Motivation Fixpoint Games Soundness and Completeness Conclusion Fixpoint Games Barbara K¨ onig Universit¨ at Duisburg-Essen Joint work with Paolo Baldan, Christina Mika-Michalski, Tommaso Padoan (POPL ’19) Barbara K¨ onig Fixpoint Games 1
Motivation Fixpoint Games Soundness and Completeness Conclusion Motivation: Solving Systems of Fixpoint Equations 1 Case of One Equation Case of Multiple Equations Fixpoint Games 2 Soundness and Completeness 3 Conclusion 4 Barbara K¨ onig Fixpoint Games 2
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving one Fixpoint Equation We are interested in techniques for solving (systems of) fixpoint equations over a lattice One-equation case Solve the equation E given as x = η f ( x ) where f : L → L is a monotone function over a complete lattice ( L , ⊑ ) η ∈ { µ, ν } , indicating whether we are interested in the least ( µ ) or greatest ( ν ) fixpoint The solution of E is denoted by sol ( E ) Applications in concurrency theory, model checking, program analysis Barbara K¨ onig Fixpoint Games 3
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving one Fixpoint Equation Solution techniques The Knaster-Tarski theorem guarantees the existence of least and greatest fixpoints for monotone functions Kleene iteration: whenever f is (co-)continuous i ∈ N f i ( ⊥ ) η = µ (least fixpoint): sol ( E ) = � η = ν (greatest fixpoint): sol ( E ) = � i ∈ N f i ( ⊤ ) In order to check whether l ⊑ sol ( E ) for some l ∈ L : η = µ (least fixpoint): use ranking functions η = ν (greatest fixpoint): construct a postfix-point l ′ ( l ′ ⊑ f ( l ′ )) such that l ⊑ l ′ Barbara K¨ onig Fixpoint Games 4
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving one Fixpoint Equation ⊤ If f is not (co-)continuous: f ( ⊤ ) ❀ Kleene iteration over the ordinals f i ( ⊤ ) (beyond ω ) . . . Pre ( f ) . . . . . . ν f . . . ω + 2 ω · 2 Fix ( f ) 3 ω + 1 µ f ω 2 Post ( f ) f i ( ⊥ ) 1 f ( ⊥ ) 0 ⊥ Barbara K¨ onig Fixpoint Games 5
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving one Fixpoint Equation Examples Bisimilarity characterized as a greatest fixpoint Behavioural metric characterized a a least fixpoint Barbara K¨ onig Fixpoint Games 6
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving (Systems of) Fixpoint Equations System of fixpoint equations Let L be a lattice. A system of equations E over L is of the following form, where f i : L m → L are monotone functions and η i ∈ { µ, ν } . x 1 = η 1 f 1 ( x 1 , . . . , x m ) . . . x m = η m f m ( x 1 , . . . , x m ) The solution of E , denoted sol ( E ) ∈ L m , is defined inductively as follows: sol ( ∅ ) = () sol ( E ) = ( sol ( E [ x m := s m ]) , s m ) where s m = η m ( λ x . f m ( sol ( E [ x m := x ]) , x )) Barbara K¨ onig Fixpoint Games 7
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving (Systems of) Fixpoint Equations Remarks: E [ x m := x ] is a system of m − 1 equations that one obtains by fixing the value of x m as x and removing the last equation. Intuitively we fix the value of x m as x , solve the remaining equation systems parameterized over x and then perform a fixpoint iteration (least or greatest) over x . The order of the equations matters. The solution is a fixpoint of the equation system (one of typically many fixpoints). Barbara K¨ onig Fixpoint Games 8
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving (Systems of) Fixpoint Equations Example: µ -calculus model checking We consider the modal µ -calculus with ✷ (“all successor states satisfy . . . ”), ✸ (“some successor state satisfies . . . ”), least and greatest fixpoints. P a b ν x 2 . ( µ x 1 . ( ✸ x 1 ∨ ( P ∧ ✸ x 2 )) ∧ ✷ x 2 ) Equations over the powerset lattice of states: = µ ✸ x 1 ∪ ( P ∩ ✸ x 2 ) x 1 x 1 ∩ ✷ x 2 x 2 = ν Barbara K¨ onig Fixpoint Games 9
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving (Systems of) Fixpoint Equations Example: µ -calculus model checking P a b Equations over the powerset lattice of states: = µ ✸ x 1 ∪ ( P ∩ ✸ x 2 ) x 1 x 1 : “there exists a path such that eventually P holds and x 2 holds for some successor” x 1 ∩ ✷ x 2 x 2 = ν x 2 : “ x 1 holds and all successors satisfy x 2 ” Combined: “from all reachable states there is a path along which P holds infinitely often” Barbara K¨ onig Fixpoint Games 10
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving (Systems of) Fixpoint Equations Efficient algorithms for µ -calculus model-checking n : number of states d : alternation depth of formula Naive approach: use the definition ❀ O ( n d ) Reduce model-checking problem to a parity game and determine whether the existential player has a winning strategy Local on-the fly algorithms [Stevens, Stirling] that perform an on-the fly search for a winning strategy of the existential player (proving that a given state satisfies a formula) d 2 ) Progress measures [Jurdzinski] ❀ O ( n Quasi-polynomial algorithms [Calude, Jain, Khoussainov, Bakhadyr, Li, Stephan] ❀ O ( n ⌈ log d ⌉ + c ) Barbara K¨ onig Fixpoint Games 11
Motivation Fixpoint Games Soundness and Completeness Conclusion Solving (Systems of) Fixpoint Equations Example: lattice-valued µ -calculi Variants: Non-boolean µ -calculi that do not check whether a formula holds in a state, but measure the “degree” with respect to which a formula is satisfied: x | = ϕ is replaced by � ϕ � : X → L Latticed µ -calculus [Kupferman, Lustig] ❀ over a lattice L Quantitative probabilistic µ -calculus [Huth, Kwiatkowska] ❀ over the real interval L = [0 , 1] � Lukasiewicz µ -calculus [Mio, Simpson] ❀ over the real interval L = [0 , 1] ❀ we require methods and techniques for solving fixpoint equations over general lattices (as opposed to powerset lattices) Barbara K¨ onig Fixpoint Games 12
Motivation Fixpoint Games Soundness and Completeness Conclusion Fixpoint Games Aim: consider a game perspective for solving systems of fixpoint equations for general lattices Let E be a system of m equations over a lattice L with a basis B L ( B L ⊆ L such that every l ∈ L can be obtained as l = � B ′ where B ′ ⊆ B L ). Let sol ( E ) = ( s 1 , . . . , s m ) be the solution. Given b ∈ B L , i ∈ { 1 , . . . , m } the existential player ( ∃ , Eve) wants to prove that b ⊑ s i . The universal player ( ∀ , Adam) is the adversary of ∃ and wants to show that b �⊑ s i . Precursor games: Parity games Unfolding games [Venema] are being played on a powerset lattice single fixpoint equation Barbara K¨ onig Fixpoint Games 13
Motivation Fixpoint Games Soundness and Completeness Conclusion Fixpoint Games Fixpoint game (first version) Position Player Moves ∃ ( l 1 , . . . , l m ) such that b ⊑ f i ( l 1 , . . . , l m ) ( b , i ) ( b ′ , j ) such that b ′ ⊑ l j ( l 1 , . . . , l m ) ∀ b , b ′ ∈ B L , ⊥ �∈ B L , ( l 1 , . . . , l m ) ∈ L m Winning condition (“parity condition”) ∃ ∀ ∀ unable to move ∃ unable to move Finite game Infinite game η h = ν η h = µ Where h ∈ { 1 , . . . , m } is the highest equation index occurring infinitely often. Barbara K¨ onig Fixpoint Games 14
Motivation Fixpoint Games Soundness and Completeness Conclusion Fixpoint Games We play the game on the powerset lattice L = P ( { a , b } ) with basis B L = {{ a } , { b }} for b = { a } , i = 2: P a b = µ ✸ x 1 ∪ ( P ∩ ✸ x 2 ) = f 1 ( x 1 , x 2 ) x 1 x 2 = ν x 1 ∩ ✷ x 2 = f 2 ( x 1 , x 2 ) Remember: the second component of the solution contains all states such that “from all reachable states there is a path along which P holds infinitely often” Barbara K¨ onig Fixpoint Games 15
Motivation Fixpoint Games Soundness and Completeness Conclusion Fixpoint Games Notation: Game positions (nodes) of ∃ : ✸ Game positions (nodes) of ∀ : ✷ Barbara K¨ onig Fixpoint Games 16
Motivation Fixpoint Games Soundness and Completeness Conclusion Fixpoint Games ( { a } , 2) ( { a } , { a , b } ) ( { a } , 1) ( { b } , 2) ( { a } , ∅ ) ( { b } , ∅ ) ( { b } , { b } ) ( { b } , 1) ( ∅ , { b } ) Only minimal moves of ∃ are given. Thick arrows: winning strategy of ∃ Barbara K¨ onig Fixpoint Games 17
Motivation Fixpoint Games Soundness and Completeness Conclusion Fixpoint Games Is the game correct and complete for all lattices? (“ ∃ has a winning strategy for ( b , i ) ⇐ ⇒ b ⊑ s i ”) Counterexample L = N ∪ { ω } , B L = L \{ 0 } f : L → L , f ( n ) = n + 1, f ( ω ) = ω ω x = µ f ( x ) We play a game to check whether ω is below the 2 solution (= least fixpoint): 1 ∃ ∀ ω ❀ ω ❀ ω . . . 0 ∀ would win this game . . . This means that something is wrong! Barbara K¨ onig Fixpoint Games 18
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