Refinement Modal Logic: Algebraic Semantics Zeinab Bakhtiari LORIA, CNRS – Universit´ e de Lorraine, France In collaboration with: Hans van Ditmarsch (LORIA), Sabine Frittella (LIFO) August 2016, TU Delft Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 1 / 19
Plan for talk Part 1: Logic Introduction to Dynamic epistemic logic Refinement modal logic Part 2: Algebra Algebraic Semantics of action model logic Algebraic Semantics of refinement modal logic Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 2 / 19
Introduction: Dynamic Epistemic Logic (DEL) Dynamic epistemic logics is a family of logics dealing with knowledge and information change. Epistemic Describing knowledge and belief... Dynamic Knowledge acquisition, belief updates... Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 3 / 19
Introduction: Dynamic Epistemic Logic (DEL) Dynamic epistemic logics is a family of logics dealing with knowledge and information change. Epistemic Describing knowledge and belief... Dynamic Knowledge acquisition, belief updates... Epistemic actions Examples : Public announcements, private announcements, ... How can we make a formula true? Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 3 / 19
Introduction: Dynamic Epistemic Logic (DEL) Dynamic epistemic logics is a family of logics dealing with knowledge and information change. Epistemic Describing knowledge and belief... Dynamic Knowledge acquisition, belief updates... Epistemic actions Examples : Public announcements, private announcements, ... How can we make a formula true? Quantifying over information change. Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 3 / 19
Di ff erent ways of quantifying over information change there is an announcement (by the agents in group G ) after which ϕ ; In arbitrary public announcement logic (APAL) we quantify over announcements. there is an action model with precondition ψ after which ϕ ; In arbitrary action model logic (AAML) we quantify over action models. In these logics the quantification is over dynamic modalities for action execution . . . Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 4 / 19
Refinement quantifier Bozzelli, et al. in 2013 proposed a new form of quantification over information change, independent from the logical language. Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 5 / 19
Refinement quantifier Bozzelli, et al. in 2013 proposed a new form of quantification over information change, independent from the logical language. It is called refinement quantification, or just refinement. Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 5 / 19
Refinement quantifier Bozzelli, et al. in 2013 proposed a new form of quantification over information change, independent from the logical language. It is called refinement quantification, or just refinement. Refinement is the dual of simulation. Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 5 / 19
What is a refinement? A refinement of a model is a submodel of a bisimilar model: Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 6 / 19
What is a refinement? A refinement of a model is a submodel of a bisimilar model: Consider this pointed model (epistemic state) M : // • // • // • � Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 6 / 19
o o // o What is a refinement? A refinement of a model is a submodel of a bisimilar model: Consider this pointed model (epistemic state) M : // • // • // • � M 1 is a bisimilar copy of the model M : // • // • • • • � • Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 6 / 19
o o o // // o What is a refinement? A refinement of a model is a submodel of a bisimilar model: Consider this pointed model (epistemic state) M : // • // • // • � M 1 is a bisimilar copy of the model M : // • // • • • • � • M 2 is a refinement of M : ( M is a simulation of M 2 :) // • • � • Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 6 / 19
o o o // // o What is a refinement? A refinement of a model is a submodel of a bisimilar model: Consider this pointed model (epistemic state) M : // • // • // • � M 1 is a bisimilar copy of the model M : // • // • • • • � • M 2 is a refinement of M : ( M is a simulation of M 2 :) // • • � • Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 6 / 19
Refinement Relation: Formal Definition Let two models M = ( S , R , V ) and M 0 = ( S 0 , R 0 , V 0 ) be given. A non-empty relation R ✓ S ⇥ S 0 is a refinement if for all ( s , s 0 ) 2 R , p 2 P : atoms s 2 V ( p ) i ff s 0 2 V 0 ( p ); 0 s 0 t 0 , there is a t such that Rst and ( t , t 0 ) 2 R . back if R $ bisimulation: atoms, forth, back ! simulation: atoms, forth refinement: atoms, back Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 7 / 19
Refinement Modal Logic — language and semantics ϕ ::= p | ¬ ϕ | ( ϕ ^ ϕ ) | ⇤ ϕ | 8 ϕ Language Structures pointed Kripke models Semantics 8 ( M 0 , s 0 ) : ( M , s ) ( M 0 , s 0 ) implies ( M 0 , s 0 ) | ( M , s ) | = 8 ϕ i ff = ϕ 9 ( M 0 , s 0 ) : ( M , s ) ( M 0 , s 0 ) and ( M 0 , s 0 ) | ( M , s ) | = 9 ϕ i ff = ϕ [Bozzelli, Laura, et al. “ Refinement modal logic.” Information and Computation 239 (2014): 303-339.] Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 8 / 19
Arbitrary action model logic and refinement modal logic Action model execution is a refinement, and (surprisingly) vice versa (on finite models). M s ( M ⌦ α ) ( s , u) Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 9 / 19
Arbitrary action model logic and refinement modal logic Action model execution is a refinement, and (surprisingly) vice versa (on finite models). M s ( M ⌦ α ) ( s , u) Refinement quantifier and action model quantifier: = ¯ M s | 9 ϕ i ff there exists an action model α u s.t. M s | = h α u i ϕ . If M s | = 9 ϕ then we can find a multi-pointed action model α S s.t. M s | = h α S i ϕ . As a result: Refinement quantifier is equivalent to Action model quantifier! [ J. Hales.“Arbitrary action model logic and action model synthesis”. 2013.] Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 9 / 19
/ ✏ ✏ Part 2: Algebra 9 M 0 M Ref . morphism / A 0 A Main Goal Dualize the notion of refinement on algebras, For any algebraic model A = ( A , V ), we want to find a Boolean algebra with operator U A and a map G : U A ! A such that for any ϕ 2 L , J 9 ϕ K A = G ( J ϕ K U A ) . Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 10 / 19
Step 1: Dualize Refinement Relation Refinement morphism Let A and A 0 be two Boolean algebra with operators. A map f : A ! A 0 is a refinement morphism if it is monotone; preserves ? and _ ; and satisfies the following inequality ⌥ A 0 � f f � ⌥ A where ⌥ a ⇤ (adjoint operator). Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 11 / 19
Step 2: Epistemic update on algebras For any algebraic model A = ( A , V ) and any formula ϕ 2 L , we define Boolean algebra with operators A ϕ , A pair of maps f ϕ : A ! A ϕ , g ϕ : A ϕ ! A . Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 12 / 19
Step 2: Epistemic update on algebras For any algebraic model A = ( A , V ) and any formula ϕ 2 L , we define Boolean algebra with operators A ϕ , A pair of maps f ϕ : A ! A ϕ , g ϕ : A ϕ ! A . For each formula ϕ , action model synthesis provides us with an action model α ϕ S = (S , R , Pre), such that for every pointed model M s we have M s | = 9 ϕ i ff M ⌦ α ϕ S | = ϕ Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 12 / 19
o o Step 2: Epistemic update on algebras For any algebraic model A = ( A , V ) and any formula ϕ 2 L , we define Boolean algebra with operators A ϕ , A pair of maps f ϕ : A ! A ϕ , g ϕ : A ϕ ! A . For each formula ϕ , action model synthesis provides us with an action model α ϕ S = (S , R , Pre), such that for every pointed model M s we have M s | = 9 ϕ i ff M ⌦ α ϕ S | = ϕ M / ` α ϕ M ? _ M ⌦ α ϕ + / / A ϕ A Q a ϕ A Ma, Sadrzadeh and Palmigiano. Algebraic semantics and model completeness for intuitionistic public announcement . Kurz and Palmigiano. Epistemic updates in algebras . Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 12 / 19
Step 2: Epistemic updates on algebras a = (S , R , Pre a ϕ ): Pre a ϕ = V � Pre α ϕ . Q S A : | S | -fold product of A , which is set-isomorphic to the a ϕ collection A S of the set maps f : S ! A . The equivalence relation ⌘ a ϕ on Q a A is defined as follows: for all h , k 2 A S , h ⌘ a ϕ k i ff h ^ Pre a ϕ = k ^ Pre a ϕ . Zeinab Bakhtiari (LORIA) Algebraic Semantics of RML August 2016, TU Delft 13 / 19
Recommend
More recommend
