SLIDE 1
NP Reasoning in the Monotone µ-Calculus
(IJCAR 2020) Daniel Hausmann and Lutz Schr¨
- der
NP Reasoning in the Monotone -Calculus (IJCAR 2020) Daniel - - PowerPoint PPT Presentation
NP Reasoning in the Monotone -Calculus (IJCAR 2020) Daniel - - PowerPoint PPT Presentation
NP Reasoning in the Monotone -Calculus (IJCAR 2020) Daniel Hausmann and Lutz Schr oder University Erlangen-Nuremberg, Germany Highlights 2020 16 September 2020 0 / 5 Satisfiability Checking Complexities of satisfiability checking for
SLIDE 2
SLIDE 3
Satisfiability Checking
Complexities of satisfiability checking for some modal logics: ◮ K PSPACE ◮ modal µ-calculus / CTL EXPTIME ◮ monotone modal logic NP [Vardi, 1989] ◮ alternation-free monotone µ-calculus NP [here]
1 / 5
SLIDE 4
Monotone Modal Logic
Standard modal formulae, interpreted over neighbourhood structures M = (W, N, I) where N : Act × W → P(P(W)) I : At → P(W) [ [[a]φ] ] = {w ∈ W | ∀S ∈ N(a, w). S ∩ [ [φ] ] = ∅} [ [aφ] ] = {w ∈ W | ∃S ∈ N(a, w). S ⊆ [ [φ] ]}
2 / 5
SLIDE 5
Monotone Modal Logic, example
x q p p q a a b
3 / 5
SLIDE 6
Monotone Modal Logic, example
x q p p q a a b x ∈ [ [ap] ]
3 / 5
SLIDE 7
Monotone Modal Logic, example
x q p p q a a b x ∈ [ [ap] ] x ∈ [ [[a](p ∨ q)] ]
3 / 5
SLIDE 8
Monotone Modal Logic, example
x q p p q a a b x ∈ [ [ap] ] x ∈ [ [[a](p ∨ q)] ] x / ∈ [ [bp] ]
3 / 5
SLIDE 9
Monotone Modal Logic, example
x q p p q a a b x ∈ [ [ap] ] x ∈ [ [[a](p ∨ q)] ] x / ∈ [ [bp] ] Cannot express e.g. “p holds in every successor state” “p holds in at least one successor state”
3 / 5
SLIDE 10
Main Result
Main Theorem
The satisfiability problem for the alternation-free monotone µ-calculus is NP-complete.
Proof sketch:
φ is satisfiable ⇔ there is tableau for φ ⇔ Eloise wins satisfiability game for φ Satisfiability games: Two-player B¨ uchi games with polynomial number of Eloise-nodes NP-algorithm for solving the games
4 / 5
SLIDE 11
Example Logics
Readings: ◮ Epistemic Logic aφ – “Agent a knows φ” ◮ Concurrent PDL (CPDL), Peleg (1987) αφ – “There is execution of program α in parallel, nondeterministic system s.t. all end states satisfy φ” ◮ Game Logic, Parikh (1983) αφ – “Player Angel has strategy to achieve φ in game α”
5 / 5
SLIDE 12