<latexit sha1_base64="P4jUJHo6g1yopyZBD74hiv3LdI=">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</latexit> Properties Fairness 3 COMP 1 5 9 3 Algorithmic Verification Safety and Liveness, Fairness Dr. Liam O’Connor CSE, UNSW (for now) Term 1 2020 1
Properties Fairness Behaviours Recall The infinite traces of a Kripke structure are called behaviours . So they are infinite sequences of state labels ⊆ (2 P ) ω . How many behaviours for these automata? •
Properties Fairness Behaviours Recall The infinite traces of a Kripke structure are called behaviours . So they are infinite sequences of state labels ⊆ (2 P ) ω . How many behaviours for these automata? • •
Properties Fairness Behaviours Recall The infinite traces of a Kripke structure are called behaviours . So they are infinite sequences of state labels ⊆ (2 P ) ω . How many behaviours for these automata? • • 4
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. • • • • • · · · σ 0 = • • • • • · · · σ 1 = • • • • • · · · σ 2 = • • • • • · · · σ 3 = • • • • • · · · σ 4 = . . . . . . . . . . . . . . .
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. σ δ = • • • • • · · · σ 0 = • • • • • · · · σ 1 = • • • • • · · · σ 2 = • • • • • · · · σ 3 = • • • • • · · · σ 4 = . . . . . . . . . . . . . . .
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. σ δ = • • • • • • · · · σ 0 = • • • • • · · · σ 1 = • • • • • · · · σ 2 = • • • • • · · · σ 3 = • • • • • · · · σ 4 = . . . . . . . . . . . . . . .
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. σ δ = • • • • • • • · · · σ 0 = • • • • • · · · σ 1 = • • • • • · · · σ 2 = • • • • • · · · σ 3 = • • • • • · · · σ 4 = . . . . . . . . . . . . . . .
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. σ δ = • • • • • • • • · · · σ 0 = • • • • • · · · σ 1 = • • • • • · · · σ 2 = • • • • • · · · σ 3 = • • • • • · · · σ 4 = . . . . . . . . . . . . . . .
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. σ δ = • • • • • • • • • · · · σ 0 = • • • • • · · · σ 1 = • • • • • · · · σ 2 = • • • • • · · · σ 3 = • • • • • · · · σ 4 = . . . . . . . . . . . . . . .
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. σ δ = • • • • • • • • • • · · · σ 0 = • • • • • · · · σ 1 = • • • • • · · · σ 2 = • • • • • · · · σ 3 = • • • • • · · · σ 4 = . . . . . . . . . . . . . . .
Properties Fairness Cantor’s Uncountability Argument Result It is impossible in general to enumerate the space of all behaviours. σ δ = • • • • • · · · Proof Suppose there ∃ a • • • • • · · · σ 0 = sequence σ 0 σ 1 σ 2 . . . that enumerates all • • • • • · · · σ 1 = behaviours. Then we can construct a • • • • • · · · σ 2 = devilish sequence σ δ that differs from any • • • • • · · · σ 3 = σ i at the i th position, and thus is • • • • • · · · σ 4 = not in our sequence. Contradiction! . . . . . . . . . . 12 . . . . .
Properties Fairness Metric for Behaviours We define the distance d ( σ, ρ ) ∈ R ≥ 0 between two behaviours σ and ρ as follows: d ( σ, ρ ) = 2 − sup { i ∈ N | σ | i = ρ | i } (we say that 2 −∞ = 0) 13
Properties Fairness Metric for Behaviours We define the distance d ( σ, ρ ) ∈ R ≥ 0 between two behaviours σ and ρ as follows: d ( σ, ρ ) = 2 − sup { i ∈ N | σ | i = ρ | i } (we say that 2 −∞ = 0) Intuitively, we consider two behaviours to be close if there is a long prefix for which they agree. 14
Properties Fairness Metric for Behaviours We define the distance d ( σ, ρ ) ∈ R ≥ 0 between two behaviours σ and ρ as follows: d ( σ, ρ ) = 2 − sup { i ∈ N | σ | i = ρ | i } (we say that 2 −∞ = 0) Intuitively, we consider two behaviours to be close if there is a long prefix for which they agree. Observations d ( x , y ) = 0 ⇔ x = y 15
Properties Fairness Metric for Behaviours We define the distance d ( σ, ρ ) ∈ R ≥ 0 between two behaviours σ and ρ as follows: d ( σ, ρ ) = 2 − sup { i ∈ N | σ | i = ρ | i } (we say that 2 −∞ = 0) Intuitively, we consider two behaviours to be close if there is a long prefix for which they agree. Observations d ( x , y ) = 0 ⇔ x = y d ( x , y ) = d ( y , x ) 16
Properties Fairness Metric for Behaviours We define the distance d ( σ, ρ ) ∈ R ≥ 0 between two behaviours σ and ρ as follows: d ( σ, ρ ) = 2 − sup { i ∈ N | σ | i = ρ | i } (we say that 2 −∞ = 0) Intuitively, we consider two behaviours to be close if there is a long prefix for which they agree. Observations d ( x , y ) = 0 ⇔ x = y d ( x , y ) = d ( y , x ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) 17
Properties Fairness Metric for Behaviours We define the distance d ( σ, ρ ) ∈ R ≥ 0 between two behaviours σ and ρ as follows: d ( σ, ρ ) = 2 − sup { i ∈ N | σ | i = ρ | i } (we say that 2 −∞ = 0) Intuitively, we consider two behaviours to be close if there is a long prefix for which they agree. Observations d ( x , y ) = 0 ⇔ x = y d ( x , y ) = d ( y , x ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) This forms a metric space and thus a topology on behaviours. 18
Properties Fairness Topology Definition A set S of subsets of U is called a topology if it contains ∅ and U , and is closed under union and finite intersection. Elements of S are called open and complements of open sets are called closed . 19
Properties Fairness Topology Definition A set S of subsets of U is called a topology if it contains ∅ and U , and is closed under union and finite intersection. Elements of S are called open and complements of open sets are called closed . Example (Sierpi´ nski Space) Let U = { 0 , 1 } and S = {∅ , { 1 } , U } . 20
Properties Fairness Topology Definition A set S of subsets of U is called a topology if it contains ∅ and U , and is closed under union and finite intersection. Elements of S are called open and complements of open sets are called closed . Example (Sierpi´ nski Space) Let U = { 0 , 1 } and S = {∅ , { 1 } , U } . Questions What are the closed sets of the Sierpi´ nski space? 21
Properties Fairness Topology Definition A set S of subsets of U is called a topology if it contains ∅ and U , and is closed under union and finite intersection. Elements of S are called open and complements of open sets are called closed . Example (Sierpi´ nski Space) Let U = { 0 , 1 } and S = {∅ , { 1 } , U } . Questions What are the closed sets of the Sierpi´ nski space? Can a set be clopen i.e. both open and closed? 22
Properties Fairness Topology for Metric Spaces Our metric space can be viewed as a topology by defining our open sets as (unions of) open balls : B ( σ, r ) = { ρ | d ( σ, ρ ) < r } This is analogous to open and closed ranges of numbers. 23
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