Focus-Style Proofs for the Two-Way Alternation-Free $\mu$-Calculus

Abstract

We introduce a cyclic proof system for the two-way alternation-free modal $\mu$-calculus. The system manipulates one-sided Gentzen sequents and locally deals with the backwards modalities by allowing analytic applications of the cut rule. The global effect of backwards modalities on traces is handled by making the semantics relative to a specific strategy of the opponent in the evaluation game. This allows us to augment sequents by so-called trace atoms, describing traces that the proponent can construct against the opponent’s strategy. The idea for trace atoms comes from Vardi’s reduction of alternating two-way automata to deterministic one-way automata. Using the multi-focus annotations introduced earlier by Marti and Venema, we turn this trace-based system into a path-based system. We prove that our system is sound for all sequents and complete for sequents not containing trace atoms.

The research of this author has been made possible by a grant from the Dutch Research Council NWO, project number 617.001.857.

A version of this paper including an appendix with full proofs can be found on arXiv.

A Parity games

Definition 10

A (two-player) game is a structure $\mathcal {G} = (B_0, B_1, E, W)$ where E is a binary relation on $B := B_0 + B_1$, and W is a map $B^\omega \rightarrow \{0, 1\}$.

The set B is called the board of $\mathcal {G}$, and its elements are called positions. Whether a position belongs to $B_0$ or $B_1$ determines which player owns that position. If a player $\varPi \in \{0, 1\}$ owns a position q, it is their turn to play and the set of their admissible moves is given by the image E[q].

Definition 11

A match in $\mathcal {G} = (B_0, B_1, E, W)$ (or simply a $\mathcal {G}$-match) is a path $\mathcal {M}$ through the graph (B, E). A match is said to be full if it is a maximal path.

Note that a full match $\mathcal {M}$ is either finite, in which case $E[\textsf{last}(\mathcal {M})] = \emptyset$, or infinite. For a $\varPi \in \{0, 1\}$, we write $\overline{\varPi }$ for the other player $\varPi + 1 \mod 2$.

Definition 12

A full match $\mathcal {M}$ in $\mathcal {G} = (B_0, B_1, E, W)$ is won by player $\varPi$ if either $\mathcal {M}$ is finite and $\textsf{last}(\mathcal {M}) \in B_{\overline{\varPi }}$, or $\mathcal {M}$ is infinite and $W(\mathcal {M}) = \varPi$.

If a full match $\mathcal {M}$ is finite, and $\textsf{last}(\mathcal {M})$ belongs to $B_\varPi$ for $\varPi \in \{0, 1\}$, we say that the player $\varPi$ got stuck. A partial match is a match which is not full.

Definition 13

In the context of a game $\mathcal {G}$, we denote by $\text {PM}_\varPi$ the set of partial $\mathcal {G}$-matches $\mathcal {M}$ such that $\textsf{last}(\mathcal {M})$ belongs to the player $\varPi$.

Definition 14

A strategy for $\varPi$ in a game $\mathcal {G}$ is a map $f : \text {PM}_\varPi \rightarrow B$. Moreover, a $\mathcal {G}$-match $\mathcal {M}$ is said to be f-guided if for any $\mathcal {M}_0 \sqsubset \mathcal {M}$ with $\mathcal {M}_0 \in \text {PM}_\varPi$ it holds that $\mathcal {M}_0 \cdot f(\mathcal {M}_0) \sqsubseteq \mathcal {M}$.

For a position q, the set $\text {PM}_\varPi (q)$ contains all $\mathcal {M} \in \text {PM}_\varPi$ such that $\textsf{first}(\mathcal {M}) = q$.

Definition 15

A strategy f for $\varPi$ in $\mathcal {G}$ is surviving at a position q if $f(\mathcal {M})$ is admissible for every $\mathcal {M} \in \text {PM}_\varPi (q)$, and winning at q if in addition all full f-guided matches starting at q are won by $\varPi$. A position q is said to be winning for $\varPi$ if $\varPi$ has a strategy winning at q. We denote the set of all positions in $\mathcal {G}$ that are winning for $\varPi$ by $\text {Win}_\varPi (\mathcal {G})$.

We write $\mathcal {G}@q$ for the game $\mathcal {G}$ initialised at the position q of $\mathcal {G}$. A strategy f for $\varPi$ is surviving (winning) in $\mathcal {G}@q$ if it is surviving (winning) in $\mathcal {G}$ at q.

Definition 16

A strategy f is positional if it only depends on the last move, i.e. if $f(\mathcal {M}) = f(\mathcal {M}')$ for all $\mathcal {M}, \mathcal {M}' \in \text {PM}_\varPi$ with $\textsf{last}(\mathcal {M}) = \textsf{last}(\mathcal {M}')$.

We will often present a positional strategy for $\varPi$ as a map $f : B_\varPi \rightarrow B$.

Definition 17

A priority map on some board B is a map $\varOmega : B \rightarrow \omega$ of finite range. A parity game is a game of which the winning condition is given by $W_\varOmega (\mathcal {M}) = \max (\textit{Inf}_\varOmega (\mathcal {M})) \mod 2$, where $\textit{Inf}_\varOmega (\mathcal {M})$ is the set of positions occuring infinitely often in $\mathcal {M}$.

The following theorem captures the key property of parity games: they are positionally determined. In fact, each player $\varPi$ has a positional strategy $f_\varPi$ that is optimal, in the sense that $f_\varPi$ is winning for $\varPi$ in $\mathcal {G}@q$ for every $q \in \text {Win}_\varPi (\mathcal {G})$.

Theorem 1

( [5, 15]). For any parity game $\mathcal {G}$, there are positional strategies $f_\varPi$ for each player $\varPi \in \{0, 1\}$, such that for every position q one of the $f_\varPi$ is a winning strategy for $\varPi$ in $\mathcal {G}@q$.