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
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
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
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
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
Outline 1. Preliminaries 2. Expressiveness 3. VLDL Verification 4. Discussion Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 3/17
Outline 1. Preliminaries 2. Expressiveness 3. VLDL Verification 4. Discussion Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 4/17
Visibly Pushdown Automata Partition input alphabet Σ into Σ c (calls), Σ r (returns), and Σ ℓ (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
Visibly Pushdown Automata Partition input alphabet Σ into Σ c (calls), Σ r (returns), and Σ ℓ (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: a n b n is a VPL, if a is a call and b a return. ww R is not a VPL. Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 5/17
Visibly Linear Dynamic Logic (VLDL) Syntax ϕ := p | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | � A � ϕ | [ A ] ϕ where p ∈ P ranges over atomic propositions and A ranges over VPA’s. All VPA’s have the same partition of 2 P into calls, returns, and local actions. Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 6/17
Visibly Linear Dynamic Logic (VLDL) Syntax ϕ := p | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | � A � ϕ | [ A ] ϕ where p ∈ P ranges over atomic propositions and A ranges over VPA’s. All VPA’s have the same partition of 2 P into calls, returns, and local actions. Semantics: ( w ∈ (2 P ) ω ) w | = � A � ϕ if there exists an n such that w 0 · · · w n − 1 is accepted by A and w n w n +1 w n +2 · · · | = ϕ . w | = [ A ] ϕ if for every n s.t. w 0 · · · w n − 1 is accepted by A we have w n w n +1 w n +2 · · · | = ϕ . Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 6/17
Example “Every request q is eventually answered by a response p and there are never more responses than requests” [ A ∗ ]( q → � A ∗ � p ) ∧ ¬� A � true 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
Outline 1. Preliminaries 2. Expressiveness 3. VLDL Verification 4. Discussion Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 8/17
Expressiveness Lemma VLDL and non-deterministic ω -VPA are expressively equivalent. Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17
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
Expressiveness Lemma VLDL and non-deterministic ω -VPA are expressively equivalent. Proof Idea VLDL Deterministic Stair Automata O (2 n ) [LMS ’04] non-deterministic ω -VPA Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17
Expressiveness Lemma VLDL and non-deterministic ω -VPA are expressively equivalent. Proof Idea VLDL O ( n 2 ) Deterministic Stair Automata O (2 n ) [LMS ’04] non-deterministic ω -VPA Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17
Expressiveness Lemma VLDL and non-deterministic ω -VPA are expressively equivalent. Proof Idea VLDL O ( n 2 ) O ( n 2 ) 1-way Alternating Deterministic Jumping Automata Stair Automata O (2 n ) [LMS ’04] non-deterministic ω -VPA Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17
Expressiveness Lemma VLDL and non-deterministic ω -VPA are expressively equivalent. Proof Idea VLDL O ( n 2 ) O ( n 2 ) 1-way Alternating Deterministic Jumping Automata Stair Automata O (2 n ) O (2 n ) [LMS ’04] [Bozelli ’07] non-deterministic ω -VPA Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17
Expressiveness Lemma VLDL and non-deterministic ω -VPA are expressively equivalent. Proof Idea VLDL O ( n 2 ) O ( n 2 ) 1-way Alternating Deterministic Jumping Automata Stair Automata O (2 n ) O (2 n ) [LMS ’04] [Bozelli ’07] non-deterministic ω -VPA Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 9/17
From Stair Automata to VLDL 1 1 2 0 2 7 3 2 2 0 3 0 5 3 0 c ω c c c c r c r r c c c r r l Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 10/17
From Stair Automata to VLDL 1 1 2 0 2 7 3 2 2 0 3 0 5 3 0 c ω c c c c r c r r c c c r r l Acceptance: maximal priority occuring at infinitely many steps even Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 10/17
From Stair Automata to VLDL 1 1 2 0 2 7 3 2 2 0 3 0 5 3 0 c ω c c c c r c r r c c c r r l 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
From Stair Automata to VLDL 1 1 2 0 2 7 3 2 2 0 3 0 5 3 0 c ω c c c c r c r r c c c r r l 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 . � � ∧ [ A ′ � q I A ′ q � [ q A ′ q ] � q A ′ q � true q ′ ] false q ∈ Q even q ′ ∈ Q > Ω( q ) Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 10/17
Outline 1. Preliminaries 2. Expressiveness 3. VLDL Verification 4. Discussion Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 11/17
Satisfiability Theorem VLDL Satisfiability is ExpTime -complete. Martin Zimmermann Saarland University Visibly Linear Dynamic Logic 12/17
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
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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