MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA Erik Pul Leipzig University August 22, 2017 QuntLA
MOTIVATION 0% 1 item x restocked in shop) every Mondy
MOTIVATION 2% 1 item x restocked in shop) every Mondy purchse events between Mondys
MOTIVATION 5% 1 item x restocked in shop) every Mondy purchse events between Mondys modeled s infinite) sequence over lphbet {restock, demnd}
MOTIVATION 8% 1 item x restocked in shop) every Mondy purchse events between Mondys modeled s infinite) sequence over lphbet {restock, demnd} How mny purchses of x ech week?
MOTIVATION 11% 1 item x restocked in shop) every Mondy purchse events between Mondys modeled s infinite) sequence over lphbet {restock, demnd} How mny purchses of x ech week? miniml demnd
MOTIVATION 14% 1 item x restocked in shop) every Mondy purchse events between Mondys modeled s infinite) sequence over lphbet {restock, demnd} How mny purchses of x ech week? miniml demnd mximl demnd
MOTIVATION 17% 1 item x restocked in shop) every Mondy purchse events between Mondys modeled s infinite) sequence over lphbet {restock, demnd} How mny purchses of x ech week? miniml demnd mximl demnd long-term verge demnd
MOTIVATION 20% 1 item x restocked in shop) every Mondy purchse events between Mondys modeled s infinite) sequence over lphbet {restock, demnd} How mny purchses of x ech week? miniml demnd mximl demnd long-term verge demnd Quntittive Monitor Automt [Chtterjee, Henzinger, Otop 16]
QUANTITATIVE MONITOR AUTOMATA 23% 2 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton
QUANTITATIVE MONITOR AUTOMATA 26% 2 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton Σ finite lphbet
QUANTITATIVE MONITOR AUTOMATA 29% 2 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton Σ finite lphbet Q, I, F sttes initil,finl)
QUANTITATIVE MONITOR AUTOMATA 32% 2 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton Σ finite lphbet Q, I, F sttes initil,finl) n number of monitor counters
QUANTITATIVE MONITOR AUTOMATA 35% 2 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton Σ finite lphbet Q, I, F sttes initil,finl) n δ Q Σ Q Z {s, t}) n number of monitor counters trnsitions
QUANTITATIVE MONITOR AUTOMATA 38% 2 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton Σ finite lphbet Q, I, F sttes initil,finl) n δ Q Σ Q Z {s, t}) n Vl: Z N R { } number of monitor counters trnsitions vlution function
QUANTITATIVE MONITOR AUTOMATA 41% 2 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton Σ finite lphbet Q, I, F sttes initil,finl) n δ Q Σ Q Z {s, t}) n Vl: Z N R { } number of monitor counters trnsitions vlution function e.g. minimum, mximum, long-term verge
QUANTITATIVE MONITOR AUTOMATA 44% 3 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton δ Q Σ Q Z {s, t}) n trnsitions ) s ) 1 b ) 1 ) t b ) 0 ) s q 0 0 0 s 2 3 t q 1 q 2 q 3 q 4 q 5...
QUANTITATIVE MONITOR AUTOMATA 47% 3 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton δ Q Σ Q Z {s, t}) n trnsitions ) s ) 1 b ) 1 ) t b ) 0 ) s q 0 0 q 1 2 0 s 2 3 t q 2 q 3 q 4 q 5...
QUANTITATIVE MONITOR AUTOMATA 50% 3 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton δ Q Σ Q Z {s, t}) n trnsitions ) s ) 1 b ) 1 ) t b ) 0 ) s q 0 0 q 1 2 0 q 2 s q 3 5 2 3 t q 4 q 5...
QUANTITATIVE MONITOR AUTOMATA 52% 3 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton δ Q Σ Q Z {s, t}) n trnsitions ) s ) 1 b ) 1 ) t b ) 0 ) s q 0 0 q 1 2 0 q 2 s q 3 5 2 q 4 3 q 5 t... 3
QUANTITATIVE MONITOR AUTOMATA 55% 3 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton δ Q Σ Q Z {s, t}) n trnsitions ) s ) 1 b ) 1 ) t b ) 0 ) s q 0 0 q 1 2 0 q 2 s q 3 5 2 q 4 3 q 5 t... 3 Weight of run: Vlz i ) i 1 )
QUANTITATIVE MONITOR AUTOMATA 58% 3 A = Σ, Q, I, F, n, δ, Vl) Quntittive Monitor Automton δ Q Σ Q Z {s, t}) n trnsitions ) s ) 1 b ) 1 ) t b ) 0 ) s q 0 0 q 1 2 0 q 2 s q 3 5 2 q 4 3 q 5 t... 3 Weight of run: Vlz i ) i 1 ) Weight of ω-word: infimum over ll runs
EXAMPLE 61% 5 demnd, 1, 0) restock, t, s) restock, s, 0) q 0 q 1 q 2 restock, s, t) demnd, 0, 1)
EXAMPLE 64% 5 demnd, 1, 0) restock, t, s) restock, s, 0) q 0 q 1 q 2 restock, s, t) demnd, 0, 1) sequence 5, 3, 7, 4,... of demnds per week
EXAMPLE 67% 5 demnd, 1, 0) restock, t, s) restock, s, 0) q 0 q 1 q 2 restock, s, t) demnd, 0, 1) sequence 5, 3, 7, 4,... of demnds per week vlution function to compute long-time verge, minimum,...
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 70% 6 β ::= P x) x y x X β β β x.β X.β
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 73% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ β? ψ 1 : ψ 2 w) = { ψ 1 w) ψ 2 w) if w = β otherwise
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 76% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ β? ψ 1 : ψ 2 w) = { ψ 1 w) ψ 2 w) if w = β otherwise
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 79% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x β? ψ 1 : ψ 2 w) = { ψ 1 w) ψ 2 w) if w = β otherwise Vl x.ζ x w) = Vl ζ x w[x i])) i 1 )
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 82% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x x,z y.ψ w) =
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 85% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x x 1 i=x+1 ψ w[y i]) x,z y.ψ w) = if x Z nd x Z : x > x otherwise
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 88% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x x 1 i=x+1 ψ w[y i]) x,z y.ψ w) = if x Z nd x Z : x > x otherwise x,z ) ) ϕ = inf Z. z.z Z P restock z))? Vl x. y.1 :
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 91% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x Muller utomt: Vl x.ψ z i = ψ w[x i]) Vl x.ψ w) = Vlz 0, z 1, z 2,...)
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 94% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x Muller utomt: Vl x.ψ z i = ψ w[x i]) Vl x.ψ w) = Vlz 0, z 1, z 2,...) w = 0 1 2 3 4... 0 z 0 ) ) ) ) ) 1 2 3 4... z 1 z 2 z 3 z 4
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA 97% 6 β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x Muller utomt: Vl x.ψ z i = ψ w[x i]) Vl x.ψ w) = Vlz 0, z 1, z 2,...) w = 0 1 2 3 4... 0 z 0 ) ) ) ) ) 1 2 3 4... z 1 z 2 z 3 z 4 correct weights is recognizble property
MONITOR LOGICS FOR QUANTITATIVE MONITOR AUTOMATA β ::= P x) x y x X β β β x.β X.β ψ ::= k β? ψ : ψ ζ x ::= β? ζ x : ζ x x,z y.ψ ϕ ::= β? ϕ : ϕ minϕ, ϕ) inf x.ϕ inf X.ϕ Vl x.ζ x QMA: Vl x.ζ x ϕ = Vl x. x,z y.1 ) restock s 1 demnd 1 0 demnd 1 0 restock t s 1 demnd 1 0 6 100%