The Ludii Game Description Language is Universal

TL;DR

The paper proves that Ludii's general game description language (L-GDL) is universal for finite extensive-form games, extending previous results that covered only deterministic, perfect-information cases to include stochastic transitions and hidden information. It presents a constructive mapping from any finite extensive-form game G to a Ludii game G^L by creating per-player copies of the game tree, markers to track true states and information sets, and start/play/end rule encodings that preserve moves, chance outcomes, and payoffs. A formal equivalence theorem shows a one-to-one correspondence between root-to-leaf trajectories in G and achievable trajectories in G^L, satisfying criteria for valid description, correct mover, branching, chance probabilities, terminal payoffs, and information indistinguishability. The result supports using Ludii as a general, efficient GGP platform and motivates further work on properties like Turing completeness and scalable analysis of L-GDL descriptions.

Abstract

There are several different game description languages (GDLs), each intended to allow wide ranges of arbitrary games (i.e., general games) to be described in a single higher-level language than general-purpose programming languages. Games described in such formats can subsequently be presented as challenges for automated general game playing agents, which are expected to be capable of playing any arbitrary game described in such a language without prior knowledge about the games to be played. The language used by the Ludii general game system was previously shown to be capable of representing equivalent games for any arbitrary, finite, deterministic, fully observable extensive-form game. In this paper, we prove its universality by extending this to include finite non-deterministic and imperfect-information games.

Figures (6)

• Figure 1: Basic structure of an L-GDL game description for Ludii. Note that curly braces are used for arrays in L-GDL.
• Figure 2: Full L-GDL description for the game of Tic-Tac-Toe.
• Figure 3: Template for the equipment definition of a Ludii game $\mathcal{G}^L$, modelling an equivalent extensive-form game $\mathcal{G}$ with $\vert \mathcal{S} \vert$ different states and $k$ players. The expressions angled brackets are used for generality, but would be replaced by the concrete result of the expression in any single concrete game description. The values used for $x$- and $y$-coordinates only affect display in Ludii's graphical user interface, and are irrelevant in terms of semantics.
• Figure 4: Start rules for a Ludii game $\mathcal{G}^L$, modelling an equivalent extensive-form game $\mathcal{G}$, with $k$ players. The expressions to compute vertex indices in angled brackets are used for generality, but would be replaced by the concrete result of the expression in any single concrete game description.
• Figure 5: Ludeme generating a move corresponding to the $n^{th}$ branch from a state $s_i$, leading to a state $s_j$, in the game tree of an extensive-form game $\mathcal{G}$ with $k$ players. The player that should be the mover in the next state $s_j$ is denoted by $\iota(s_j)$---except we replace it by any arbitrary integer in $[1, k]$ if $\iota(s_j) = \eta$.

Theorems & Definitions (3)

• Definition 1
• Theorem 1
• proof