Logics with Probability Operators L5:Extensions and Modi...
Transcript of Logics with Probability Operators L5:Extensions and Modi...
Logics with Probability OperatorsL5: Extensions and Modi�cations of Probability Logic
Zoran Ognjanovi¢ and Dragan Doder
Mathematical Institute SANU and IRIT, Universite Paul Sabatier
August 10, 2018
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 1 / 33
Probability Logics: Beyond LPP1/LPP2 and FHM
1 Extensions:
First order logic (L4)Extending syntax with other probability operators (independence,qualitative probability,. . . ) (L4)Combining probability with other modalities (temporal, epistemic,dynamic,dots) � extending syntax and semantics
2 Modi�cations:
Changing the type of probability operators (conditional probability,expectation operator, lower and upper probability,. . . )Changing the range of probability measures (�nite, rational,nonstandard (in�nitesimals), complex-valued,. . . )
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 2 / 33
Probability Logics: Beyond LPP1/LPP2 and FHM
1 Extensions:
First order logic (L4)Extending syntax with other probability operators (independence,qualitative probability,. . . ) (L4)Combining probability with other modalities (temporal, epistemic,dynamic,dots) � extending syntax and semantics
2 Modi�cations:
Changing the type of probability operators (conditional probability,expectation operator, lower and upper probability,. . . )Changing the range of probability measures (�nite, rational,nonstandard (in�nitesimals), complex-valued,. . . )
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 2 / 33
Upper and lower probabilities
Upper and lower probabilities
Halpern, Pucella.
A Logic for Reasoning about Upper Probabilities.
J. Artif. Intell. Res. 17: 57-81 (2002)
Savic, Doder, Ognjanovic.
Logics with lower and upper probability operators.
Int. J. Approx. Reasoning 88: 148-168 (2017)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 3 / 33
Upper and lower probabilities
Example
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Set of probabilities P = {µα | α ∈ {0, 0.1, . . . , 0.7}}, where µα gives
green-event probability 0.3, blue-event probability α, and red-event
probability 0.7− α.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 4 / 33
Upper and lower probabilities
Example
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Set of probabilities P = {µα | α ∈ {0, 0.1, . . . , 0.7}}, where µα gives
green-event probability 0.3, blue-event probability α, and red-event
probability 0.7− α.Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 4 / 33
Upper and lower probabilities
Example
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P?(X ) = inf{µ(X ) | µ ∈ P} P?(X ) = sup{µ(X ) | µ ∈ P}
P?(X ) = 1− P?(X c)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 5 / 33
Upper and lower probabilities
Example
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P?(X ) = inf{µ(X ) | µ ∈ P} P?(X ) = sup{µ(X ) | µ ∈ P}
P?(X ) = 1− P?(X c)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 5 / 33
Upper and lower probabilities
Example
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P?(R) = 0, P?(R) = 0.7, P?(B) = 0, P?(B) = 0.7,P?(G ) = P?(G ) = 0.3.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 6 / 33
Upper and lower probabilities
Example
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P?(R) = 0, P?(R) = 0.7,
P?(B) = 0, P?(B) = 0.7,P?(G ) = P?(G ) = 0.3.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 6 / 33
Upper and lower probabilities
Example
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P?(R) = 0, P?(R) = 0.7, P?(B) = 0, P?(B) = 0.7,
P?(G ) = P?(G ) = 0.3.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 6 / 33
Upper and lower probabilities
Example
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P?(R) = 0, P?(R) = 0.7, P?(B) = 0, P?(B) = 0.7,P?(G ) = P?(G ) = 0.3.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 6 / 33
Upper and lower probabilities
General remarks
Models:
uncertainty is modeled by a set of probabilities (on possible worlds)
Syntax:
P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)
Axiomatization:
How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?Characterization result needed.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 7 / 33
Upper and lower probabilities
General remarks
Models:
uncertainty is modeled by a set of probabilities (on possible worlds)
Syntax:
P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)
Axiomatization:
How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?Characterization result needed.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 7 / 33
Upper and lower probabilities
General remarks
Models:
uncertainty is modeled by a set of probabilities (on possible worlds)
Syntax:
P≥r is not appropriate
µ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)
Axiomatization:
How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?Characterization result needed.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 7 / 33
Upper and lower probabilities
General remarks
Models:
uncertainty is modeled by a set of probabilities (on possible worlds)
Syntax:
P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?
We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)
Axiomatization:
How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?Characterization result needed.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 7 / 33
Upper and lower probabilities
General remarks
Models:
uncertainty is modeled by a set of probabilities (on possible worlds)
Syntax:
P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)
Axiomatization:
How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?Characterization result needed.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 7 / 33
Upper and lower probabilities
General remarks
Models:
uncertainty is modeled by a set of probabilities (on possible worlds)
Syntax:
P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)
Axiomatization:
How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?
Characterization result needed.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 7 / 33
Upper and lower probabilities
General remarks
Models:
uncertainty is modeled by a set of probabilities (on possible worlds)
Syntax:
P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)
Axiomatization:
How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?Characterization result needed.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 7 / 33
Upper and lower probabilities
Theorem (Anger and Lembcke 1985)
Let W be a set, H an algebra of subsets of W , and f a function
f : H −→ [0, 1]. There exists a set P of probability measures such that
f = P? i� f satis�es the following three properties:
(1) f (∅) = 0,
(2) f (W ) = 1,
(3) for all natural numbers m, n, k and all subsets A1, . . . ,Am in H, if the
multiset {{A1, . . . ,Am}} is an (n, k)-cover of (A,W ), thenk + nf (A) ≤
∑mi=1 f (Ai ).
De�nition ((n, k)-cover)
A set A is said to be covered n times by a multiset {{A1, . . . ,Am}} of setsif every element of A appears in at least n sets from A1, . . . ,Am, i.e., for all
x ∈ A, there exists i1, . . . , in in {1, . . . ,m} such that for all j ≤ n, x ∈ Aij .
An (n, k)-cover of (A,W ) is a multiset {{A1, . . . ,Am}} that covers W ktimes and covers A n + k times.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 8 / 33
Upper and lower probabilities
Theorem (Anger and Lembcke 1985)
Let W be a set, H an algebra of subsets of W , and f a function
f : H −→ [0, 1]. There exists a set P of probability measures such that
f = P? i� f satis�es the following three properties:
(1) f (∅) = 0,
(2) f (W ) = 1,
(3) for all natural numbers m, n, k and all subsets A1, . . . ,Am in H, if the
multiset {{A1, . . . ,Am}} is an (n, k)-cover of (A,W ), thenk + nf (A) ≤
∑mi=1 f (Ai ).
De�nition ((n, k)-cover)
A set A is said to be covered n times by a multiset {{A1, . . . ,Am}} of setsif every element of A appears in at least n sets from A1, . . . ,Am, i.e., for all
x ∈ A, there exists i1, . . . , in in {1, . . . ,m} such that for all j ≤ n, x ∈ Aij .
An (n, k)-cover of (A,W ) is a multiset {{A1, . . . ,Am}} that covers W ktimes and covers A n + k times.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 8 / 33
Upper and lower probabilities
Possible-world semantics
Every world is equipped with
- an evaluation function on propositional letters, and
- one generalized probability space for each agent
De�nition
An ILUPP-structure is a tuple 〈W , LUP, υ〉, where:W is a nonempty set of worlds,
LUP assigns, to every w ∈W and every a ∈ Σ, a spaceLUP(w) = 〈W (w),H(w),P(w)〉, where:
∅ 6= W (w) ⊆W ,H(w) is an algebra of subsets of W (w)P(w) is a set of �nitely additive probability measures on H(w)
υ : W × P −→ {true, false}.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 9 / 33
Upper and lower probabilities
Possible-world semantics
Every world is equipped with
- an evaluation function on propositional letters, and
- one generalized probability space for each agent
De�nition
An ILUPP-structure is a tuple 〈W , LUP, υ〉, where:W is a nonempty set of worlds,
LUP assigns, to every w ∈W and every a ∈ Σ, a spaceLUP(w) = 〈W (w),H(w),P(w)〉, where:
∅ 6= W (w) ⊆W ,H(w) is an algebra of subsets of W (w)P(w) is a set of �nitely additive probability measures on H(w)
υ : W × P −→ {true, false}.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 9 / 33
Upper and lower probabilities
Satis�ability relation
Notation:
P?(w)([α]M,w ) = inf{µ([α]M,w ) | µ ∈ P(w)} andP?(w)([α]M,w ) = sup{µ([α]M,w ) | µ ∈ P(w)},
Satis�ability relation:
M,w |= U≥sα i� P?(w)([α]M,w ) ≥ s,
M,w |= L≥sα i� P?(w)([α]M,w ) ≥ s.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 10 / 33
Upper and lower probabilities
Satis�ability relation
Notation:
P?(w)([α]M,w ) = inf{µ([α]M,w ) | µ ∈ P(w)} andP?(w)([α]M,w ) = sup{µ([α]M,w ) | µ ∈ P(w)},
Satis�ability relation:
M,w |= U≥sα i� P?(w)([α]M,w ) ≥ s,
M,w |= L≥sα i� P?(w)([α]M,w ) ≥ s.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 10 / 33
Upper and lower probabilities
Axiom schemes
(1) all instances of the classical propositional tautologies
(2) U≤1α ∧ L≤1α
(3) U≤rα→ U<sα, s > r
(4) U<sα→ U≤sα
(5) (U≤r1α1 ∧ · · · ∧ U≤rmαm)→ U≤rα, ifα→
∨J⊆{1,...,m},|J|=k+n
∧j∈J αj and
∨J⊆{1,...,m},|J|=k
∧j∈J αj are
propositional tautologies, where r =∑m
i=1 ri−kn , n 6= 0
(6) ¬(U≤r1α1 ∧ · · · ∧ U≤rmαm), if∨
J⊆{1,...,m},|J|=k
∧j∈J αj is a
propositional tautology and∑m
i=1 ri < k
(7) L=1(α→ β)→ (U≥sα→ U≥sβ)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 11 / 33
Upper and lower probabilities
Inference Rules
(1) From α and α→ β infer β
(2) From α infer L≥1α
(3) From the set of premises
{α→ U≥s− 1kβ | k ≥ 1
s}
infer α→ U≥sβ
(4) From the set of premises
{α→ L≥s− 1kβ | k ≥ 1
s}
infer α→ L≥sβ.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 12 / 33
Non-standard Ranges
Probabilities with Non-standard Ranges
Raskovic, Markovic, Ognjanovic.
A logic with approximate conditional probabilities that can model
default reasoning.
Int. J. Approx. Reasoning 49(1): 52-66 (2008)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 13 / 33
Non-standard Ranges
System P
Kraus, Lehmann, Magidor.
Nonmonotonic Reasoning, Preferential Models and Cumulative Logics.
Artif. Intell. 44(1-2): 167-207 (1990)
α |∼ β � �if α, then generally β"
REF :α|∼α
; LLE :` α↔ β, α|∼γ
β|∼γ;
RW :` α→ β, γ|∼α
γ|∼β; AND :
α|∼β, α|∼γα|∼β ∧ γ
;
OR :α|∼γ, β|∼γα ∨ β|∼γ
; CM :α|∼β, α|∼γα ∧ β|∼γ
.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 14 / 33
Non-standard Ranges
System P
Kraus, Lehmann, Magidor.
Nonmonotonic Reasoning, Preferential Models and Cumulative Logics.
Artif. Intell. 44(1-2): 167-207 (1990)
α |∼ β � �if α, then generally β"
REF :α|∼α
; LLE :` α↔ β, α|∼γ
β|∼γ;
RW :` α→ β, γ|∼α
γ|∼β; AND :
α|∼β, α|∼γα|∼β ∧ γ
;
OR :α|∼γ, β|∼γα ∨ β|∼γ
; CM :α|∼β, α|∼γα ∧ β|∼γ
.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 14 / 33
Non-standard Ranges
Rational preferential relations
Lehmann, Magidor.
What does a Conditional Knowledge Base Entail?
Artif. Intell. 55(1): 1-60 (1992)
Rational relation = System P + Rational Monotonicity (RM)
RM :α |∼ γ, α |6∼ ¬β
α ∧ β |∼ γ
Nonstandard probabilistic semantics:
|∼ is rational i� there exists a neat �nitely additive non-standard probability
measure µ such that
|∼=|∼µ,
where
α |∼µ β i� µ(β|α) ≈ 1 or µ(α) = 0.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 15 / 33
Non-standard Ranges
Rational preferential relations
Lehmann, Magidor.
What does a Conditional Knowledge Base Entail?
Artif. Intell. 55(1): 1-60 (1992)
Rational relation = System P + Rational Monotonicity (RM)
RM :α |∼ γ, α |6∼ ¬β
α ∧ β |∼ γ
Nonstandard probabilistic semantics:
|∼ is rational i� there exists a neat �nitely additive non-standard probability
measure µ such that
|∼=|∼µ,
where
α |∼µ β i� µ(β|α) ≈ 1 or µ(α) = 0.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 15 / 33
Non-standard Ranges
Logic with non-standard probability values
Modeling α |∼ β � we need:
In syntax:
conditional probability operators
�approximately 1�
In semantics:
measures with non-standard values
Axiomatization:
Archimedean rule is not appropriate
we need a new rule to control the range
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 16 / 33
Non-standard Ranges
Syntax and Semantics
Hardy �eld Q(ε) is a recursive non-archimedean �eld which contains:
a �xed positive in�nitesimal ε
all standard rational numbers
Q(ε) = all rational functions of ε
eg. ε3+ε4
ε2−5ε6
countable
Semantics: the range of µ is [0, 1]Q(ε) � the unit interval of Q(ε).
Syntax:
CP≥s , CP≤s where s ∈ [0, 1]Q(ε)
CP≈r , where r ∈ [0, 1]Q
Abbreviations: P∗sα =def CP∗s(α,>), where ∗ ∈ {≥,≤,≈}No iterations, no mixing with classical formulas
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 17 / 33
Non-standard Ranges
Axioms
1 all ForC -instances of classical propositional tautologies
2 all ForP -instances of classical propositional tautologies
3 CP≥0(α, β)
4 CP≤s(α, β)→ CP<t(α, β), t > s
5 CP<s(α, β)→ CP≤s(α, β)
6 P≥1(α↔ β)→ (P=sα→ P=sβ)
7 P≤sα↔ P≥1−s¬α8 (P=sα ∧ P=tβ ∧ P≥1¬(α ∧ β))→ P=min(1,s+t)(α ∨ β)
9 P=0β → CP=1(α, β)
10 (P=tβ ∧ P=s(α ∧ β))→ CP=s/t(α, β), t 6= 0
11 CP≈r (α, β)→ CP≥r1(α, β), for every rational r1 ∈ [0, r)
12 CP≈r (α, β)→ CP≤r1(α, β), for every rational r1 ∈ (r , 1]
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 18 / 33
Non-standard Ranges
Inference rules
1 From ϕ and ϕ→ ψ infer ψ.
2 If α ∈ ForC , from α infer P≥1α.
3 From A→ P 6=sα, for every s ∈ [0, 1]Q(ε), infer A→ ⊥.4 For every r ∈ [0, 1]Q, from A→ CP≥r−1/n(α, β), for every integer
n ≥ 1/r , and A→ CP≤r+1/n(α, β) for every integer n ≥ 1/(1− r),infer A→ CP≈r (α, β).
Strong completeness for the class LPCP[0,1]Q(ε),≈2,Meas,Neat.
µ([α]) = s ⇔def T∗ ` P=sα.
(neat: If µ([α]) = 0, then there is no w in which α holds.)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 19 / 33
Non-standard Ranges
Inference rules
1 From ϕ and ϕ→ ψ infer ψ.
2 If α ∈ ForC , from α infer P≥1α.
3 From A→ P 6=sα, for every s ∈ [0, 1]Q(ε), infer A→ ⊥.4 For every r ∈ [0, 1]Q, from A→ CP≥r−1/n(α, β), for every integer
n ≥ 1/r , and A→ CP≤r+1/n(α, β) for every integer n ≥ 1/(1− r),infer A→ CP≈r (α, β).
Strong completeness for the class LPCP[0,1]Q(ε),≈2,Meas,Neat.
µ([α]) = s ⇔def T∗ ` P=sα.
(neat: If µ([α]) = 0, then there is no w in which α holds.)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 19 / 33
Temporal probability logics
Probability and Time
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 20 / 33
Temporal probability logics About temporal logic
Temporal operators
Basic:
© � next, U � until
©α: α has to hold at the next stateαUβ: α has to hold at least until β, which holds at the current or afuture moment
A � universal path operator (branching time)
Other:
G � always, F � sometime
E � existential path operator (branching time)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 21 / 33
Temporal probability logics About temporal logic
Temporal operators
Basic:
© � next, U � until
©α: α has to hold at the next stateαUβ: α has to hold at least until β, which holds at the current or afuture moment
A � universal path operator (branching time)
Other:
G � always, F � sometime
E � existential path operator (branching time)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 21 / 33
Temporal probability logics About temporal logic
Temporal operators
Basic:
© � next, U � until
©α: α has to hold at the next state
αUβ: α has to hold at least until β, which holds at the current or afuture moment
A � universal path operator (branching time)
Other:
G � always, F � sometime
E � existential path operator (branching time)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 21 / 33
Temporal probability logics About temporal logic
Temporal operators
Basic:
© � next, U � until
©α: α has to hold at the next stateαUβ: α has to hold at least until β, which holds at the current or afuture moment
A � universal path operator (branching time)
Other:
G � always, F � sometime
E � existential path operator (branching time)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 21 / 33
Temporal probability logics About temporal logic
Temporal operators
Basic:
© � next, U � until
©α: α has to hold at the next stateαUβ: α has to hold at least until β, which holds at the current or afuture moment
A � universal path operator (branching time)
Other:
G � always, F � sometime
E � existential path operator (branching time)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 21 / 33
Temporal probability logics About temporal logic
Temporal operators
Basic:
© � next, U � until
©α: α has to hold at the next stateαUβ: α has to hold at least until β, which holds at the current or afuture moment
A � universal path operator (branching time)
Other:
G � always, F � sometime
E � existential path operator (branching time)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 21 / 33
Temporal probability logics About temporal logic
Semantics of LTL
©α ∧ G (α→ β)→©β
Semantics for LTL � the set of paths Σ
σ = s0, s1, s2, . . .
si � the i-th time instance of σ � a subset of P,
Abbreviations:
σ≥i is the path si , si+1, si+2, . . .
σi is the state si .
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 22 / 33
Temporal probability logics How to combine them?
How to combine probabilistic and temporal logics
1 Temporal reasoning about probabilistic information
2 Probabilistic reasoning about temporal information
3 Modal approach - random nesting of both types of modalities
Examples:
1 Halpern, Pucella: A logic for reasoning about evidence (JAIR, 2006)
2 Grant, Parisi, Parker, Subrahmanian: An agm-style belief revision
mechanism for probabilistic spatio-temporal logics (AIJ, 2010)
3 Ognjanovic: Discrete linear-time probabilistic logics: Completeness,
decidability and complexity (JLC, 2006)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 23 / 33
Temporal probability logics How to combine them?
How to combine probabilistic and temporal logics
1 Temporal reasoning about probabilistic information
2 Probabilistic reasoning about temporal information
3 Modal approach - random nesting of both types of modalities
Examples:
1 Halpern, Pucella: A logic for reasoning about evidence (JAIR, 2006)
2 Grant, Parisi, Parker, Subrahmanian: An agm-style belief revision
mechanism for probabilistic spatio-temporal logics (AIJ, 2010)
3 Ognjanovic: Discrete linear-time probabilistic logics: Completeness,
decidability and complexity (JLC, 2006)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 23 / 33
Temporal probability logics How to combine them?
How to combine probabilistic and temporal logics
1 Temporal reasoning about probabilistic information
2 Probabilistic reasoning about temporal information
3 Modal approach - random nesting of both types of modalities
Examples:
1 Halpern, Pucella: A logic for reasoning about evidence (JAIR, 2006)
2 Grant, Parisi, Parker, Subrahmanian: An agm-style belief revision
mechanism for probabilistic spatio-temporal logics (AIJ, 2010)
3 Ognjanovic: Discrete linear-time probabilistic logics: Completeness,
decidability and complexity (JLC, 2006)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 23 / 33
The logic PLLTL
Doder, Ognjanovic. A Probabilistic Logic for Reasoningabout Uncertain Temporal Information. UAI 2015
We extend both:
Linear time logic LTL
Probabilistic logic (FHM)
We allow formulas like
�A will always hold"
�the probability that A will hold in next moment is at least the
probability that B will always hold"
The results:
Strongly complete axiomatization for countably additive semantics
Decidability: our logic is PSPACE-complete, no worse than LTL
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 24 / 33
The logic PLLTL
Doder, Ognjanovic. A Probabilistic Logic for Reasoningabout Uncertain Temporal Information. UAI 2015
We extend both:
Linear time logic LTL
Probabilistic logic (FHM)
We allow formulas like
�A will always hold"
�the probability that A will hold in next moment is at least the
probability that B will always hold"
The results:
Strongly complete axiomatization for countably additive semantics
Decidability: our logic is PSPACE-complete, no worse than LTL
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 24 / 33
The logic PLLTL
Doder, Ognjanovic. A Probabilistic Logic for Reasoningabout Uncertain Temporal Information. UAI 2015
We extend both:
Linear time logic LTL
Probabilistic logic (FHM)
We allow formulas like
�A will always hold"
�the probability that A will hold in next moment is at least the
probability that B will always hold"
The results:
Strongly complete axiomatization for countably additive semantics
Decidability: our logic is PSPACE-complete, no worse than LTL
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 24 / 33
The logic PLLTL Syntax and semantics
Syntax
Two types of formulas:
Certain knowledge: linear temporal formulas (ForLTL)
Uncertain knowledge: linear weighted formulas over ForLTL (ForP)
Example: P(p ∨ q) = P(©p)→ P(Gq) ≤ 1
2
�if the probability that either p or q hold in this moment is equal to theprobability that p will hold in the next moment, then the probabilitythat q will always hold is at most one half"
For = ForLTL ∪ ForP .
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 25 / 33
The logic PLLTL Syntax and semantics
Syntax
Two types of formulas:
Certain knowledge: linear temporal formulas (ForLTL)
Uncertain knowledge: linear weighted formulas over ForLTL (ForP)
Example: P(p ∨ q) = P(©p)→ P(Gq) ≤ 1
2
�if the probability that either p or q hold in this moment is equal to theprobability that p will hold in the next moment, then the probabilitythat q will always hold is at most one half"
For = ForLTL ∪ ForP .
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 25 / 33
The logic PLLTL Syntax and semantics
Syntax
Two types of formulas:
Certain knowledge: linear temporal formulas (ForLTL)
Uncertain knowledge: linear weighted formulas over ForLTL (ForP)
Example: P(p ∨ q) = P(©p)→ P(Gq) ≤ 1
2
�if the probability that either p or q hold in this moment is equal to theprobability that p will hold in the next moment, then the probabilitythat q will always hold is at most one half"
For = ForLTL ∪ ForP .
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 25 / 33
The logic PLLTL Syntax and semantics
Syntax
Two types of formulas:
Certain knowledge: linear temporal formulas (ForLTL)
Uncertain knowledge: linear weighted formulas over ForLTL (ForP)
Example: P(p ∨ q) = P(©p)→ P(Gq) ≤ 1
2
�if the probability that either p or q hold in this moment is equal to theprobability that p will hold in the next moment, then the probabilitythat q will always hold is at most one half"
For = ForLTL ∪ ForP .
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 25 / 33
The logic PLLTL Syntax and semantics
Semantics
M = 〈W ,H, µ, π〉:W � a nonempty set of worlds,
〈W ,H, µ〉 � a probability space, i.e.
H � an algebra of subsets of Wµ � a countably additive probability measure on H
π : W −→ Σ provides for each world w ∈W a path π(w).
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 26 / 33
The logic PLLTL Syntax and semantics
Satis�ability relation � LTL formulas
The evaluation function v : Σ× ForLTL −→ {0, 1}:
- for p ∈ P, v(σ, p) = 1 i� p ∈ σ0,- v(σ,©α) = 1 i� v(σ≥1, α) = 1,
- v(σ, αUβ) = 1 i� there is some i ∈ ω such that v(σ≥iβ) = 1, and for
each j ∈ ω, if 0 ≤ j < i then v(σ≥j , β) = 1.
M |= α i� v(π(w), α) = 1 for every w ∈W ,
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 27 / 33
The logic PLLTL Syntax and semantics
Satis�ability relation � LTL formulas
The evaluation function v : Σ× ForLTL −→ {0, 1}:
- for p ∈ P, v(σ, p) = 1 i� p ∈ σ0,- v(σ,©α) = 1 i� v(σ≥1, α) = 1,
- v(σ, αUβ) = 1 i� there is some i ∈ ω such that v(σ≥iβ) = 1, and for
each j ∈ ω, if 0 ≤ j < i then v(σ≥j , β) = 1.
M |= α i� v(π(w), α) = 1 for every w ∈W ,
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 27 / 33
The logic PLLTL Syntax and semantics
Satis�ability relation � probabilistic formulas
M = 〈W ,H, µ, π〉- [α]M = {w ∈W | v(π(w), α) = 1}- M is measurable, if [α]M ∈ H for every α ∈ ForLTL
M |= P(α) ≥ r i� µ([α]M) ≥ r ,
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 28 / 33
The logic PLLTL Syntax and semantics
Satis�ability relation � probabilistic formulas
M = 〈W ,H, µ, π〉- [α]M = {w ∈W | v(π(w), α) = 1}- M is measurable, if [α]M ∈ H for every α ∈ ForLTL
M |= P(α) ≥ r i� µ([α]M) ≥ r ,
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 28 / 33
The logic PLLTL Syntax and semantics
On countable additivity
αUnβ ≡ αUβ + β has realization at least at time n
[αUβ]M =⋃n∈ω
[αUnβ]M
Countable additivity necessary!
T = {P(αUβ) = 1} ∪ {P(αUnβ) = 0 | n ∈ ω}
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 29 / 33
The logic PLLTL Syntax and semantics
On countable additivity
αUnβ ≡ αUβ + β has realization at least at time n
[αUβ]M =⋃n∈ω
[αUnβ]M
Countable additivity necessary!
T = {P(αUβ) = 1} ∪ {P(αUnβ) = 0 | n ∈ ω}
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 29 / 33
The logic PLLTL Syntax and semantics
On countable additivity
αUnβ ≡ αUβ + β has realization at least at time n
[αUβ]M =⋃n∈ω
[αUnβ]M
Countable additivity necessary!
T = {P(αUβ) = 1} ∪ {P(αUnβ) = 0 | n ∈ ω}
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 29 / 33
The logic PLLTL Completeness and Decidability
Axioms
1 All instances of classical propositional tautologies for both LTL and
probabilistic formulas.
2 3 standard LTL axioms
3 Probabilisitc axioms a la FHM
4 Axioms for reasoning about linear inequalities a la FHM
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 30 / 33
The logic PLLTL Completeness and Decidability
Inference rules
1 2 x Modus Ponens
2 From α infer ©α (restricted to theorems)
3 From α infer P(α) = 1
4 From {γ → ¬(αUnβ) | n ∈ ω} infer γ → ¬(αUβ).
5 From {φ→ f ≥ r − 1n | n ∈ ω \ {0}} infer φ→ f ≥ r .
6 From {φ→ P(αUnβ) ≤ r | n ∈ ω} infer φ→ P(αUβ) ≤ r .
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 31 / 33
The logic PLLTL Completeness and Decidability
Completeness theorem
Theorem (Strong completeness)
A set of formulas T ⊆ For is consistent i� it is satis�able.
Proof. Henkin-like construction; extending T to a maximal consistent set
T ∗, and then using T ∗ to de�ne the canonical model.
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 32 / 33
The logic PLLTL Completeness and Decidability
Decidability and complexity
Theorem
The problem of deciding whether a formula of the logic PLLTL is satis�able
in a measurable structure from PLMeasLTL is PSPACE -complete. (no worse
than LTL)
Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 33 / 33