Game Theory in Formal Verification Scribe Notes for Lecture 5

1.1. Definition. Consider a graph (S, E). Let d be a non-negative integer and let p : S → {0, 1,. .. , d} be a function which we call the priority function. The corresponding parity objective Parity(p) is defined to be the set of (infinite) paths in which the minimal priority of infinitely often visited nodes is even. In a more symbolic notation, this reads (1) Parity(p) = π ∈ Π : min s∈Inf(π) p(s) is even where Π denotes the set of all infinite paths in (S, E) and Inf(π) is the set of nodes visited infinitely often in path π. d} is a priority function, then we call the pair (G, p) a parity game. 1.2. Parity Objectives Generalize Büchi and co-Büchi. It turns out that Büchi and co-Büchi objectives are in fact parity objectives: Given a Büchi set B ⊆ S, the priority function p : S → {0, 1}, (2) p(s) = 0 if s ∈ B 1 if s ∈ B yields Parity(p) = 23B. (3) p (s) = 1 if s ∈ B 2 if s ∈ B gives rise to Parity(p) = 32¬B, which covers co-Büchi objectives. The passage from p to p is part of a more general principle: Given a priority function p, we can calculate a p such that Parity(p) is the complement of Parity(p); we achieve this by setting p (s) = p(s) + 1. The property that the complement of a parity objective is again a parity objective is called self-duality. 2 1.3. ω-Regular Languages, Büchi Automata, and Parity Automata. Recall that a regular language is a set of finite words over the alphabet Σ that can be described by one of the following means which are mutually equivalent. • a non-deterministic finite automaton (NFA) • a deterministic finite automaton (DFA)