Reasoning about causal belief Kaibo Xie Institute for Logic, - PowerPoint PPT Presentation

Reasoning about causal belief Kaibo Xie Institute for Logic, Language and Computation July 27, 2018 Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 1 / 20

Causal Logic in Halpern (2016) Signature a signature is defined as a triple ( U , V , R ) where U is the set of exogenous variables and V is the set of endogenous variables, and R is a function that indicates the range of possible values of each causal variables. Causal Model Given such a signature S , a causal model is a pair ( S , F ) where F associates with every endogenous variable X a function denoted F X which characterize the value of X given the value of all the other variables in U ∪ V . Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 2 / 20

An Example U 1 U 2 A B P A and B stand for two assassins. P stands for whether the president is killed. U 1 and U 2 represent external factors that determine whether assassin A or B will shoot the president. Exogenous variables: U = { U 1 , U 2 } ; Endogenous variables: V = { A , B , P } . Structural equations: A = U 1 , B = U 2 , P = A ∨ B Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 3 / 20

What if we prevent A from shooting Suppose in the real world, A receives order to shoot the president and B does not, in this case the president is killed ( U 1 = A = 1, U 2 = B = 0, P = 1). What will happen if we prevent A from shooting? Is the president alive in this case? Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 4 / 20

What if we prevent A from shooting Suppose in the real world, A receives order to shoot the president and B does not, in this case the president is killed ( U 1 = A = 1, U 2 = B = 0, P = 1). What will happen if we prevent A from shooting? Is the president alive in this case? Intervention 1 Set the value of A to 0: replace F with F A = 0 where F A = 0 is the result of replacing the equation for A in F by A = 0 (by turning F A into constant functions whose output is 0) and leaving the remaining equations untouched. 2 Check: whether in all possible solutions to the structural equations obtained after setting A to 0 (namely F A = 0 ), P = 0 holds whenever U 1 = 1, U 2 = 0 Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 4 / 20

Halpern’s Causal Language (Basic) causal formula A basic causal formula is in the form of [ Y 1 = y 1 , ..., Y k = y k ] φ where Y 1 , ..., Y k are distinct causal variables, and φ is a boolean combination of formulas in the form of X = x . A causal formula is a boolean combination of basic causal formulas. Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 5 / 20

Halpern’s Causal Language (Basic) causal formula A basic causal formula is in the form of [ Y 1 = y 1 , ..., Y k = y k ] φ where Y 1 , ..., Y k are distinct causal variables, and φ is a boolean combination of formulas in the form of X = x . A causal formula is a boolean combination of basic causal formulas. For instance, [ Y 1 = y 1 , ..., Y k = y k ] X = x means: in all possible solutions to the structural equations obtained after setting Y i to y i , i = 1 , ..., k , The random variable X has value x Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 5 / 20

Reasoning about causal belief 1 If Y = y had been the case, X = x would been the case 2 It is believed that setting the value of Y to y results in X having the value x 3 After revising my belief state with Z = z , it is believed that setting the value of Y to y results in X having the value x Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 6 / 20

Epistemic operators Knowledge: K φ (the agent knows φ ) Belief: Bel φ (The agent believes φ ) Conditional Belief: Bel ψ φ (The agent believes φ after revising its belief with ψ ) Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 7 / 20

Epistemic operators Knowledge: K φ (the agent knows φ ) Belief: Bel φ (The agent believes φ ) Conditional Belief: Bel ψ φ (The agent believes φ after revising its belief with ψ ) For instance ∧ x ∈R ( X ) ∨ y ∈R ( Y ) K [ X = x ]( Y = y ) Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 7 / 20

Epistemic model (basic) An epistemic model is a tuple ( W , � , Π) 1 W is a set of possible worlds 2 � is a plausibility ordering over W 3 Π is an information partition over W : for each w ∈ W , Π( w ) tells us which possible worlds are indistinguishable for the agent Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 8 / 20

Epistemic Model w 1 ¯ PQ w 2 PQ w 3 P ¯ ¯ Q w 4 P ¯ Q w 1 � w 2 � w 3 � w 4 Π( w 1 ) = Π( w 2 ) = { w 1 , w 2 } ; Π( w 3 ) = Π( w 4 ) = { w 3 , w 4 } Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 9 / 20

Causal Epistemic Model A causal epistemic model is a tuple �S , F , Π , ≤� 1 S is a tuple ( U , V , R ) 2 F is a set of structural equations, for each X ∈ V , F X is a function from ( × Z ∈U R ( Z ) × ( × Y ∈V−{ X } R ( Y )) to R ( X ) . F has no causal loops. 3 Π is an information partition over W where W = × X ∈U∪V R ( X ) . 4 � ⊂ W × W is a total pre-order on W satisfying certain constraints. � is known as the plausibility ordering. Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 10 / 20

Combine the languages Let S = ( U , V , R ) , the language for S , write L ( S ) , is defined as follows: X = x (if X ∈ U ∪ V and x ∈ R ( X ) | φ ∧ ψ |¬ φ | [ X 1 = x 1 , ..., X i = x i ] φ (if X 1 = x 1 , ..., X i = x i is a sequence of distinct atomic sentences with X 1 , ..., X i ∈ V and φ is a formula without intervention operators | Bel φ ∈ L ( S ) | Bel ψ φ | K φ Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 11 / 20

Semantics Let S = ( U , V , R ) be a signature and M = �S , F , Π , � � is a causal epistemic model. Boolean Cases Let w be a possible world in W in the form of ( y 1 , ..., y n ) . M , ( y 1 , ..., y n ) | = X i = x i ( 1 � i � n ) if and only if x i = y i . The boolean combinations are defined in the usual way. Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 12 / 20

Epistemic Operators (Baltag and Smets (2006)) Belief M , w | = Bel φ if and only if φ holds on the most plausible worlds in Π( w ) Believing φ means φ is true at the most plausible worlds Conditional Belief = Bel ψ φ if and only if for any M , w | s ∈ Min � ( { t ∈ W | M , t | = ψ } ∩ Π( w )) , M , s | = φ . It means that the agent has the conditional belief “given ψ , then φ ” if and only if φ holds on the most plausible ψ worlds. Believing φ conditional on ψ means φ is true at the most plausible ψ -worlds. Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 13 / 20

Find the counterpart of Halpern’s intervention operator We need to define the output of setting the value of X to x in ( � u , � v ) , write a , � f X = x (( � u , � v )) = ( � b ) The output of setting the value of X to x F X = x is defined as the result of replacing the equation for X 1 , ..., X n in F by X 1 = x 1 , ... X n = x n (namely F X 1 , ..., F X n become constant functions whose output are x 1 , ..., x n and leaving the remaining equations untouched). Let � v = v 1 , ..., v n , � v ′ = v ′ n , � 1 , ..., v ′ V = V 1 , ..., V n , � b = b 1 , ..., b n . Define a , � u ; for any 1 ≤ i ≤ n , b i = v ′ ( � b ) as: � a = � i if X � V i , otherwise b i = v i . Y � Z means “ Y affects Z ” as an abbreviation for the formula u ∈× U ∈U R ( U ) , z � = z ′ ∈R ( Z ) ([ � ∨ � X = � x , Y = y ] Z = X ⊂V ,� x ∈× X ∈V R ( X ) , y ∈R ( Y ) ,� z ′ ∧ [ � X = � x , Y = y ] Z = z ) Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 14 / 20

The intervention operator The truth condition of the a sentence with intervention operator Given a signature S = ( U , V , R ) and a causal epistemic model M = �S , F , Π , ≤� . Let w be a possible world in W in the form of ( y 1 , ..., y n ) . = [ � M , w | X = � x ] φ if and only if M , f � x ( w ) | = φ where f � x is defined as X = � X = � before. Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 15 / 20

Plausibility ordering is not arbitrary Belief should consistent with agents’ causal information. Constraint on the plausibility ordering For any w 1 , w 2 ∈ W if w 1 ≺ w 2 and w 2 ⊀ w 1 then w 1 < w 2 (where w 1 ≺ w 2 is defined as there is X ∈ V such that w 1 complies F X and w 2 does not.) Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 16 / 20