Social Interaction – A Formal Exploration Dominik Klein University of Bayreuth PhD Colloquium of the DVMLG, Hamburg, October 10 th 2016 Klein: Social Interaction – A Formal Exploration 1/39
Social Interaction – An Example Klein: Social Interaction – A Formal Exploration 2/39
Another Example Klein: Social Interaction – A Formal Exploration 3/39
Information in Social Situations ◮ Success of situations depends upon information of the agents ◮ Not too little belief ◮ Not too much belief ◮ Higher order belief matters Klein: Social Interaction – A Formal Exploration 4/39
Our Perspective: Logics for Social Interaction ◮ Qualitative Modelling of Information ◮ Descriptive: Adequate representation of the situation ◮ Goal State: Distribution of Information that should be achieved ◮ Protocols: Achieving a certain type of Information Klein: Social Interaction – A Formal Exploration 5/39
Information in Interaction – The logic Fix a set of atomic propositions P and a set of agent At. Define the epistemic language L K as: ϕ := p | ϕ ∧ ϕ |¬ ϕ | K i ϕ : i ∈ At Klein: Social Interaction – A Formal Exploration 6/39
Information in Interaction – The logic Fix a set of atomic propositions P and a set of agent At. Define the epistemic language L K as: ϕ := p | ϕ ∧ ϕ |¬ ϕ | K i ϕ : i ∈ At Axioms P All propositional validities N K ( ϕ → ψ ) → ( K ϕ → K ψ ) T K ϕ → ϕ PI K ϕ → KK ϕ NI ¬ K ϕ → K ¬ K ϕ Klein: Social Interaction – A Formal Exploration 6/39
The Semantics An epistemic model is a tripel � W , ( R i ) i ∈ At , V � where ◮ W is a set of worlds p ◮ R i is an equivalence q p,q relation on W p,q p ◮ V : P → P ( W ) is an p,q p atomic valuation Klein: Social Interaction – A Formal Exploration 7/39
The Semantics An epistemic model is a tripel � W , ( R i ) i ∈ At , V � where ◮ W is a set of worlds p ◮ R i is an equivalence q p,q relation on W p,q p ◮ V : P → P ( W ) is an p,q p atomic valuation Evaluate the epistemic language on model-world pairs by ◮ M , w � p iff w ∈ V ( p ) M , w � ¬ ϕ iff M , w � � ϕ . . . ◮ M , w � K i ψ iff for all v with vR i w : M , v � ψ Klein: Social Interaction – A Formal Exploration 7/39
The Semantics An epistemic model is a tripel � W , ( R i ) i ∈ At , V � where ◮ W is a set of worlds p ◮ R i is an equivalence q p,q relation on W p,q p ◮ V : P → P ( W ) is an p,q p atomic valuation Evaluate the epistemic language on model-world pairs by ◮ M , w � p iff w ∈ V ( p ) M , w � ¬ ϕ iff M , w � � ϕ . . . ◮ M , w � K i ψ iff for all v with vR i w : M , v � ψ L K is sound and complete w.r.t the class of epistemic models Klein: Social Interaction – A Formal Exploration 7/39
An Example Car ϕ ϕ Ped ¬ ϕ ϕ = Both approaching at the same time Klein: Social Interaction – A Formal Exploration 8/39
Information in Interaction – The belief case Fix a set of atomic propositions P and a set of agent At. Define the doxastic language L B as: ϕ := p | ϕ ∧ ϕ |¬ ϕ | B i ϕ Klein: Social Interaction – A Formal Exploration 9/39
Information in Interaction – The belief case Fix a set of atomic propositions P and a set of agent At. Define the doxastic language L B as: ϕ := p | ϕ ∧ ϕ |¬ ϕ | B i ϕ Axioms All propositional validities N B ( ϕ → ψ ) → ( B ϕ → B ψ ) PI B ϕ → BB ϕ NI ¬ B ϕ → B ¬ B ϕ Klein: Social Interaction – A Formal Exploration 9/39
The Semantics A doxastic model is a tripel � W , ( R i ) i ∈ At , V � where ◮ W is a set of worlds ◮ R i is transitive and p q p,q Euclidean (i.e. aRb ∧ aRc ⇒ bRc ) p,q p p,q ◮ V : P → P ( W ) is an p atomic valuation Klein: Social Interaction – A Formal Exploration 10/39
The Semantics A doxastic model is a tripel � W , ( R i ) i ∈ At , V � where ◮ W is a set of worlds ◮ R i is transitive and p q p,q Euclidean (i.e. aRb ∧ aRc ⇒ bRc ) p,q p p,q ◮ V : P → P ( W ) is an p atomic valuation Evaluate the epistemic language on model-world pairs by ◮ M , w � p iff w ∈ V ( p ) ◮ M , w � K i ψ iff for all v with vR i w : M , v � ψ Klein: Social Interaction – A Formal Exploration 10/39
The Semantics A doxastic model is a tripel � W , ( R i ) i ∈ At , V � where ◮ W is a set of worlds ◮ R i is transitive and p q p,q Euclidean (i.e. aRb ∧ aRc ⇒ bRc ) p,q p p,q ◮ V : P → P ( W ) is an p atomic valuation Evaluate the epistemic language on model-world pairs by ◮ M , w � p iff w ∈ V ( p ) ◮ M , w � K i ψ iff for all v with vR i w : M , v � ψ L B is sound and complete w.r.t the class of doxastic models Klein: Social Interaction – A Formal Exploration 10/39
The Central Question Which language should we use ◮ Knowledge: L K ? ◮ Belief: L B ? ◮ Knowledge & Belief? ◮ Common Knowledge? Everybody knows ϕ , Everybody knows everybody knows ϕ . . . ◮ Only Interested in special propositions ◮ Only fragments of the language? Only bounded information. Only positive belief . . . Klein: Social Interaction – A Formal Exploration 11/39
Some Considerations ◮ Needs of the situation ◮ Poor languages can’t represent the situation adequately ◮ Too rich languages might have complexity issues • Compactness? • (Finite) Realizability? • . . . Klein: Social Interaction – A Formal Exploration 12/39
The Questions for Today ◮ Expressive power • When does a description language allow to distinguish only few different situations Klein: Social Interaction – A Formal Exploration 13/39
The Questions for Today ◮ Expressive power • When does a description language allow to distinguish only few different situations (few = countably many) Klein: Social Interaction – A Formal Exploration 13/39
The Questions for Today ◮ Expressive power • When does a description language allow to distinguish only few different situations (few = countably many) ◮ Realizability • Can I guarantee that every consistent state description is realizable in a finite model? Klein: Social Interaction – A Formal Exploration 13/39
The Questions for Today ◮ Expressive power • When does a description language allow to distinguish only few different situations (few = countably many) ◮ Realizability • Can I guarantee that every consistent state description is realizable in a finite model? ◮ Dynamics • How do state descriptions change under information dynamics • How to bring about a certain situation? Klein: Social Interaction – A Formal Exploration 13/39
Let’s make things a bit more precise Let L be the language with a single atom x ϕ = x | ϕ ∧ ϕ |¬ ϕ | K i ϕ Definition A reasoning language is any fragment L res of L . Klein: Social Interaction – A Formal Exploration 14/39
Let’s make things a bit more precise Let L be the language with a single atom x ϕ = x | ϕ ∧ ϕ |¬ ϕ | K i ϕ Definition A reasoning language is any fragment L res of L . For example L K , the reasoning language generated by x , K 1 , K 2 contains all formulas of the form K 1 K 2 K 2 K 1 x Klein: Social Interaction – A Formal Exploration 14/39
Let’s make things a bit more precise Let L be the language with a single atom x ϕ = x | ϕ ∧ ϕ |¬ ϕ | K i ϕ Definition A reasoning language is any fragment L res of L . For example L K , the reasoning language generated by x , K 1 , K 2 contains all formulas of the form K 1 K 2 K 2 K 1 x Definition For a reasoning language L res , a level of L res information is a set T ⊆ L res such that the set T ∪ {¬ ϕ | ϕ ∈ L res \ T } is consistent. Klein: Social Interaction – A Formal Exploration 14/39
The first Question: When does a reasoning language allow for only few (countably many) levels of information? Klein: Social Interaction – A Formal Exploration 15/39
Why is this a thing ◮ Take the reasoning language generated by K 1 , K 2 , ¬ All formulas of the form K 1 ¬ K 2 ¬ K 2 K 1 x Klein: Social Interaction – A Formal Exploration 16/39
Why is this a thing ◮ Take the reasoning language generated by K 1 , K 2 , ¬ All formulas of the form K 1 ¬ K 2 ¬ K 2 K 1 x ◮ There are infinitely many such formulas, hence uncountable many sets of formulas Klein: Social Interaction – A Formal Exploration 16/39
Why is this a thing ◮ Take the reasoning language generated by K 1 , K 2 , ¬ All formulas of the form K 1 ¬ K 2 ¬ K 2 K 1 x ◮ There are infinitely many such formulas, hence uncountable many sets of formulas Consider the set { x , K 1 x , ¬ K 2 K 1 x , ¬ K 1 ¬ K 2 K 1 x , K 2 ¬ K 1 ¬ K 2 K 1 x } Klein: Social Interaction – A Formal Exploration 16/39
Why is this a thing ◮ Take the reasoning language generated by K 1 , K 2 , ¬ All formulas of the form K 1 ¬ K 2 ¬ K 2 K 1 x ◮ There are infinitely many such formulas, hence uncountable many sets of formulas Consider the set { x , K 1 x , ¬ K 2 K 1 x , ¬ K 1 ¬ K 2 K 1 x , K 2 ¬ K 1 ¬ K 2 K 1 x } ¬ K 1 x → ¬ K 2 K 1 x K 1 ¬ K 1 x → K 1 ¬ K 2 K 1 x ¬ K 1 x → K 1 ¬ K 2 K 1 x Negative Introsp ¬ K 1 ¬ K 2 K 1 x → K 1 x Counterpos. K 2 ¬ K 1 ¬ K 2 K 1 x → K 2 K 1 x Klein: Social Interaction – A Formal Exploration 16/39
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