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A positional $\mathbfΠ^0_3$-complete objective

Antonio Casares, Pierre Ohlmann, Pierre Vandenhove

Abstract

We study zero-sum turn-based games on graphs. In this note, we show the existence of a game objective that is $\mathbfΠ^0_3$-complete for the Borel hierarchy and that is positional, i.e., for which positional strategies suffice for the first player to win over arenas of arbitrary cardinality. To the best of our knowledge, this is the first known such objective; all previously known positional objectives are in $\mathbfΣ^0_3$. The objective in question is a qualitative variant of the well-studied total-payoff objective, where the goal is to maximise the sum of weights.

A positional $\mathbfΠ^0_3$-complete objective

Abstract

We study zero-sum turn-based games on graphs. In this note, we show the existence of a game objective that is -complete for the Borel hierarchy and that is positional, i.e., for which positional strategies suffice for the first player to win over arenas of arbitrary cardinality. To the best of our knowledge, this is the first known such objective; all previously known positional objectives are in . The objective in question is a qualitative variant of the well-studied total-payoff objective, where the goal is to maximise the sum of weights.

Paper Structure

This paper contains 6 sections, 6 theorems, 8 equations, 1 figure.

Table of Contents

  1. Context and contribution
  2. Pi_3-completeness of SumToInfinity
  3. Positionality of SumToInfinity
  4. Almost-universality.
  5. Definition of $U$.
  6. Almost-universality of $U$.

Key Result

Theorem 1

The objective $\mathsf{SumToInfinity}$ is $Borel$-complete and positional.

Figures (1)

  • Figure 1: Tree $T$ used in Example \ref{['ex:morphismExample']}. Remember that the weights (in black) label edges. This tree satisfies $\mathsf{SumToInfinity}$ (as any infinite path ends with $1^\omega$). The value in red inside each vertex is the value $n(\cdot)$ defined in the proof of Lemma \ref{['lem:embedSmallGraphs']}. The top vertex $v_0$ is such that $n(v_0) = -1 < 0$, so we can assume it is the vertex given by Claim \ref{['claim:v0']}. Every path from $v_0$ not reaching another vertex with value $-1$ is tight. Observe that there is exactly one infinite tight path from $v_0$ (staying on the left branch), indeed satisfying the property of Claim \ref{['claim:tightpaths']}. The tuples in blue next to vertices correspond to the morphism to $U$ built in the proof of Claim \ref{['claim:morphism']}.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2: Ohlmann23
  • Lemma 3
  • Example 4
  • Theorem 5
  • Lemma 6
  • Lemma 7