Turing Completeness and Sid Meier's Civilization

TL;DR

The paper demonstrates that Sid Meier's Civilization games can simulate universal computation by embedding UTMs inside their mechanics under infinite-map and infinite-turn assumptions. It constructs explicit $ (10,3) $-UTMs for Civ:BE and Civ:V and a $ (48,2) $-UTM for Civ:VI, detailing tape and state encodings using in-game primitives such as Roads, Railroads, Terrascapes, and City occupations. A Busy Beaver example in Civ:BE substantiates the operational capacity of these machines, and the results imply undecidability of the games under the stated limits. The work highlights the deep connections between complex game dynamics and foundational computability theory, while discussing limitations, alternative constructions, and potential automation via game APIs.

Abstract

We prove that three strategy video games from the Sid Meier's Civilization series: Sid Meier's Civilization: Beyond Earth, Sid Meier's Civilization V, and Sid Meier's Civilization VI, are Turing complete. We achieve this by building three universal Turing machines-one for each game-using only the elements present in the games, and using their internal rules and mechanics as the transition function. The existence of such machines imply that under the assumptions made, the games are undecidable. We show constructions of these machines within a running game session, and we provide a sample execution of an algorithm-the three-state Busy Beaver-with one of our machines.

Figures (5)

• Figure 1: A $(10, 3)$-UTM built within Civ:BE. The tape is located to the left of the image, and runs from the top middle of the screen to the bottom left. It currently has Roads (symbols in $\Gamma$) on every hex, except one pillaged Road. The head (the tape Worker and the Rover) are positioned in the middle of the screen, reading the symbol "Road"--the read is implicit, but can also be seen from the "Build Actions" menu, which does not allow the player to build a Road on an existing Road. The UTM is in state $q_2$, since the total Culture yield in this turn is $C_t = 7$ (the purple number, top right on the image). Each Terrascape (lush green tiles on the right) contributes $3$ Culture, and the base Culture yield is $C_* = 1$, so $q_2 = (C_t - C_*)/3 = 2$. For this particular map, desert tiles appear as solid ice.
• Figure 2: A $(10, 3)$-UTM built within Civ:V. It is in the state $q_2$ since the City (left) has two Railroads built: one on the tile with the elephant and another on the plantation. In this case, the area reserved for state tracking is every tile owned by the City. The head (the rightmost Worker, near the top) is positioned over a Railroad hex with two Roads adjacent to it.
• Figure 3: Diagram of a $(48, 2)$-UTM built with Civ:VI components and rules. The bottom two Cities act as the state of the machine, and each of the Cities on the top strip contain two cells from the tape. The head position is tracked by the Worker. Refer to Figure \ref{['fig:igcivvi']} for a working in-game construction.
• Figure 4: Sample execution of BB-$3$ with a Civ:BE Turing machine. The machine executes $11$ instructions $t_1, \dots, t_{11}$ before halting. Top left: state corresponding to $t_1 = q_00;1Rq_1$. Top right: state for the machine at $t_2 = q_10;1Lq_0$. Bottom left: state for the machine at $t_{10} = q_01;1Lq_2$. Bottom right: state for the machine at $t_{11} = q_21;HALT$.
• Figure 5: A $(48, 2)$-UTM built within Civ:VI. Top: the UTM in-game, with three Cities acting as the states and two as the tape. Bottom right: one of the Cities holding the state, with $8$ Monasteries and $9$ Farms. Bottom left: The tape being read (right hex; Worker not pictured) with a state of "Is Being Worked".

Theorems & Definitions (12)

• theorem thmcountertheorem
• proof
• remark thmcounterremark
• theorem thmcountertheorem
• proof
• remark thmcounterremark
• theorem thmcountertheorem
• proof
• remark thmcounterremark
• corollary thmcountercorollary