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Optimal Bounds in Parametric LTL Games Martin Zimmermann Universit at des Saarlandes October 28th, 2013 AVACS Meeting Freiburg, Germany Martin Zimmermann Universit at des Saarlandes Parametric LTL Games 1/13 Motivation Linear

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Optimal Bounds in Parametric LTL Games

Martin Zimmermann

Universit¨ at des Saarlandes

October 28th, 2013

AVACS Meeting Freiburg, Germany

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 1/13

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Motivation

Linear Temporal Logic (LTL) as specification language: Simple and variable-free syntax and intuitive semantics. Expressively equivalent to first-order logic on words. LTL model-checking routinely applied in industrial settings. But LTL cannot express timing constraints.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/13

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Motivation

Linear Temporal Logic (LTL) as specification language: Simple and variable-free syntax and intuitive semantics. Expressively equivalent to first-order logic on words. LTL model-checking routinely applied in industrial settings. But LTL cannot express timing constraints. Possible remedies: Add F≤k for k ∈ N. Problem: finding “right” k impracticable.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/13

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Motivation

Linear Temporal Logic (LTL) as specification language: Simple and variable-free syntax and intuitive semantics. Expressively equivalent to first-order logic on words. LTL model-checking routinely applied in industrial settings. But LTL cannot express timing constraints. Possible remedies: Add F≤k for k ∈ N. Problem: finding “right” k impracticable. Alur et. al, Kupferman et. al: add F≤x for variable x. Now: does there exist a value x such that F≤xϕ holds? what is the best value x such that F≤xϕ holds? In Model-Checking: adding variable time bounds does not increase complexity.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/13

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Infinite Games

Arena A = (V , V0, V1, E): finite directed graph (V , E), V0 ⊆ V positions of Player 0 (circles), V1 = V \ V0 positions of Player 1 (squares). 1 2

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 3/13

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Infinite Games

Arena A = (V , V0, V1, E): finite directed graph (V , E), V0 ⊆ V positions of Player 0 (circles), V1 = V \ V0 positions of Player 1 (squares). 1 2 Play: path ρ0ρ1 · · · through A. Strategy for Player i: σ: V ∗Vi → V s.t. (v, σ(wv)) ∈ E. ρ0ρ1 · · · consistent with σ: ρn+1 = σ(ρ0 · · · ρn) for all n s.t. ρn ∈ Vi. Finite-state strategy: implemented by finite automaton with

  • utput.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 3/13

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PLTL: Syntax and Semantics

Parametric LTL: p atomic proposition, x ∈ X, y ∈ Y (X ∩ Y = ∅). ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | Xϕ | ϕUϕ | ϕRϕ | F≤xϕ | G≤yϕ

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/13

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PLTL: Syntax and Semantics

Parametric LTL: p atomic proposition, x ∈ X, y ∈ Y (X ∩ Y = ∅). ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | Xϕ | ϕUϕ | ϕRϕ | F≤xϕ | G≤yϕ Semantics w.r.t. variable valuation α: X ∪ Y → N: As usual for LTL operators. (ρ, n, α) | = F≤xϕ: ρ n n + α(x) ϕ (ρ, n, α) | = G≤yϕ: ρ n n + α(y) ϕ ϕ ϕ ϕ ϕ

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/13

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PLTL: Syntax and Semantics

Parametric LTL: p atomic proposition, x ∈ X, y ∈ Y (X ∩ Y = ∅). ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | Xϕ | ϕUϕ | ϕRϕ | F≤xϕ | G≤yϕ Semantics w.r.t. variable valuation α: X ∪ Y → N: As usual for LTL operators. (ρ, n, α) | = F≤xϕ: ρ n n + α(x) ϕ (ρ, n, α) | = G≤yϕ: ρ n n + α(y) ϕ ϕ ϕ ϕ ϕ Fragments: PLTLF: no parameterized always operators G≤y. PLTLG: no parameterized eventually operators F≤x.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/13

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PLTL Games

PLTL game: G = (A, v0, ϕ) with arena A (labeled by ℓ: V → 2P), initial vertex v0, and PLTL formula ϕ.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13

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PLTL Games

PLTL game: G = (A, v0, ϕ) with arena A (labeled by ℓ: V → 2P), initial vertex v0, and PLTL formula ϕ. Rules: all plays start in v0. Player 0 wins ρ0ρ1 · · · w.r.t. α, if (ℓ(ρ0)ℓ(ρ1) · · · , α) | = ϕ. Player 1 wins ρ0ρ1 · · · w.r.t. α, if (ℓ(ρ0)ℓ(ρ1) · · · , α) | = ϕ.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13

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PLTL Games

PLTL game: G = (A, v0, ϕ) with arena A (labeled by ℓ: V → 2P), initial vertex v0, and PLTL formula ϕ. Rules: all plays start in v0. Player 0 wins ρ0ρ1 · · · w.r.t. α, if (ℓ(ρ0)ℓ(ρ1) · · · , α) | = ϕ. Player 1 wins ρ0ρ1 · · · w.r.t. α, if (ℓ(ρ0)ℓ(ρ1) · · · , α) | = ϕ. σ is winning strategy for Player i w.r.t. α, if every consistent play is winning for Player i w.r.t. α. Winning valuations for Player i Wi(G) = {α | Player i has winning strategy for G w.r.t. α}

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13

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PLTL Games

PLTL game: G = (A, v0, ϕ) with arena A (labeled by ℓ: V → 2P), initial vertex v0, and PLTL formula ϕ. Rules: all plays start in v0. Player 0 wins ρ0ρ1 · · · w.r.t. α, if (ℓ(ρ0)ℓ(ρ1) · · · , α) | = ϕ. Player 1 wins ρ0ρ1 · · · w.r.t. α, if (ℓ(ρ0)ℓ(ρ1) · · · , α) | = ϕ. σ is winning strategy for Player i w.r.t. α, if every consistent play is winning for Player i w.r.t. α. Winning valuations for Player i Wi(G) = {α | Player i has winning strategy for G w.r.t. α}

Lemma

Determinacy: W0(G) is the complement of W1(G).

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13

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An Example

v0 {q0} {q1} {q0, q1} {d} {p0} {p1}

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 6/13

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An Example

v0 {q0} {q1} {q0, q1} {d} {p0} {p1} ϕ1 = FGd ∨

i∈{0,1} G(qi → Fpi) : W1(G1) = ∅.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 6/13

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An Example

v0 {q0} {q1} {q0, q1} {d} {p0} {p1} ϕ1 = FGd ∨

i∈{0,1} G(qi → Fpi) : W1(G1) = ∅.

ϕ2 = FGd ∨

i∈{0,1} G(qi → F≤xipi) : W0(G2) = ∅.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 6/13

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More Example Properties

Bounded B¨ uchi: GF≤xp

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 7/13

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More Example Properties

Bounded B¨ uchi: GF≤xp Finitary parity (Chatterjee, Henzinger, Horn): FG

  • c odd

  c → F≤x

  • c′>c

c′ even

c′   

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 7/13

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More Example Properties

Bounded B¨ uchi: GF≤xp Finitary parity (Chatterjee, Henzinger, Horn): FG

  • c odd

  c → F≤x

  • c′>c

c′ even

c′    Finitary Streett (CHH): FG

k

  • j=1

(Rj → F≤xGj)

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 7/13

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Decision Problems

Membership: given G, i ∈ {0, 1}, and α, is α ∈ Wi(G)? Emptiness: given G and i ∈ {0, 1}, is Wi(G) empty? Finiteness: given G and i ∈ {0, 1}, is Wi(G) finite? Universality: given G and i ∈ {0, 1}, is Wi(G) universal?

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 8/13

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Decision Problems

Membership: given G, i ∈ {0, 1}, and α, is α ∈ Wi(G)? Emptiness: given G and i ∈ {0, 1}, is Wi(G) empty? Finiteness: given G and i ∈ {0, 1}, is Wi(G) finite? Universality: given G and i ∈ {0, 1}, is Wi(G) universal? The benchmark:

Theorem (Pnueli, Rosner 1989)

Solving LTL games is 2Exptime-complete.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 8/13

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Decision Problems

Membership: given G, i ∈ {0, 1}, and α, is α ∈ Wi(G)? Emptiness: given G and i ∈ {0, 1}, is Wi(G) empty? Finiteness: given G and i ∈ {0, 1}, is Wi(G) finite? Universality: given G and i ∈ {0, 1}, is Wi(G) universal? The benchmark:

Theorem (Pnueli, Rosner 1989)

Solving LTL games is 2Exptime-complete. Adding parameterized operators does not increase complexity:

Theorem

All four decision problems are 2Exptime-complete.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 8/13

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Proof Idea

Emptiness for PLTLF games, i.e., only F≤x in ϕ.

  • 1. Duplicate arena, color one copy red, the other green. Player 0

can change between copies after every move.

  • 2. Inductively replace every F≤xψ by

(red → (redU(greenUψ)))∧(green → (greenU(redUψ)))

  • 3. Add conjunct GFred ∧ GFgreen to ϕ, obtain ϕ′.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 9/13

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Proof Idea

Emptiness for PLTLF games, i.e., only F≤x in ϕ.

  • 1. Duplicate arena, color one copy red, the other green. Player 0

can change between copies after every move.

  • 2. Inductively replace every F≤xψ by

(red → (redU(greenUψ)))∧(green → (greenU(redUψ)))

  • 3. Add conjunct GFred ∧ GFgreen to ϕ, obtain ϕ′.
  • 4. Player 0 wins LTL game (A′, ϕ′) iff there exists α s.t. Player 0

wins (A, ϕ) w.r.t. α.

  • 5. Proof relies on finite-state determinacy of LTL games.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 9/13

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Proof Idea

Emptiness for PLTLF games, i.e., only F≤x in ϕ.

  • 1. Duplicate arena, color one copy red, the other green. Player 0

can change between copies after every move.

  • 2. Inductively replace every F≤xψ by

(red → (redU(greenUψ)))∧(green → (greenU(redUψ)))

  • 3. Add conjunct GFred ∧ GFgreen to ϕ, obtain ϕ′.
  • 4. Player 0 wins LTL game (A′, ϕ′) iff there exists α s.t. Player 0

wins (A, ϕ) w.r.t. α.

  • 5. Proof relies on finite-state determinacy of LTL games.

Corollary: doubly-exponential upper bound on α.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 9/13

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Proof Idea

Emptiness for PLTLF games, i.e., only F≤x in ϕ.

  • 1. Duplicate arena, color one copy red, the other green. Player 0

can change between copies after every move.

  • 2. Inductively replace every F≤xψ by

(red → (redU(greenUψ)))∧(green → (greenU(redUψ)))

  • 3. Add conjunct GFred ∧ GFgreen to ϕ, obtain ϕ′.
  • 4. Player 0 wins LTL game (A′, ϕ′) iff there exists α s.t. Player 0

wins (A, ϕ) w.r.t. α.

  • 5. Proof relies on finite-state determinacy of LTL games.

Corollary: doubly-exponential upper bound on α. Full PLTL and other problems: use monotonicity and duality of F≤x and G≤y

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 9/13

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Optimization Problems

For PLTLF and PLTLG winning conditions, synthesis is an

  • ptimization problem: what is the best variable valuation in Wi(G)?

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 10/13

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Optimization Problems

For PLTLF and PLTLG winning conditions, synthesis is an

  • ptimization problem: what is the best variable valuation in Wi(G)?

Theorem

Let GF be a PLTLF game with winning condition ϕF and let GG be a PLTLG game with winning condition ϕG. The following values (and winning strategies realizing them) can be computed in triply-exponential time.

  • 1. minα∈W0(GF) minx∈var(ϕF) α(x).

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 10/13

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Optimization Problems

For PLTLF and PLTLG winning conditions, synthesis is an

  • ptimization problem: what is the best variable valuation in Wi(G)?

Theorem

Let GF be a PLTLF game with winning condition ϕF and let GG be a PLTLG game with winning condition ϕG. The following values (and winning strategies realizing them) can be computed in triply-exponential time.

  • 1. minα∈W0(GF) minx∈var(ϕF) α(x).
  • 2. minα∈W0(GF) maxx∈var(ϕF) α(x).

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 10/13

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Optimization Problems

For PLTLF and PLTLG winning conditions, synthesis is an

  • ptimization problem: what is the best variable valuation in Wi(G)?

Theorem

Let GF be a PLTLF game with winning condition ϕF and let GG be a PLTLG game with winning condition ϕG. The following values (and winning strategies realizing them) can be computed in triply-exponential time.

  • 1. minα∈W0(GF) minx∈var(ϕF) α(x).
  • 2. minα∈W0(GF) maxx∈var(ϕF) α(x).
  • 3. maxα∈W0(GG) maxy∈var(ϕG) α(y).
  • 4. maxα∈W0(GG) miny∈var(ϕG) α(y).

All values are at most doubly-exponential in the size of the game.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 10/13

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Proof Idea

  • 1. All problems reducible to minα∈W0(G) α(x) for ϕ with

var(ϕ) = {x}.

  • 2. Recall: algorithm for emptiness of W0(G) yields

doubly-exponential upper bound b on minα∈W0(G) α(x).

  • 3. For every n ∈ [0, b] test whether n is optimum:

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 11/13

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Proof Idea

  • 1. All problems reducible to minα∈W0(G) α(x) for ϕ with

var(ϕ) = {x}.

  • 2. Recall: algorithm for emptiness of W0(G) yields

doubly-exponential upper bound b on minα∈W0(G) α(x).

  • 3. For every n ∈ [0, b] test whether n is optimum:

3.1 Translate ϕ into B¨ uchi automaton Aϕ (treat F≤x as F). 3.2 Add a counter with range [0, n] for every occurence of x to simulate semantics of F≤x, obtain A′

ϕ of size 2|ϕ| · n|ϕ|.

3.3 Determize A′

ϕ to obtain parity automaton Pϕ of size

2O(|ϕ|2·(2n)2|ϕ|) and O(|ϕ| · n|ϕ|) colors. 3.4 Solve the parity game A × Pϕ.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 11/13

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Proof Idea

  • 1. All problems reducible to minα∈W0(G) α(x) for ϕ with

var(ϕ) = {x}.

  • 2. Recall: algorithm for emptiness of W0(G) yields

doubly-exponential upper bound b on minα∈W0(G) α(x).

  • 3. For every n ∈ [0, b] test whether n is optimum:

3.1 Translate ϕ into B¨ uchi automaton Aϕ (treat F≤x as F). 3.2 Add a counter with range [0, n] for every occurence of x to simulate semantics of F≤x, obtain A′

ϕ of size 2|ϕ| · n|ϕ|.

3.3 Determize A′

ϕ to obtain parity automaton Pϕ of size

2O(|ϕ|2·(2n)2|ϕ|) and O(|ϕ| · n|ϕ|) colors. 3.4 Solve the parity game A × Pϕ. Algorithm has triply exponential running time, since n is at most doubly-exponential.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 11/13

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Lower Bounds

For PLTLF games: doubly-exponential lower bound

Theorem

For every n ≥ 1, there exists a PLTLF game Gn with winning condition ϕn with |Gn| ∈ O(n2) and var(ϕn) = {x} such that W0(Gn) = ∅, but Player 1 wins Gn with respect to every variable valuation α such that α(x) ≤ 22n.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 12/13

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Lower Bounds

For PLTLF games: doubly-exponential lower bound

Theorem

For every n ≥ 1, there exists a PLTLF game Gn with winning condition ϕn with |Gn| ∈ O(n2) and var(ϕn) = {x} such that W0(Gn) = ∅, but Player 1 wins Gn with respect to every variable valuation α such that α(x) ≤ 22n. For PLTLG games: doubly-exponential lower bound (by duality)

Theorem

For every n ≥ 1, there exists a PLTLG game Gn with winning condition ϕn with |Gn| ∈ O(n2) and var(ϕn) = {y} such that W0(Gn) is not universal, but Player 0 wins Gn with respect to every variable valuation α such that α(x) ≤ 22n.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 12/13

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Open Problems

Optimization problems in 2Exptime?

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 13/13

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Open Problems

Optimization problems in 2Exptime? Tradeoff quality vs. size of finite-state strategies.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 13/13

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Open Problems

Optimization problems in 2Exptime? Tradeoff quality vs. size of finite-state strategies. Change order of quantifiers: emptiness ≡ ∃σ∃α∀ρ. What about ∃σ∀ρ∃α (non-uniform bounds)?

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 13/13

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Open Problems

Optimization problems in 2Exptime? Tradeoff quality vs. size of finite-state strategies. Change order of quantifiers: emptiness ≡ ∃σ∃α∀ρ. What about ∃σ∀ρ∃α (non-uniform bounds)? Explicit representation of Wi(G) for PLTLF and PLTLG games (upwards-closed and semi-linear)? How big has such a representation to be?

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 13/13

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Open Problems

Optimization problems in 2Exptime? Tradeoff quality vs. size of finite-state strategies. Change order of quantifiers: emptiness ≡ ∃σ∃α∀ρ. What about ∃σ∀ρ∃α (non-uniform bounds)? Explicit representation of Wi(G) for PLTLF and PLTLG games (upwards-closed and semi-linear)? How big has such a representation to be? Once again: Optimization problems in 2Exptime?

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 13/13

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The Game for the Lower Bounds

s {s} h0 {0} l0 {1} h1 {0} l1 {1} · · · · · · · · · hn−1 {0} ln−1 {1} hn {0′} ln {1′} e {e} f0 ∅ f1 {f } d0 ∅ d1 {$}

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 1/4

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A Play in Gn

We start in d1. The trace of a play looks as follows: {$}{s}{b0

0} · · · {b0 n−1}{b0 n}{e}F0D0

{s}{b1

0} · · · {b1 n−1}{b1 n}{e}F1D1

{s}{b2

0} · · · {b2 n−1}{b2 n}{e}F2D2 · · ·

where bj

0, . . . , bj n−1 ∈ {0, 1} ⇒ encoding of cj ∈ {0, 1, . . . , 2n − 1}

bj

n ∈ {0′, 1′}

Fj is {f } or ∅ (a flag for Player 0) Dj is {$} or ∅ (a flag for Player 1)

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/4

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A Play in Gn

We start in d1. The trace of a play looks as follows: {$}{s}{b0

0} · · · {b0 n−1}{b0 n}{e}F0D0

{s}{b1

0} · · · {b1 n−1}{b1 n}{e}F1D1

{s}{b2

0} · · · {b2 n−1}{b2 n}{e}F2D2 · · ·

where bj

0, . . . , bj n−1 ∈ {0, 1} ⇒ encoding of cj ∈ {0, 1, . . . , 2n − 1}

bj

n ∈ {0′, 1′}

Fj is {f } or ∅ (a flag for Player 0) Dj is {$} or ∅ (a flag for Player 1) Infinitely many $: primed bits encode numbers dℓ ∈ N

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/4

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The Winning Condition

Recall: numbers cj (adresses) and numbers dℓ whose bits are adressed by the cj There is an LTL formula ψ1 which expresses:

  • 1. Structure: Infinitely many $
  • 2. Initialization: after each $, the next cj is zero.
  • 3. Increment: if cj < 2n − 1, then cj+1 = cj + 1.
  • 4. Reset: if cj is 2n − 1, then it is followed by $.

ψ1 uses n nested next-operators to check condition 3.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 3/4

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The Winning Condition

Recall: numbers cj (adresses) and numbers dℓ whose bits are adressed by the cj There is an LTL formula ψ1 which expresses:

  • 1. Structure: Infinitely many $
  • 2. Initialization: after each $, the next cj is zero.
  • 3. Increment: if cj < 2n − 1, then cj+1 = cj + 1.
  • 4. Reset: if cj is 2n − 1, then it is followed by $.

ψ1 uses n nested next-operators to check condition 3.

Lemma

ϕ1 ⇒ dℓ ∈ {0, 1, . . . , 22n − 1}.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 3/4

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The Winning Condition, Part 2

ϕn = ψ1 → (ψf ∧ ψerr ∧ F≤xf ) where ψf : exactly one f ψerr: Player 0 used f to mark a single bit that is incorrectly updated from dℓ to dℓ+1 (formula uses adresses to verify this), or a dℓ with dℓ = 22n − 1 (no primed 0 between two $).

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/4

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The Winning Condition, Part 2

ϕn = ψ1 → (ψf ∧ ψerr ∧ F≤xf ) where ψf : exactly one f ψerr: Player 0 used f to mark a single bit that is incorrectly updated from dℓ to dℓ+1 (formula uses adresses to verify this), or a dℓ with dℓ = 22n − 1 (no primed 0 between two $). Player 0 wins, since Player 1 has to reach 22n − 1 or has to introduce an increment-error. But this can take more than 22n − 1 moves using correct updates.

Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/4