Visibly Linear Dynamic Logic Joint work with Alexander Weinert - - PowerPoint PPT Presentation

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Visibly Linear Dynamic Logic Joint work with Alexander Weinert - - PowerPoint PPT Presentation

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Visibly Linear Dynamic Logic Joint work with Alexander Weinert (Saarland University) Martin Zimmermann Saarland University December 14th, 2016 FSTTCS 2016, Chennai, India Martin Zimmermann Saarland University Visibly Linear Dynamic Logic

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SLIDE 1

Visibly Linear Dynamic Logic

Joint work with Alexander Weinert (Saarland University)

Martin Zimmermann

Saarland University

December 14th, 2016

FSTTCS 2016, Chennai, India

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 1/17

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SLIDE 2

The Everlasting Quest for Expressiveness

Consider an arbiter granting access to a shared resource. Requirements: “Every request q is eventually answered by a response p” “Every request q is eventually answered by a response p after an even number of steps” “There are never more responses than requests”

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 2/17

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SLIDE 3

The Everlasting Quest for Expressiveness

Consider an arbiter granting access to a shared resource. Requirements: “Every request q is eventually answered by a response p” Linear Temporal Logic: G(q → F p) “Every request q is eventually answered by a response p after an even number of steps” “There are never more responses than requests”

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 2/17

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SLIDE 4

The Everlasting Quest for Expressiveness

Consider an arbiter granting access to a shared resource. Requirements: “Every request q is eventually answered by a response p” Linear Temporal Logic: G(q → F p) “Every request q is eventually answered by a response p after an even number of steps” Linear Dynamic Logic: [true∗]( q → (true · true)∗p ) “There are never more responses than requests”

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 2/17

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SLIDE 5

The Everlasting Quest for Expressiveness

Consider an arbiter granting access to a shared resource. Requirements: “Every request q is eventually answered by a response p” Linear Temporal Logic: G(q → F p) “Every request q is eventually answered by a response p after an even number of steps” Linear Dynamic Logic: [true∗]( q → (true · true)∗p ) “There are never more responses than requests” Expressible with pushdown automata/context-free grammars as guards ⇒ Visibly Linear Dynamic Logic

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 2/17

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SLIDE 6

Outline

  • 1. Preliminaries
  • 2. Expressiveness
  • 3. VLDL Verification
  • 4. Discussion

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 3/17

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SLIDE 7

Outline

  • 1. Preliminaries
  • 2. Expressiveness
  • 3. VLDL Verification
  • 4. Discussion

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 4/17

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SLIDE 8

Visibly Pushdown Automata

Partition input alphabet Σ into Σc (calls), Σr (returns), and Σℓ (local actions). A visibly pushdown automaton (VPA) has to push when processing a call, pop when processing a return, and leave the stack unchanged when processing a local action. Stack height determined by input word ⇒ closure under union, intersection, and complement.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 5/17

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SLIDE 9

Visibly Pushdown Automata

Partition input alphabet Σ into Σc (calls), Σr (returns), and Σℓ (local actions). A visibly pushdown automaton (VPA) has to push when processing a call, pop when processing a return, and leave the stack unchanged when processing a local action. Stack height determined by input word ⇒ closure under union, intersection, and complement. Examples: anbn is a VPL, if a is a call and b a return. wwR is not a VPL.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 5/17

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Visibly Linear Dynamic Logic (VLDL)

Syntax ϕ := p | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | Aϕ | [A]ϕ where p ∈ P ranges over atomic propositions and A ranges over VPA’s. All VPA’s have the same partition of 2P into calls, returns, and local actions.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 6/17

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Visibly Linear Dynamic Logic (VLDL)

Syntax ϕ := p | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | Aϕ | [A]ϕ where p ∈ P ranges over atomic propositions and A ranges over VPA’s. All VPA’s have the same partition of 2P into calls, returns, and local actions. Semantics: (w ∈ (2P)ω) w | = Aϕ if there exists an n such that w0 · · · wn−1 is accepted by A and wnwn+1wn+2 · · · | = ϕ. w | = [A]ϕ if for every n s.t. w0 · · · wn−1 is accepted by A we have wnwn+1wn+2 · · · | = ϕ.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 6/17

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Example

“Every request q is eventually answered by a response p and there are never more responses than requests” [A∗]( q → A∗p ) ∧ ¬Atrue where A∗ accepts every word, and A accepts those words with more responses than requests. Both languages are visibly pushdown, if {q} is a call, {p} is a return, and ∅ and {p, q} are local actions.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 7/17

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SLIDE 13

Outline

  • 1. Preliminaries
  • 2. Expressiveness
  • 3. VLDL Verification
  • 4. Discussion

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 8/17

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SLIDE 14

Expressiveness

Lemma

VLDL and non-deterministic ω-VPA are expressively equivalent.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17

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SLIDE 15

Expressiveness

Lemma

VLDL and non-deterministic ω-VPA are expressively equivalent. Proof Idea VLDL non-deterministic ω-VPA

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17

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SLIDE 16

Expressiveness

Lemma

VLDL and non-deterministic ω-VPA are expressively equivalent. Proof Idea VLDL non-deterministic ω-VPA Deterministic Stair Automata [LMS ’04] O(2n)

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17

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SLIDE 17

Expressiveness

Lemma

VLDL and non-deterministic ω-VPA are expressively equivalent. Proof Idea VLDL non-deterministic ω-VPA Deterministic Stair Automata [LMS ’04] O(2n) O(n2)

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17

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SLIDE 18

Expressiveness

Lemma

VLDL and non-deterministic ω-VPA are expressively equivalent. Proof Idea VLDL non-deterministic ω-VPA Deterministic Stair Automata [LMS ’04] O(2n) O(n2) 1-way Alternating Jumping Automata O(n2)

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17

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SLIDE 19

Expressiveness

Lemma

VLDL and non-deterministic ω-VPA are expressively equivalent. Proof Idea VLDL non-deterministic ω-VPA Deterministic Stair Automata [LMS ’04] O(2n) O(n2) 1-way Alternating Jumping Automata O(n2) [Bozelli ’07] O(2n)

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17

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SLIDE 20

Expressiveness

Lemma

VLDL and non-deterministic ω-VPA are expressively equivalent. Proof Idea VLDL non-deterministic ω-VPA Deterministic Stair Automata [LMS ’04] O(2n) O(n2) 1-way Alternating Jumping Automata O(n2) [Bozelli ’07] O(2n)

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17

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From Stair Automata to VLDL

c c c c r c r r c l c c r r cω 3 7 1 3 2 2 5 2 1 2 3

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 10/17

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From Stair Automata to VLDL

c c c c r c r r c l c c r r cω 3 7 1 3 2 2 5 2 1 2 3 Acceptance: maximal priority occuring at infinitely many steps even

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 10/17

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SLIDE 23

From Stair Automata to VLDL

c c c c r c r r c l c c r r cω 3 7 1 3 2 2 5 2 1 2 3 Acceptance: maximal priority occuring at infinitely many steps even Equivalently: For some state q of even priority c there is step with state q s.t.

  • 1. after this step, no larger priority appears at a step, and
  • 2. for every step with state q, there is a later one with state q.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 10/17

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SLIDE 24

From Stair Automata to VLDL

c c c c r c r r c l c c r r cω 3 7 1 3 2 2 5 2 1 2 3 Acceptance: maximal priority occuring at infinitely many steps even Equivalently: For some state q of even priority c there is step with state q s.t.

  • 1. after this step, no larger priority appears at a step, and
  • 2. for every step with state q, there is a later one with state q.
  • q∈Qeven

qI A′

q

 

  • q′∈Q>Ω(q)

[qA′

q′ ]false

  ∧ [A′

q ]qA′ qtrue

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 10/17

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SLIDE 25

Outline

  • 1. Preliminaries
  • 2. Expressiveness
  • 3. VLDL Verification
  • 4. Discussion

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 11/17

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SLIDE 26

Satisfiability

Theorem

VLDL Satisfiability is ExpTime-complete.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 12/17

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Satisfiability

Theorem

VLDL Satisfiability is ExpTime-complete. Proof Sketch Membership: Construct equivalent ω-VPA and check it for emptiness. Hardness: Adapt ExpTime-hardness proof of LTL model-checking of pushdown systems [BEM ’97]

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 12/17

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Model Checking

Theorem

VLDL model checking of visibly pushdown systems is ExpTime-complete.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 13/17

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Model Checking

Theorem

VLDL model checking of visibly pushdown systems is ExpTime-complete. Proof Sketch Membership: To check S | = ϕ, construct ω-VPA equivalent to ¬ϕ and check intersection with S for emptiness. Hardness: Follows immediately from ExpTime-hardness of satisfiability.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 13/17

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Synthesis

Theorem

Solving infinite games on visibly pushdown graphs with VLDL winning conditions is 3ExpTime-complete.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 14/17

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Synthesis

Theorem

Solving infinite games on visibly pushdown graphs with VLDL winning conditions is 3ExpTime-complete. Proof Sketch Membership: To determine the winner, construct an ω-VPA that accepts the winning condition and solve the resulting game with VPA winning condition [LMS ’04]. Hardness: Adapt 3ExpTime-hardness proof of pushdown games with LTL winning condition [LMS ’04].

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 14/17

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SLIDE 32

Outline

  • 1. Preliminaries
  • 2. Expressiveness
  • 3. VLDL Verification
  • 4. Discussion

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 15/17

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The Competitors

“If p holds true immediately after entering module m, it shall hold immediately after the corresponding return from m as well”

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 16/17

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The Competitors

“If p holds true immediately after entering module m, it shall hold immediately after the corresponding return from m as well” VLDL: [Ac ](p → Arp) with Ac Σc, ↓A Σr, ↑A Σℓ, → Σc, ↓A Ar Σc, ↓A Σr, ↑A Σℓ, → Σr, ↑⊥

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 16/17

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The Competitors

“If p holds true immediately after entering module m, it shall hold immediately after the corresponding return from m as well” ω-VPA: Σc, ↓P Σr, ↑P Σc, ↓P Σr, ↑P Σℓ, → Σp

ℓ , →

Σp

c, ↓ ¯

P Σp

r , ↑ ¯

P Σp

c, ↓P

Σp

r , ↑P

Σ¬p

ℓ , →

Σ¬p

c , ↓P

Σ¬p

r , ↑P

Σ¬p

c , ↓ ¯

P Σ¬p

r , ↑ ¯

P

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 16/17

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The Competitors

“If p holds true immediately after entering module m, it shall hold immediately after the corresponding return from m as well” VLTL: [Bozzelli ’14] (α; true)|αfalse with visibly rational expression α below: [(p ∪ q)∗callm [(q) ∪ (pp)] returnm(p ∪ q)∗] (p ∪ q)∗

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 16/17

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Conclusion

Results: VLDL as expressive as ω-VPA

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 17/17

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Conclusion

Results: VLDL as expressive as ω-VPA validity model-checking infinite games LTL PSpace PSpace 2ExpTime LDL PSpace PSpace 2ExpTime

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 17/17

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Conclusion

Results: VLDL as expressive as ω-VPA validity model-checking infinite games LTL PSpace PSpace 2ExpTime LDL PSpace PSpace 2ExpTime VLDL ExpTime ExpTime 3ExpTime

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 17/17

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Conclusion

Results: VLDL as expressive as ω-VPA validity model-checking infinite games LTL PSpace PSpace 2ExpTime LDL PSpace PSpace 2ExpTime VLDL ExpTime ExpTime 3ExpTime VLTL ExpTime ExpTime ? Using (deterministic) pushdown automata as guards leads to undecidability, i.e., A1# ∧ A2# ∧ “exactly one #” is satisfiable ⇔ L(A1) ∩ L(A2) = ∅.

Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 17/17