Łukasz Kaiser Joint work with Diana Fischer and Erich Grädel
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1 Quantitative Systems, Modal Logics and Games Łukasz Kaiser Joint work with Diana Fischer and Erich Grädel Mathematische Grundlagen der Informatik RWTH Aachen AlgoSyn, February 28 Quantitative Systems (Łukasz Kaiser) / 32
2 Overview Introduction Quantitative µ-calculus Quantitative Parity Games Crash Games Future Work Quantitative Systems (Łukasz Kaiser) 2 / 32
3 What is a Quantitative System and Logic? Different kinds of quantitative questions What is the probability of an event? (stochastic systems, PCTL) How much time does an action take? (timed systems, TCTL) What is the total cost of actions? (weighted timed systems, WCTL) What is the value of a variable that evolves by a differential equation? (hybrid systems, e.g. ICTL) How much money is gained given a discount rate? (quantitative systems) Questions to consider (in addition to decidability) Are there evolving variables in the system or just predicates? Is probability a part of the system or the logic? Is continuous time inherently special or just another variable? What properties of qualitative logic should carry over? Negation? Quantitative Systems (Łukasz Kaiser) 3 / 32
4 Basic Example P = Q = Infinite transition system with two quantitative predicates P and Q. Quantitative Systems (Łukasz Kaiser) 4 / 32
5 Basic Example P = Q = Infinite transition system with two quantitative predicates P and Q. Task: Evaluate P until Q in this setting! Quantitative Systems (Łukasz Kaiser) 4 / 32
6 The µ-calculus Syntax: φ = P i φ φ φ φ φ φ µx.φ νx.φ Example: P until Q = µx.(q (P X)) Motivation for µ-calculus The µ-calculus is more expressive than LTL and CTL Can express all bisimulation invariant MSO properties The connection between µ-calculus and parity games can be exploited algorithmically Quantitative Systems (Łukasz Kaiser) 5 / 32
7 Overview Introduction Quantitative µ-calculus Quantitative Parity Games Crash Games Future Work Quantitative Systems (Łukasz Kaiser) 6 / 32
8 Introduction De Alfaro, Faella, and Stoelinga introduced a quantitative µ-calculus, which is interpreted over metric transition systems, where predicates can take values in arbitrary metric spaces allows discounting in modalities is studied as a logical characterisation of quantitative versions of basic system relations like bisimulation We modify it in the following ways: depart from arbitrary metric spaces and stick to non-negative reals with infinity, R +, and fix the usual topology on this space allow discount factors to be greater than one allow additional discounts on the edges in transition systems decouple discounts from the diamond and box operators Quantitative Systems (Łukasz Kaiser) 7 / 32
9 Quantitative Transition Systems Quantitative Transition System (QTS): Q = (V, E, δ, {P i } i I ) δ E R +, P i V R +, Example: QTS Q, predicates P, Q, discount δ /3,/2 2 5, 2 3 2, Quantitative Systems (Łukasz Kaiser) 8 / 32
10 Quantitative µ-calculus Syntax: φ = P i c φ φ φ φ φ φ d φ µx.φ νx.φ Semantics: evaluation on QTS φ K V R + min max Example: v 2,4 P Q (v) = 2 Quantitative Systems (Łukasz Kaiser) 9 / 32
11 Quantitative µ-calculus Syntax: φ = P i c φ φ φ φ φ φ d φ µx.φ νx.φ Formally: Semantics: evaluation on QTS φ K V R + min max P i c K (v) = P i (v) c φ φ 2 K = min{ φ K, φ 2 K } φ φ 2 K = max{ φ K, φ 2 K } Example: v 2,4 P Q (v) = 2 Quantitative Systems (Łukasz Kaiser) 9 / 32
12 Evaluation of and : Non-Discounted Case Intuition: inf sup min and max for finitely branching systems Example: v v 2 2 P (v) = P (v) = Quantitative Systems (Łukasz Kaiser) / 32
13 Evaluation of and : Non-Discounted Case Intuition: inf sup min and max for finitely branching systems Example: v v 2 2 P (v) = Formally: φ K (v) = sup v ve φ K (v ) φ K (v) = inf v ve φ K (v ) P (v) = Quantitative Systems (Łukasz Kaiser) / 32
14 and over Discounted Systems Evaluation of modalities takes discount on edges into account: φ = sup δ φ φ = inf δ φ Example: v 2 P (v) = /2 P (v) = 2 2 P (v) = Quantitative Systems (Łukasz Kaiser) / 32
15 Evaluation of Fixed Points Intuition: Lattice (F, ) F = { f f is function from V to R + } top element, bottom element Theorem of Knaster and Tarski applies Inductive evaluation of µ: g = g α = { φ ε[x g α ] lim β<α φ ε[x gβ ] for α successor ordinal, for α limit ordinal, µx.φ K ε = g γ where g γ = g γ+ Quantitative Systems (Łukasz Kaiser) 2 / 32
16 Evaluation of Fixed Points Intuition: Lattice (F, ) F = { f f is function from V to R + } top element, bottom element Theorem of Knaster and Tarski applies Inductive evaluation of µ: g = g α = { φ ε[x g α ] lim β<α φ ε[x gβ ] for α successor ordinal, for α limit ordinal, µx.φ K ε = g γ where g γ = g γ+ Formally: µx.φ K ε = inf{ f F f = φ K ε[x f ] } νx.φ K ε = sup{ f F f = φ K ε[x f ] } Quantitative Systems (Łukasz Kaiser) 2 / 32
17 Quantitative µ-calculus Syntax: φ = P φ φ φ φ φ φ d φ µx.φ νx.φ Semantics: Evaluation on quantitative transitions system, φ K V R + Quantitative Systems (Łukasz Kaiser) 3 / 32
18 Quantitative µ-calculus Syntax: φ = P φ φ φ φ φ φ d φ µx.φ νx.φ Semantics: Evaluation on quantitative transitions system, φ K V R + φ ψ = min{ φ, ψ } φ ψ = max{ φ, ψ } φ = sup δ φ (succ) φ = inf δ φ (succ) d φ = d φ inductive evaluation of fixed points over lattice (F, ) QTS Q = (V, E, δ, P, Q) /3,/2 2 5, 2 3 2, Quantitative Systems (Łukasz Kaiser) 3 / 32
19 Basic Example in Quantitative µ-calculus P = Q = P until Q = µx.(q (P X)) Inductive Evaluation: P until Q (v) =, P until Q (v) = Q(v), P until Q 2 (v) = max{q(v), min{p(v), max w ve {Q(w)}}... Intuition: Value of P at the last time P is greater than Q Quantitative Systems (Łukasz Kaiser) 4 / 32
20 Negation Operator Reminder: So far, there is no negation and no duality of operators Goal: Find a suitable negation operator, include it in the syntax Properties expected: () φ φ (2) (φ ψ) φ ψ and (φ ψ) φ ψ (3) φ φ and φ φ (4) d φ β(d) φ for some β independent of φ (5) µx.φ νx. φ[x/ X] and νx.φ µx. φ[x/ X] Quantitative Systems (Łukasz Kaiser) 5 / 32
21 A Suitable Negation Operator The only possible negation operator for Qµ is Theorem f x R + R + x /x for x, x, for x =, for x =, f satisfies all the properties on the previous slide x there is no other negation operator for Qµ, even if interpreted only over non-discounted QTSs Quantitative Systems (Łukasz Kaiser) 6 / 32
22 Overview Introduction Quantitative µ-calculus Quantitative Parity Games Crash Games Future Work Quantitative Systems (Łukasz Kaiser) 7 / 32
23 Model Checking Games for Qµ? Motivation: connection between modal µ-calculus and parity games Goal: definition of suitable games to establish a similar connection in the quantitative setting Requirements: two players play on quantitative transition systems parity winning condition for infinite games quantitative outcomes for finite games Quantitative Systems (Łukasz Kaiser) 8 / 32
24 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
25 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
26 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
27 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
28 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
29 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) /2 4 2 Outcome: p(π) = = 4 2 Quantitative Systems (Łukasz Kaiser) 9 / 32
30 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
31 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
32 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
33 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
34 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
35 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
36 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
37 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Quantitative Systems (Łukasz Kaiser) 9 / 32
38 Quantitative Parity Games A quantitative parity game G = (V, V, V, E, δ, λ, Ω) / Outcome: p(π) = since lowest priority on the cycle is Quantitative Systems (Łukasz Kaiser) 9 / 32
39 Differences to Classical Parity Games No positional or strictly optimal strategy for Player /2 Quantitative Systems (Łukasz Kaiser) 2 / 32
40 Differences to Classical Parity Games No positional or strictly optimal strategy for Player /2 No bounded-memory strategy (not even ε-optimal strategy for fixed ε) /2 2 Quantitative Systems (Łukasz Kaiser) 2 / 32
41 Model Checking Games for Qµ Model Checking Game: MC[K, φ] = (V, V, V, E, δ, λ, Ω) Positions: (ψ, s), ψ is a subformula of φ, and s state of K, or (),( ) Player moves: (ψ φ, s) (ψ, s) (φ, s) ( φ, s) δ(s, t) (φ, t), t se (), se = (µx.φ, s) (φ, s) d (d φ, s) (φ, s) (X, s) (φ, s) Quantitative Systems (Łukasz Kaiser) 2 / 32
42 Model Checking Games for Qµ Player moves: (ψ φ, s) (ψ, s) (φ, s) ( φ, s) δ(s,t) (φ, t), t se ( ), se = (νx.φ, s) (φ, s) (X, s) (φ, s) Terminal Positions: ( P i c, s), (), and ( ) with payoff λ accordingly Priorities: Ω(X, s) is even if X is a ν-variable, Ω(X, s) is odd otherwise; priority chosen according to alternation level of X, all other positions get alternation depth of φ. Quantitative Systems (Łukasz Kaiser) 22 / 32
43 Example Model Checking Game MC[Q, φ] for Q and φ = µx.(p 2 X) µx.(p 2 X),a a /2 b P,a P 2 X,a 2 P,a Q X,b 2 /2 X,a X,a P 2 X,b µx.(p 2 X),b P,b 2 2 X,b X,b Quantitative Systems (Łukasz Kaiser) 23 / 32
44 Main Result Theorem For every formula φ in Qµ, a quantitative transition system K, and v K, the game MC[K, φ] is determined, i.e. for all positions v, sup inf p(π σ,ρ (v)) = inf sup p(π σ,ρ (v)) = valmc[k, φ](v) σ Γ ρ Γ ρ Γ σ Γ and valmc[k, φ](φ, v) = φ K (v). Proof: By induction on number of priorities using game unfolding one can show determinacy by providing strategies for both players correctness by showing the connection between computing the value of a game via unfolding and the inductive evaluation of fixed points Quantitative Systems (Łukasz Kaiser) 24 / 32
45 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority /2 v w q G λ(q) = Quantitative Systems (Łukasz Kaiser) 25 / 32
46 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority G /2 v w q λ(q) = Quantitative Systems (Łukasz Kaiser) 25 / 32
47 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority G /2 v w q λ (v) = λ(q) = Quantitative Systems (Łukasz Kaiser) 25 / 32
48 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority G /2 v w q λ (v) = valg (w) = λ(q) = Quantitative Systems (Łukasz Kaiser) 25 / 32
49 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority G /2 v w q λ (v) = valg (w) = λ(q) = λ (v) = Quantitative Systems (Łukasz Kaiser) 25 / 32
50 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority G v w q λ (v) = λ (v) = /2 valg (w) = valg (w) = 2 λ(q) = Quantitative Systems (Łukasz Kaiser) 25 / 32
51 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority G2 /2 v w q λ (v) = valg (w) = λ(q) = λ (v) = valg (w) = 2 λ 2 (v) = 2 Quantitative Systems (Łukasz Kaiser) 25 / 32
52 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority G2 /2 v w q λ (v) = valg (w) = λ(q) = λ (v) = valg (w) = 2 λ 2 (v) = 2 valg 2 (w) = 4 Quantitative Systems (Łukasz Kaiser) 25 / 32
53 Unfolding Quantitative Parity Games Unfolding of a game G: sequence of games G α with one less priority Gγ /2 v w q λ (v) = valg (w) = λ(q) = λ (v) = valg (w) = 2 λ 2 (v) = 2 valg2 (w) = 4 λ (v) = lim i< valg i (w) = = valg (v) Quantitative Systems (Łukasz Kaiser) 25 / 32
54 Strategy Construction Prove that valg (v) = valg(v) by providing strategies for both players Quantitative Systems (Łukasz Kaiser) 26 / 32
55 Strategy Construction Player plays σ ε 2 from G G v Player plays ρ ε 4 from G G m m m Quantitative Systems (Łukasz Kaiser) 26 / 32
56 Strategy Construction Player plays σ ε 2 from G G v Player plays ρ ε 4 from G G m m m Quantitative Systems (Łukasz Kaiser) 26 / 32
57 Strategy Construction Player plays unique move in G G v Player plays G m m m Quantitative Systems (Łukasz Kaiser) 26 / 32
58 Strategy Construction Player plays unique move in G G v Player plays G m m m Quantitative Systems (Łukasz Kaiser) 26 / 32
59 Strategy Construction Player plays σ ε 4D from G G G v Player plays ρ ε 8D from G k k choose k s.t. valg valg k ε 4 m m m Quantitative Systems (Łukasz Kaiser) 26 / 32
60 Strategy Construction Player plays σ ε 4D from G G G v Player plays ρ ε 8D from G k k choose k s.t. valg valg k ε 4 m m m Quantitative Systems (Łukasz Kaiser) 26 / 32
61 Strategy Construction Player plays σ ε i from G G v Player plays ε i 2 ρk i from G k i ε i = ε i 2D G m m m Quantitative Systems (Łukasz Kaiser) 26 / 32
62 Overview Introduction Quantitative µ-calculus Quantitative Parity Games Crash Games Future Work Quantitative Systems (Łukasz Kaiser) 27 / 32
63 Motivation Reasons for including evolving variables Standard way of thinking in engineering (e.g. hybrid systems) Compact representation of infinite state space Direct mapping of the kind of memory needed to win the game May be exploited algorithmically Moving to games played with numbers Not just a token is moved around, value is accumulated in a variable The numbers on the edges are now added instead of multiplied The payoff at the end node is exactly the accumulated value Quantitative Systems (Łukasz Kaiser) 28 / 32
64 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
65 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
66 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
67 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
68 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
69 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 Outcome: p(π) = + 2 = 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
70 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
71 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
72 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
73 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
74 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
75 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
76 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
77 Crash Games A crash game G = (V, V, V, E, δ, Ω) val = 2 2 Quantitative Systems (Łukasz Kaiser) 29 / 32
78 Crash Games A crash game G = (V, V, V, E, δ, Ω) 2 2 Outcome: p(π) = since lowest priority on the cycle is Quantitative Systems (Łukasz Kaiser) 29 / 32
79 Results on Crash Games Algorithmic properties of finite crash games (T. Gawlitza and H. Seidl) Game value characterized by hierarchical systems of simple equations Algorithm for solving such systems Run-time independent of the sizes of occurring numbers Based on strategy improvement and with the same complexity Connections to Quantitative Parity Games It is possible to translate between QPGs and Crash Games QPG Crash: take logarithms of discounts, get logarithm of value Corollary: algorithm for quantitative µ-calculus on finite systems Quantitative Systems (Łukasz Kaiser) 3 / 32
80 Results on Crash Games Algorithmic properties of finite crash games (T. Gawlitza and H. Seidl) Game value characterized by hierarchical systems of simple equations Algorithm for solving such systems Run-time independent of the sizes of occurring numbers Based on strategy improvement and with the same complexity Connections to Quantitative Parity Games It is possible to translate between QPGs and Crash Games QPG Crash: take logarithms of discounts, get logarithm of value Corollary: algorithm for quantitative µ-calculus on finite systems Crash QPG: take exponents Question: what is the µ-calculus for systems with variables? Quantitative Systems (Łukasz Kaiser) 3 / 32
81 Overview Introduction Quantitative µ-calculus Quantitative Parity Games Crash Games Future Work Quantitative Systems (Łukasz Kaiser) 3 / 32
82 Future Work Starting point: for simple systems with quantitative predicates there is a natural quantitative µ-calculus. First step: one evolving variable either as discount or changing additively. Open Questions: Can we formulate a µ-calculus with the variable in syntax? Will this solve the problem of inverted discounts? What if we allow to check the value of the variable for? Are systems with more variables and resets still decidable? How can real systems be approximated with such models? Quantitative Systems (Łukasz Kaiser) 32 / 32
83 Future Work Starting point: for simple systems with quantitative predicates there is a natural quantitative µ-calculus. First step: one evolving variable either as discount or changing additively. Open Questions: Can we formulate a µ-calculus with the variable in syntax? Will this solve the problem of inverted discounts? What if we allow to check the value of the variable for? Are systems with more variables and resets still decidable? How can real systems be approximated with such models? Thank You Quantitative Systems (Łukasz Kaiser) 32 / 32
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