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ESSLLI 2018 course Logics for Epistemic and Strategic Reasoning in Multi-Agent Systems Supplementary slides on Operators on binary relations

Valentin Goranko Stockholm University ESSLLI 2018 August 6-10, 2018 Sofia University, Bulgaria

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Auxiliary slides: some operations on binary relations

Let W be a set of possible worlds and Ra ⊆ W 2, Rb ⊆ W 2 be two binary (accessibility) relations in W . Thus, each Ra, Rb is a set of ordered pairs in W . Then we can define union, intersection, and composition of Ra and Rb as the following binary relations in W .

• Ra∪b := Ra ∪ Rb = {(u, w) | uRaw or uRbw}.
• Ra∩b := Ra ∩ Rb = {(u, w) | uRaw and uRbw}.
• Ra◦b := Ra ◦ Rb = {(u, w) | uRav and vRbw for some v ∈W }.

Examples follow.

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Operations on binary relations: union

s1 s2 s3 s4 b b a a a b b a b a a

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Operations on binary relations: union

Ra∪b = Ra ∪ Rb s1 s2 s3 s4

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Operations on binary relations: intersection

s1 s2 s3 s4 b b b b b b a a a a a a

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Operations on binary relations: intersection

Ra ∩ Rb s1 s2 s3 s4 b b b a a a

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Operations on binary relations: intersection

Ra∩b = Ra ∩ Rb s1 s2 s3 s4

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Operations on binary relations: composition

s1 s2 s3 s4 a a b b a b a b

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Operations on binary relations: composition

Ra◦b = Ra ◦ Rb s1 s2 s3 s4 a a b b b a b a

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Operations on binary relations: composition

Ra◦b = Ra ◦ Rb s1 s2 s3 s4

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Operations on binary relations: composition

Ra◦b = Ra ◦ Rb s1 s2 s3 s4

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Operations on binary relations: composition 2

s1 s2 s3 s4 a a b b a b a b

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Operations on binary relations: composition 2

Rb◦a = Rb ◦ Ra s1 s2 s3 s4 a a b b b a b a

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Operations on binary relations: composition

Rb◦a = Rb ◦ Ra s1 s2 s3 s4

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Operations on binary relations: composition 2

Rb◦a = Rb ◦ Ra s1 s2 s3 s4

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Auxiliary slides: reflexive and transitive closure of a relation

Recall that, if W be a set and R ⊆ W 2 is a binary relation in W , then: R2 = R ◦ R = {(u, w) | uRv and vRw for some v ∈W }. Likewise, we define R3 = R ◦ R2 = R ◦ (R ◦ R), etc.... Rn = R ◦ Rn−1 = R ◦ (... n times)... ◦ R)...). Now, we define the reflexive and transitive closure of R as R∗ :=

n≥0 Rn, where R0 = {(w, w) | w ∈ W }.

Intuitively, R∗ consists of all pairs (u, v) in W such that v can be reached from u in a finite number of R-steps. Formally,

R∗ = {(u, v) ∈ W 2 | uRu1...Ruk = v for some u1, ..., uk ∈ W , k ≥ 0}

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Operations on binary relations: transitive closure example

R s1 s2 s3 s4

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Operations on binary relations: transitive closure example

R2 s1 s2 s3 s4

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Operations on binary relations: transitive closure example

R ∪ R2 s1 s2 s3 s4

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Operations on binary relations: transitive closure example

R3 s1 s2 s3 s4

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Operations on binary relations: transitive closure example

R∗ = R ∪ R2 ∪ R3 s1 s2 s3 s4

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