M = hΩ, F , P, [[·]]i where hΩ, F , Pi is a probability space, and [[·]] is a mapping of Prop into F which is expanded to all propositional formulae by putting
[[ψ1 ∧ ψ2 ]] := [[ψ1 ]] ∩ [[ψ2 ]]
and
[[¬ψ1 ]] := Ω \ [[ψ1 ]] .
The semantics of quanti er-free QPL-sentences is fairly standard and provided in [FHM] (where they are called `polynomial weight formulas'): the idea is to replace all occurrences of the sort µ (ψ) with P ([[ψ]]) and then verify the resulting expression in the ordered eld R.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
The semantics can be easily extended to all QPL-sentences: assume that ∀α Φ1 (α) and ∃α Φ2 (α) are viewed (respectively) as ^ _ Φ1 (ψ) and Φ2 (ψ) ψ∈ForCL
ψ∈ForCL
where ForCL is the set of all propositional formulas. From this perspective, one may think of QPL-sentences as forming a special sort of in nite probabilistic formulas, like those appearing in H. J. Keisler, `Probability quanti ers', in J. Barwise, S. Feferman (eds.), Model-theoretic Logics, Springer-Verlag, Berlin, 1985, 509556. Though, in contrast, our syntactical (formula) expressions are all nite.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
The semantics can be easily extended to all QPL-sentences: assume that ∀α Φ1 (α) and ∃α Φ2 (α) are viewed (respectively) as ^ _ Φ1 (ψ) and Φ2 (ψ) ψ∈ForCL
ψ∈ForCL
where ForCL is the set of all propositional formulas. From this perspective, one may think of QPL-sentences as forming a special sort of in nite probabilistic formulas, like those appearing in H. J. Keisler, `Probability quanti ers', in J. Barwise, S. Feferman (eds.), Model-theoretic Logics, Springer-Verlag, Berlin, 1985, 509556. Though, in contrast, our syntactical (formula) expressions are all nite.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Note that the validity problem for quanti er-free QPL-sentences is known to be decidable (cf. [FHM]). However, the following holds Theorem
The validity problem for ∃∀-sentences in QPL is undecidable, though it is decidable for ∀∃-sentences in QPL. The proof of the rst part combines the elementary de nability technique with the contributions of A. Nies, `Undecidable fragments of elementary theories', Algebra Universalis, 35 (1996), 833. To obtain the second, we eventually reduce the problem to that of deciding whether a certain rst-order sentence belongs to Th (R).
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Note that the validity problem for quanti er-free QPL-sentences is known to be decidable (cf. [FHM]). However, the following holds Theorem
The validity problem for ∃∀-sentences in QPL is undecidable, though it is decidable for ∀∃-sentences in QPL. The proof of the rst part combines the elementary de nability technique with the contributions of A. Nies, `Undecidable fragments of elementary theories', Algebra Universalis, 35 (1996), 833. To obtain the second, we eventually reduce the problem to that of deciding whether a certain rst-order sentence belongs to Th (R).
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Corollary
The problem of deciding whether a QPL-sentences is satis ed in all nite probability structures is Π01 -complete. Moreover, the complexity for the general validity is not arithmetical: Theorem
The validity problem for QPL-sentences is Π11 -complete. Remark that the lower bound proof makes use of the work by J. Y. Halpern, `Presburger arithmetic with unary predicates is Π11 complete', The Journal of Symbolic Logic, 56:2 (1991), 637642.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Corollary
The problem of deciding whether a QPL-sentences is satis ed in all nite probability structures is Π01 -complete. Moreover, the complexity for the general validity is not arithmetical: Theorem
The validity problem for QPL-sentences is Π11 -complete. Remark that the lower bound proof makes use of the work by J. Y. Halpern, `Presburger arithmetic with unary predicates is Π11 complete', The Journal of Symbolic Logic, 56:2 (1991), 637642.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Corollary
The problem of deciding whether a QPL-sentences is satis ed in all nite probability structures is Π01 -complete. Moreover, the complexity for the general validity is not arithmetical: Theorem
The validity problem for QPL-sentences is Π11 -complete. Remark that the lower bound proof makes use of the work by J. Y. Halpern, `Presburger arithmetic with unary predicates is Π11 complete', The Journal of Symbolic Logic, 56:2 (1991), 637642.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
References
Of current results S. O. Speranskii, `Quanti cation over propositional formulas in probability logic: decidability issues', Algebra and Logic, 50:4 (2011), 365374. Additional reading M. Abadi, J. Y. Halpern, `Decidability and expressiveness for rst-order logics of probability', Information and Computation, 112:1 (1994), 136.
S. O. Speranski
QPL:
complexity issues
Complexity for probability logic with quanti ers over propositions Stanislav O. Speranski
Novosibirsk State University Mechanics and Mathematics Department Maltsev Meeting 2011
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Abstract Underlying idea
We describe the generalization of the language proposed in [FHM] R. Fagin, J. Y. Halpern, N. Megiddo, `A logic for reasoning about probabilities', Information and Computation, 87:1,2 (1990), 78128. by allowing meta-variables in propositions under the probability sign together with quanti ers bounding these variables. After the presentation of its syntax and semantics, we proceed by investigating the complexity issues.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Abstract Underlying idea
The characteristic feature of the semantic constructions in [FHM]: for a xed probability structure M , a measurable [[ψ]] is associated with each ψ . Intuitively, the meaning is that
ψ is true exactly at the worlds from [[ψ]] . Thus, for every ψ , one may consider the Bernulli random variable ξψ given by the property
ξψ (ω) = 1
S. O. Speranski
⇐⇒
ω ∈ [[ψ]] .
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Abstract Underlying idea
The characteristic feature of the semantic constructions in [FHM]: for a xed probability structure M , a measurable [[ψ]] is associated with each ψ . Intuitively, the meaning is that
ψ is true exactly at the worlds from [[ψ]] . Thus, for every ψ , one may consider the Bernulli random variable ξψ given by the property
ξψ (ω) = 1
S. O. Speranski
⇐⇒
ω ∈ [[ψ]] .
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
Let Prop and X be (in nite) computable collections of propositional symbols and variables, respectively; the logical connectives are ∧, ¬. Propositional formulae are introduced in the usual way. Analogously parametric propositional formulae are built up from Prop ∪ X using logical connectives. De nition The notion of QPL-term is given by induction: 1
2
for a parametric propositional formula ϕ, the expression µ (ϕ) is an (atomic) QPL-term; if f (x1 , . . . , xn ) is a polynomial with rational coe cients, and t1 , . . . , tn are QPL-terms, then the expression f (t1 , . . . , tn ) is a QPL-term.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
Let Prop and X be (in nite) computable collections of propositional symbols and variables, respectively; the logical connectives are ∧, ¬. Propositional formulae are introduced in the usual way. Analogously parametric propositional formulae are built up from Prop ∪ X using logical connectives. De nition The notion of QPL-term is given by induction: 1
2
for a parametric propositional formula ϕ, the expression µ (ϕ) is an (atomic) QPL-term; if f (x1 , . . . , xn ) is a polynomial with rational coe cients, and t1 , . . . , tn are QPL-terms, then the expression f (t1 , . . . , tn ) is a QPL-term.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
De nition For any QPL-terms t1 and t2 , the expression t1 6 t2 is a QPL-atom. The set of QPL-formulas is de ned inductively as the smallest class containing all QPL-atoms, subject to the constraint: if Φ1 and Φ2 are QPL-formulas , α is a variable from X , then ¬Φ1 , (Φ1 ∧ Φ2 ) , ∀α Φ1 and ∃α Φ1 are also QPL-formulas . The related notions (e.g., ∀-formulas, ∃-formulas) are introduced by analogy with the well-known de nitions for the rst-order logic.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
According to [FHM], a (measurable) probability structure is a tuple
M = hΩ, F , P, [[·]]i where hΩ, F , Pi is a probability space, and [[·]] is a mapping of Prop into F which is expanded to all propositional formulae by putting
[[ψ1 ∧ ψ2 ]] := [[ψ1 ]] ∩ [[ψ2 ]]
and
[[¬ψ1 ]] := Ω \ [[ψ1 ]] .
The semantics of quanti er-free QPL-sentences is fairly standard and provided in [FHM] (where they are called `polynomial weight formulas'): the idea is to replace all occurrences of the sort µ (ψ) with P ([[ψ]]) and then verify the resulting expression in the ordered eld R.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
According to [FHM], a (measurable) probability structure is a tuple
M = hΩ, F , P, [[·]]i where hΩ, F , Pi is a probability space, and [[·]] is a mapping of Prop into F which is expanded to all propositional formulae by putting
[[ψ1 ∧ ψ2 ]] := [[ψ1 ]] ∩ [[ψ2 ]]
and
[[¬ψ1 ]] := Ω \ [[ψ1 ]] .
The semantics of quanti er-free QPL-sentences is fairly standard and provided in [FHM] (where they are called `polynomial weight formulas'): the idea is to replace all occurrences of the sort µ (ψ) with P ([[ψ]]) and then verify the resulting expression in the ordered eld R.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
The semantics can be easily extended to all QPL-sentences: assume that ∀α Φ1 (α) and ∃α Φ2 (α) are viewed (respectively) as ^ _ Φ1 (ψ) and Φ2 (ψ) ψ∈ForCL
ψ∈ForCL
where ForCL is the set of all propositional formulas. From this perspective, one may think of QPL-sentences as forming a special sort of in nite probabilistic formulas, like those appearing in H. J. Keisler, `Probability quanti ers', in J. Barwise, S. Feferman (eds.), Model-theoretic Logics, Springer-Verlag, Berlin, 1985, 509556. Though, in contrast, our syntactical (formula) expressions are all nite.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Syntax Semantics
The semantics can be easily extended to all QPL-sentences: assume that ∀α Φ1 (α) and ∃α Φ2 (α) are viewed (respectively) as ^ _ Φ1 (ψ) and Φ2 (ψ) ψ∈ForCL
ψ∈ForCL
where ForCL is the set of all propositional formulas. From this perspective, one may think of QPL-sentences as forming a special sort of in nite probabilistic formulas, like those appearing in H. J. Keisler, `Probability quanti ers', in J. Barwise, S. Feferman (eds.), Model-theoretic Logics, Springer-Verlag, Berlin, 1985, 509556. Though, in contrast, our syntactical (formula) expressions are all nite.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Note that the validity problem for quanti er-free QPL-sentences is known to be decidable (cf. [FHM]). However, the following holds Theorem
The validity problem for ∃∀-sentences in QPL is undecidable, though it is decidable for ∀∃-sentences in QPL. The proof of the rst part combines the elementary de nability technique with the contributions of A. Nies, `Undecidable fragments of elementary theories', Algebra Universalis, 35 (1996), 833. To obtain the second, we eventually reduce the problem to that of deciding whether a certain rst-order sentence belongs to Th (R).
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Note that the validity problem for quanti er-free QPL-sentences is known to be decidable (cf. [FHM]). However, the following holds Theorem
The validity problem for ∃∀-sentences in QPL is undecidable, though it is decidable for ∀∃-sentences in QPL. The proof of the rst part combines the elementary de nability technique with the contributions of A. Nies, `Undecidable fragments of elementary theories', Algebra Universalis, 35 (1996), 833. To obtain the second, we eventually reduce the problem to that of deciding whether a certain rst-order sentence belongs to Th (R).
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Corollary
The problem of deciding whether a QPL-sentences is satis ed in all nite probability structures is Π01 -complete. Moreover, the complexity for the general validity is not arithmetical: Theorem
The validity problem for QPL-sentences is Π11 -complete. Remark that the lower bound proof makes use of the work by J. Y. Halpern, `Presburger arithmetic with unary predicates is Π11 complete', The Journal of Symbolic Logic, 56:2 (1991), 637642.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Corollary
The problem of deciding whether a QPL-sentences is satis ed in all nite probability structures is Π01 -complete. Moreover, the complexity for the general validity is not arithmetical: Theorem
The validity problem for QPL-sentences is Π11 -complete. Remark that the lower bound proof makes use of the work by J. Y. Halpern, `Presburger arithmetic with unary predicates is Π11 complete', The Journal of Symbolic Logic, 56:2 (1991), 637642.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
Corollary
The problem of deciding whether a QPL-sentences is satis ed in all nite probability structures is Π01 -complete. Moreover, the complexity for the general validity is not arithmetical: Theorem
The validity problem for QPL-sentences is Π11 -complete. Remark that the lower bound proof makes use of the work by J. Y. Halpern, `Presburger arithmetic with unary predicates is Π11 complete', The Journal of Symbolic Logic, 56:2 (1991), 637642.
S. O. Speranski
QPL:
complexity issues
Short introduction Syntax and semantics Complexity results
References
Of current results S. O. Speranskii, `Quanti cation over propositional formulas in probability logic: decidability issues', Algebra and Logic, 50:4 (2011), 365374. Additional reading M. Abadi, J. Y. Halpern, `Decidability and expressiveness for rst-order logics of probability', Information and Computation, 112:1 (1994), 136.
S. O. Speranski
QPL:
complexity issues
