Home Assignment 1a: Multi-focus Games for LTL

We are interested in a game-theoretic approach to proving satisfiability of LTL formulæ. We consider for this LTL formulæ in negative normal form, i.e. using the abstract syntax ' ::= p ∣ ¬p ∣ ' ∨ ' ∣ ' ∧ ' ∣ X' ∣ ' U ' ∣ ' R ' , where p ranges over a non-empty finite set of atomic propositions AP. Let ⊤ ≡ p ∨ ¬p and ⊥ ≡ p ∧ ¬p for some atomic proposition p. The F and G modalities are defined as usual by F' ≡ ⊤ U ' and G' ≡ ⊥ R '. Given a LTL formula ', the closure cl(') of ' is the smallest set of LTL formulæ such that ∙ ' ∈ cl('), ∙ if ¬ ∈ cl(') or X ∈ cl('), then ∈ cl('), ∙ if 1 ∨ 2 ∈ cl('), 1 ∧ 2 ∈ cl('), 1 U 2 ∈ cl('), or 1 R 2 ∈ cl('), then 1 ∈ cl(') and 2 ∈ cl('), ∙ if 1 U 2 ∈ cl('), then X( 1 U 2) ∈ cl('), and ∙ if 1 R 2 ∈ cl('), then X( 1 R 2) ∈ cl(').