### Magic The Gathering
Magic: The Gathering is a trading card gam...
## Playing Magic
The full rules of Magic are extensive - you can f...
Rengo Kriegspiel is a "blindfolded", team variant of Go. There are ...
Zero-sum games are 2-player games where the sum of the payoffs of t...
Example of a Turing Machine built by a player in Minecraft
### Rogozhin Turing Machines
When Alan Turing came up with the ide...
Cards mentioned by the authors:
Rogozhin's (2,18) Turing Machine is a Turing machine with 2 states ...
 is a popular
trading card game owned by Wizards of the Coast. Formally, it
is a two-player zero-sum stochastic card game with imperfect
informa tion, putting it in the same category as games like
poker and hearts. Unlike those games, players desig n their
own custom decks ou t of a card-pool of over 20,000 cards.
Magic’s multifaceted strategy has made it a p opular topic in
artificial intelligence researc h.
In this paper, we examine Magic: The Gathering from the
point of view of algorithmic game theory, looking at the
computational complexity of evaluating who will win a game.
As most games have finite limits o n their complexity (such
as the size of a game board) most research in algorithm ic
game theory of real-world games has primarily looked at
generalisations of commo nly played games rather than the
real-world versions of the games. A few real-world games have
been found to have non-trivial complexity, inc luding Dots-and-
Boxes, Jenga and Tetris [8]. We believe that no real-world
game is known to be h arder than NP previous to this work.
Even when looking at generalised games, very few examples
of undecid a ble games are known. On an abstract level, the
Team Computation Game [ 9] shows that some games can be
undecid able, if they are a particular kind of team game with
imperfect information. The authors also present an equivalent
construction in their Constraint Logic fra mewo rk that was used
by Coulombe and Lynch (2018) [7] to show th a t some video
games, including Super Smash Bros Melee and Mario Kart,
have u ndecidable gen eralisations. Constraint Logic is a highly
successful and highly flexible framework for modelling games
as computations.
The core of this paper is the c onstruction presente d in
Section IV: a universal Turing machin e embedded into a game
of Magic: The Gathering. As we can arrange for the victor
of the game to be determined by the halting behaviour of
the Turing machine, this construction establishes th e following
theorem:
Theorem 1: Determining the outcome of a game of Magic:
The Gathering in which all remain ing moves are forced is
undecid able.
A. Previous Work
Prior to this work, no u ndecidable real games were known
to exist. Demaine an d Hearn (2009) [10] note that almost every
real-world game is trivially decidable, as they produce game
trees with only computable paths. They further note that Rengo
Kriegspiel
1
is “a game humans play that is not obvio usly
decidable; we are not aware of any other such game.” It
is conjectured by Auger and Teytaud (2012) [1] that Rengo
Kriegspiel is in fact undecidable, and it is posed as an open
problem to demonstrate a ny real game that is undecida ble.
The approach of embedding a Turing machine inside a
game directly is generally not considered to be feasible for
real-world games [10]. Althoug h some open-world sandbox
games such as Minecraft and Dwarf Fortress can support
the construc tion of Turing machines, those machines have no
strategic relevance and those games are deliberately design e d
to su pport large-scale simulation. In contrast, leading formal
theory of strategic game s claims tha t the unbounded memory
required to simulate a Turing machine entirely in a game
would be a violation of the very natur e of a game [9].
The computational complexity of Magic: Th e Gathe rin g in
has been studied previously by several authors. Our work is
inspired by [4], in which it was show n that four-player Magic
can simulate a Turing machine under certain assumptions
about player behaviour. In that work, Churchill conjectures
that these limitations can be removed and preliminary work
along those lines is discussed in [5]. The com putational
complexity of checkin g the legality of a particular decision
in Magic (blocking) is investigated in [3] and is found to
be coNP-complete. There have also been a number of papers
1
Rengo Kriegspiel is a combination of two variations on Go: Rengo, in
which two players play on a team alternating turns, and Shadow Go, in which
players are only able to see their own moves.
investigating algorithmic and artificial intelligence approaches
to playin g Magic, including Ward and Cowling (2009) [15],
Cowling et al. (2012) [6], and Esche (2018) [11]. Esche (2018)
briefly considers the theoretical computational complexity of
Magic and states an op en problem that has a positive answer
only if Magic e nd-games are decidable.
B. Our Contribution
This paper complete s the project started by Churchill [4]
and continued by Churchill et a l. [5] of embed ding a u niversal
Turing machine in Magic: The Gathering such that determin-
ing the outcome of the game is eq uivalent to determining the
halting of the Tur ing machine. This is the first result showing
that there exists a real-world game for which deter mining
the winning strategy is non-computable, answering an open
question of Demaine and Hearn [10] and Auger and Teytaud
[1] in the positive. This result, combined with Rice’s Theorem
[13], also answers an open problem from Esche [11] in the
negative by showing that the equivalen c e of two strategies for
playing Magic is undecidable.
This result raises important f oundational questions about
the nature of a game itself. As we have alr eady discussed, the
leading formal theory of games holds that this construc tion
is unreasonable, if not impossible, and so a reconsideration
of those assumptions is called for. In section V-A we discuss
additional foundational assumptions of Constraint Logic that
Magic: The Gathering violates, and present our interpreta tion
of the implications for a unified theory of ga mes.
C. Overview
The paper is structured as follows. In Section I I we provide
backgr ound information on this work , including previous work
on Magic Turing machines. In Section III we present a sketch
of the construction and its key pieces. In Section IV we provide
the full construction of a universal Tu ring machine embedded
in a two-player game of Magic. In Section V we discu ss the
game-theoretic and real-world implications of our result.
II. PRELIMINARIES
One initial challenge with Magic: The Gathering is the
encodin g of infor mation. Som e cards ask players to choose
a number. Although rules for how to specify a number are not
discussed in the Compreh e nsive Rules [16], convention is that
players are allowed to specify n umbers in any way that both
players can agree to. For example, you are allowed to choose
the number 2
100
or ⌈log 177⌉. This presents an issue brough t
to our attention by Fortanely [12]. Consid e r the following
situation: both playe rs control Lich, Transcendence, an d
Laboratory Maniac. O ne player then casts Menacing Ogre.
The net effect of this play is the “Who Can Name the Bigger
Number” game – whoever picks the biggest number wins on
the spot. This makes identifying the next board state non-
computable [2], so we require that any numbers specified by
a playe r mu st be expressed in stand ard binary notation.
We believe that with this r e stric tion Magic: The Gathering is
transition-computable, meaning that the function that maps a
board state and a move to the next board state is computable
2
.
However, it is unclear how to prove this beyond exhaustive
analysis of th e over 20,000 card s in the game. We leave that
question open for future work:
Conjecture 1: The function that takes a board state and
a legal move and returns the next b oard state in Magic: The
Gathering is c omputable.
In this conjecture we say “a legal move” because it is also not
obvious that checking to see if a move is legal is computable.
Chatterjee and Ibsen-Jensen [3] show that checking the legality
of a particular kind of game action is coNP-complete, but the
question ha s not be e n othe rwise considered. Ag ain, we leave
this for future work:
Conjecture 2: There does not exist an algorithm for
checking the legality of a move in Magic: The Gathering .
A. Previous Magic Turing Machines
In [4], the author presents a Magic: The Gathering end-
game that embeds a universal Turing machine. However, this
work ha s a major issue: it’s not qu ite deterministic. At several
points in the simulatio n, players have the ability to stop the
computation at any time by opting to decline to use effects that
say “may.” For example, Kazuul Warlord read s “Whenever
Kazuul Warlord or another Ally enters the battlefield under
your control, you may p ut a +1/ + 1 counter on each Ally
you con trol.” Declining to u se this ability will interfe re with
the Turing machine, either causing it to stop or cau sin g it
to perform a different calculation from the one intende d.
The construction as given in Churchill [4] works under the
assumption that all players that are given the option to do
something actually do it, but as the a uthor notes it fails w ithout
this assumption. Attempts to correc t this issue are discussed
in Churchill et al. [5].
In this work, we solve this problem by reformulating the
construction to exclusively use cards with mandatory effects.
We also substan tially simplify the most co mplicated aspect
of the construction, the recording of th e tape, and reduce the
construction from one involving four players to one involving
two, and whic h only places constraints on one player’s d eck,
matching the format in which Magic is most commonly played
in the real world (two-player duels). Like the previous work,
we will embed Rog ozhin’s (2, 18) universal Turing machine
[14].
III. AN OVERVIEW OF THE CONSTRUCTION
In this section we give a big picture view of the Turing
machine, with full d etails deferred to the next section. The
two players in the game are named Alice and Bob.
To construct a Turing machine in Magic: The Gathering
requires thr ee main elements: the tape which encodes the
computation, the c ontroller which determines what action to
take next ba sed on the current state and the last read cell, and
the read/write head which interacts with the tape under the
control of the controller.
2
We avoid the term “computable game” which is more commonly used to
mean that the game has a computable winning strategy.
2
A. The Tape
As the rules of Magic: The Gathering do not contain a ny
concept of geometry or adjacency, encoding the tape itself
is tr ic ky. Our solution is to have many creature tokens with
carefully controlled power and toughness, with each token’s
power and toughness r epresenting the distance fr om the head
of the Tur ing machine. The tape to the left of the Turing
machine’s current read head position is represented by a
series of creature to kens which all have the game colour
green, while th e tape to the right is represented by white
tokens. Our distan c e-counting starts at 2, so there is one 2/2
token representing th e space currently under the head of the
Turing machine; a green 3/3 token represents the tape space
immediately to the left of the Turing head, a gr een 4/4 is
the space to the left of that, and so on. Rogozh in’s universal
Turing mac hine starts with the read head in the middle of the
tape [14].
To represent the symbols on the tape, we use creature types.
We choose 18 creature types from the list of creature types in
Magic to correspond to the 18 symbols in Rogozhin’s (2, 18)
UTM. We can choose th e se creature types to begin with suc-
cessive letters of the alphabet: Aetherborn, Basilisk, Cephalid,
Demon, Elf, Faerie, Giant, Harpy, I llusion, Juggernaut, Kavu,
Leviathan, Myr, Noggle, Orc, Pegasus, Rhino, and Sliver. For
example, a green 5/5 A etherborn token represents that the 1
st
symbol is w ritten on the 3
rd
cell to the left of the head, and a
white 10/10 Sliver represents that the 18
th
symbol is written
on the 9
th
cell to the right of the head. These tokens are all
controlled by Bob, except the most recently c reated token (the
space the Tu ring head has just left) which is controlled by
Alice.
B. The Controller
Control instructions in a Turing machine are repr esented by
a table of conditio nal statements of the form “if the machine
is in state s, and the last re ad cell is symbol k, then do such-
and-such.” Many Magic cards h ave triggered abilities which
can function like conditional statements. The two we shall use
are Rotlung Reanimator (“Whenever Rotlu ng Reanimator or
another Cleric dies, create a 2/2 black Zombie creature token”)
and Xathrid Necromancer (“Whenever Xathrid Necromancer
or another Human creature you c ontrol dies, create a tapped
2/2 black Zombie creature token”). We will use both, and the
difference between tapped and unta pped c reature tokens will
contribute to the design of the Tu ring m achine.
Each Rotlung Reanimator
3
needs to trig ger from a dif-
ferent state being read – that is, a different creature type
dying – and needs to encode a different result. Fortunately,
Magic includes cards that can be used to ed it the text of other
cards. The card Artificial Evolution is uniquely powerful for
our purposes, as it reads “Change the text of target spell or
permane nt by replacing all instanc e s of one creature type with
another. The new creature type can’t be Wall. (This effect
3
For now we will speak about Rotlung Reanimator for simplicity. Some of
these will in fact be Xathrid Necromancers as explained in the next section.
lasts indefinitely.)” So we crea te a large number of copies
of Rotlung Reanimator and edit e ach one. A similar card
Glamerdye can be used to modify the colour word s with in
card text.
Thus, we edit a Rotlung Reanimator by casting two c opies
of Artificial Evolution replacing ‘Cleric’ with ‘Aethe rborn’
and ‘Zombie’ with ‘Sliver’ and one copy of Glamerdye to
replace ‘blac k’ with ‘white’, so that this Rotlung Reani-
mator now reads “Whenever Rotlung Rean imator or another
Aetherborn dies, create a 2/2 white Sliver creature token ”
4
.
This Rotlung Reanimator now encodes the first rule of the
q
1
program of the (2, 18) UTM: “When reading symbol 1
in state A, write symbol 18 and move lef t.” The Aetherborn
creature token represents symbol 1, the Sliver crea ture token
represents symbol 18, and the fact that the token is white leads
to processing that will cause th e head to move left.
We similarly have seventeen more Rotlung Reanimators
encodin g the rest of the q
1
program from [14]. Between them
they say:
1) Whenever an Aetherborn dies, create a 2 /2 white Sliver.
2) Whenever a Basilisk dies, create a 2/2 green Elf.
Whenever a . . . dies, create a 2/2 . . .
18) Whenever a Sliver dies, create a 2/2 green Cephalid.
See Table II for the full encoding of the program.
C. The Read /Write Head
The operation “read the current cell of the tape” is rep-
resented in-game by forcing Alice to c a st Infest to give all
creatures
−
2/
−
2. This causes the unique token with 2 tou ghness
to die. It had a colour (green or white) which is irreleva nt,
and a creature type which corresponds to the symbol written
on that cell. That creature type is noticed by a Rotlung
Reanimator, which has a triggered ability that is used to carr y
out the logic encoded in the head of the Turing machine . It
produces a new 2/2 token, co ntaining the information written
to the cell that was just read.
The Turing machine then moves either left or right and
modifies the tokens to keep the tape in order by ad ding
+1/ + 1 counters to all tokens on one side of the head and
−
1/
−
1 counters to all tokens on the other sid e. M oving left o r
right will be accomplishe d by casting first Cleansing Beam
and then So ul Snuffers.
D. Adding a Second State
Everything described so far outlines the operation of one
state of the Turing machine. However, our Turing machine
requires two states. To ac complish this, we leverage phasing:
an object with phasing can ‘phase in’ or ‘phase out’, and
while it’s phased out, it’s treated as though it doesn’t exist.
We can grant phasing to our Rotlung Reanimators usin g the
enchantment Cloak of Invisibility (“Enchanted creature has
phasing and can’t be blocked except by Walls”) and create a
second set of Ro tlung Reanimat ors to encode th e program
4
Throughout this paper, card text that has been modified using cards such
as A rtificial Evolution is written in italics.
3
TABLE I
GAME STATE WHEN THE (2, 18) UTM BEGINS
Card
Controller Changed text / choices / attachment
29 Rotlung Reanimator Bob See Table II
7 Xathrid Necromancer Bob See Table II
29 Cloak of Invisibility
Alice attached to each Rotlung Reanimator
7 Cloak of Invisibility Alice attached to each Xathrid Necromancer
Wheel of Sun and Moon Alice attached to Alice
Illusory Gains Alice attached to latest tape token
Steely Resolve
Alice Assembly-Worker
2 Dread of Night Alice Black
Fungus Sliver Alice Incarnation
Rotlung Reanimator
Alice Lhurgoyf, black, Cephalid
Rotlung Reanimator Bob Lhurgoyf, green, Lhurgoyf
Shared Triumph Alice Lhurgoyf
Rotlung Reanimator Alice Rat, black, Cephalid
Rotlung Reanimator
Bob Rat, white, Rat
Shared Triumph Alice Rat
Card Controller
Wild Evocation Bob
Recycle Bob
Privileged Position
Bob
Vigor Alice
Vigor Bob
Mesmeric Orb Alice
Ancient Tomb
Alice
Prismatic Omen Alice
Choke Alice
Blazing Archon
Alice
Blazing Archon Bob
q
2
. At the moment we read the current cell, exactly one set of
Rotlung Reanimato rs will be phased in.
Objects with phasing phase in or out at the beginning of
their c ontroller’s turn, effectively toggling between two states.
Accordingly we will arr a nge for the turn cycle to last 4 turns
for each player when no state chan ge occurs, but just 3 turns
when we need to chan ge state.
IV. THE FULL CONSTRUCTION
Now we will provide the full construction of the Magic: The
Gathering Turing machine and walk thr ough a computational
step. The outline of one step of the comp utation is as follows
(Bob’s tur ns ar e omitted as nothing happens during them):
T1 Alice c asts Infest. Turing processing occurs: a white or
green token dies, a new white or gre en token is c reated.
T2 Alice casts Cleansing Beam, putting two +1/ + 1 c ount-
ers on the side of the tape we are moving away fro m.
T3 If the Turing machine is remaining in the same state,
Alice casts Coalit ion Victor y. If it is changing state,
Alice casts Soul Snuffers, putting a
−
1/
−
1 counter on
each creature.
T4 If the Turing machine is remaining in the same state, this
is the point where Alice casts Soul Snuffer s. Otherwise,
the next computational step begins.
A. Beginning a Computational Step and Casting Spells
At the beginning of a computational step, it is Alice’s turn
and she has the card Infest in hand. Her library consists of the
other cards she will cast during the co mputation (Cleansing
Beam, Coalition Victory, and Soul Snuffers, in tha t order).
Bob’s hand and library are both empty. The Turing machine
is in its starting state and th e tape has already been initialised.
At the start of each of Alice’s turns, she has one card in
hand. She’s for ced to cast it due to Bob controlling Wild
Evocation, which reads “At the beginning of each player’s
upkeep, that player reveals a card at random from their hand.
If it’s a land card, the player pu ts it onto th e battlefield.
Otherwise, the player casts it without paying its mana cost
if able.” When the card resolves, it would normally be put
into her graveyard, but Alice is enchanted by Wheel of Sun
and Moon, which causes it to be placed at the bottom of her
library instead, allowing her to redraw it in the future and
keeping the cards she needs to cast in order. After her upkeep
step, Alice proceeds to her draw step and draws the c ard that
she will ca st next turn.
Alice has no choices throughout this process: she do es
control one land, but it remains permanently tapped because
of Choke (“Islands don’t untap during their con trollers untap
steps”), so she is unable to cast any of the spells she draws
except via Wild Evocation’s ability. Neither player is able to
attack because they both control a Blazing Archon, “Creatures
can’t attack you.”
Bob has no cards in hand and controls Recycle, which reads
(in par t) “Skip your draw step”. This prevents Bob from losing
due to drawing fro m an empty library.
B. Reading the Current Cell
On the first turn of the cycle, Alice is forced to cast Infest,
“All creatures get
−
2/
−
2 until end of turn.” This kills one
creature: the tape token at the position of the current read
head, controlled by Bob. This will cause prec isely o ne creature
of Bob’s to trigger – either a Rotlung Reanimator or a
Xathrid Necromancer. Which precise one triggers is b ased
on that token’s cre ature type and the machine’s current state,
correspo nding to the a ppropriate rule in the definition of the
(2, 18) UTM . This Reanimator or Necromancer will create a
new 2/2 token to replace the one that died. T he new token’s
creature type represents the symbol to be written to the current
cell, and the new token’s colour indicates the direction for the
machine to move: white for left or green for right.
Alice controls Illusory Gains, an Aura which reads “You
control enchanted creature. Whenever a creature enters the
battlefield under an opponent’s control, attach Illusory Gains to
that creature.” Each time one of Bo b’s Rotlung Reanimato r s
or Xathrid Necromancers creates a new token, Illusory
Gains triggers, granting Alice co ntrol of the newest token o n
the tape, and reverting contr ol of the previous token to Bob.
4
TABLE II
FULL TEXT OF THE ROTLUNG REANIMATORS AND XATHRID NECROMANCERS ENCODING THE (2, 18) UTM
Rogozhin’s program Card text
q
1
1 c
2
Lq
1
Whenever an Aetherborn dies, create a 2/2 white Sliver
q
1
−→
1
←−
1
1
Rq
1
Whenever a Basilisk dies, create a 2/2 green Elf
q
1
←−
1 c
2
Lq
1
Whenever a Cephalid dies, create a 2/2 white Sliver
q
1
−→
1
1
1 Rq
1
Whenever a Demon dies, create a 2/2 green Aetherborn
q
1
←−
1
1
−→
1
1
Lq
1
Whenever an Elf dies, create a 2/2 white Demon
q
1
b
←−
b Rq
1
Whenever a Faerie dies, create a 2/2 green Harpy
q
1
−→
b
←−
b
1
Rq
1
Whenever a Giant dies, create a 2/2 green Juggernaut
q
1
←−
b b Lq
1
Whenever a Harpy dies, create a 2/2 white Faerie
q
1
−→
b
1
b Rq
1
Whenever an Illusion dies, create a 2/2 green Faerie
q
1
←−
b
1
−→
b
1
Lq
1
Whenever a Juggernaut dies, create a 2/2 white Illusion
q
1
b
2
b
3
Lq
2
Whenever a Kavu dies, create a tapped 2/2 white Leviathan
q
1
b
3
−→
b
1
Lq
2
Whenever a Leviathan dies, create a tapped 2/2 white Illusion
q
1
c
−→
1 Lq
2
Whenever a Myr dies, create a tapped 2/2 white Basilisk
q
1
−→
c
←−
c Rq
1
Whenever a Noggle dies, create a 2/2 green Orc
q
1
←−
c
−→
c
1
Lq
1
Whenever an Orc dies, create a 2/2 white Pegasus
q
1
−→
c
1
←−
c
1
Rq
2
Whenever a Pegasus dies, create a tapped 2/2 green Rhino
q
1
←−
c
1
HALT
Whenever a Rhino dies, create a 2/2 blue Assassin
q
1
c
2
←−
1 Rq
1
Whenever a Sliver dies, create a 2/2 green Cephalid
q
2
1
←−
1 Rq
2
Whenever an Aetherborn dies, create a 2/2 green Cephalid
q
2
−→
1
←−
1 Rq
2
Whenever a Basilisk dies, create a 2/2 green Cephalid
q
2
←−
1
−→
1 Lq
2
Whenever a Cephalid dies, create a 2/2 white Basilisk
q
2
−→
1
1
←−
1
1
Rq
2
Whenever a Demon dies, create a 2/2 green Elf
q
2
←−
1
1
1 Lq
2
Whenever an Elf dies, create a 2/2 white Aetherborn
q
2
b b
2
Rq
1
Whenever a Faerie dies, create a tapped 2/2 green Kavu
q
2
−→
b
←−
b Rq
2
Whenever a Giant dies, create a 2/2 green H arpy
q
2
←−
b
−→
b Lq
2
Whenever a Harpy dies, create a 2/2 white Giant
q
2
−→
b
1
←−
b
1
Rq
2
Whenever an Illusion dies, create a 2/2 green Juggernaut
q
2
←−
b
1
−→
b Lq
2
Whenever a Juggernaut dies, create a 2/2 white Giant
q
2
b
2
b Rq
1
Whenever a Kavu dies, create a tapped 2/2 green Faerie
q
2
b
3
←−
b
1
Rq
2
Whenever a Leviathan dies, create a 2/2 green Juggernaut
q
2
c
←−
c Rq
2
Whenever a Myr dies, create a 2/2 green Orc
q
2
−→
c
←−
c Rq
2
Whenever a Noggle dies, create a 2/2 green Orc
q
2
←−
c
−→
c Lq
2
Whenever an Orc dies, create a 2/2 white Noggle
q
2
−→
c
1
c
2
Rq
2
Whenever a Pegasus dies, create a 2/2 green Sliver
q
2
←−
c
1
c
2
Lq
1
Whenever a Rhino dies, create a tapped 2/2 white Sliver
q
2
c
2
c Lq
2
Whenever a Sliver dies, create a 2/2 white Myr
So at any point Bob controls all of the tape except for the
most rece ntly written sy mbol, which is controlled by Alice .
C. Moving Left or Right
If the new token is white, the Turing machine needs to
move left. To do this we n eed to take two actions: put a
+1/ + 1 counter on a ll white creatures (move the tape away
from white), and put a
−
1/
−
1 counter on all green creatures
(move the tape towards green). We rephrase this instead as:
put two +1/ + 1 counters on a ll white creatures, and put a
−
1/
−
1 counter on all creatur es.
On Alice’s second turn, she ca sts Cleansing Bea m, which
reads “Cleansing Beam deals 2 d a mage to target creature and
each other crea ture that shares a color with it.” Bob controls
Privileged Position so none of Bob’s creatures can be targeted
by any spell Alice casts. Alice controls some creature s other
than the tape token, but they have all been gr a nted creature
type Assembly-Worker by a hacked Olivia Voldaren, and
Alice controls a Steely Resolve namin g Assembly-Worker
(“Creatures of the chosen type have shroud. (They can’t b e
the targets of spells or abilities.)”) This makes it so that the
only legal target for Cleansing Beam is the one tape token
that Alice controls thanks to her Illusory Gains.
Recall that this token is white if we’re moving left, or green
if we’re moving right. Cleansing Beam is about to deal 2
damage to ea ch white crea ture if we’re moving left, or to
each green creature if we’re moving rig ht. Alice an d Bob
each control a copy of Vigor – “If damage would be dealt
to another creature you control, prevent tha t damage. Put a
+1/ + 1 counter on that creature for each 1 damage prevented
this way.” So Cleansing Beam ends up putting two +1/ + 1
counters on eith e r each white cr e ature or each gr e en creature.
On the last tu rn of the cycle, Alice casts Soul Snuffers, a
3/3 black creature which reads “When Soul Snuffers enters the
battlefield, put a
−
1/
−
1 counter on eac h creature.” Th e re are
two copies of Dread of Night hacked to each say “Black
creatures get
−
1/
−
1”, which mean that the Soul Snuffers’
triggered ability will kill itself, as well as shrinking every other
5
creature. The creatures comprising the tape have n ow received
either a single
−
1/
−
1 counter, or two +1/ + 1 counter s and a
−
1/
−
1 counter.
To ensure that the creatures providing the infrastructure
(such as Rotlung Rea nima tor) aren’t killed by the succession
of
−
1/
−
1 counters each computational step, we arrange that
they also have game colours gre en, white, red and black,
using Prismatic Lace, “Target permanent becomes the color
or colors of your choice. (This effect lasts indefinitely.)”
Accordingly, each cycle Cleansing Beam will put two +1/+1
counters on them, growing them faster than the
−
1/
−
1 counters
shrink them. This applies to each creature excep t Vigor
itself; to keep each player’s Vigor from dwindling, there is
a Fungus Sliver hacked to read “All Incarnation creatures
have ‘Whenever this crea ture is dealt damage, put a +1/ + 1
counter on it.’ ”
D. Changing State
The instruction to change state is handled by replacing seven
of Bob’s Rotlung Reanimators with Xathrid Necromancer.
These two c ards have very similar text, except that Xathrid
Necromancer only notices Bob’s creatures dying (this is not
a problem, as the active cell of the tape is always contro lled
by Bob), and that Xathrid Necromancer creates its token
tapped.
For example, when the q
1
program (State A) sees symbol 1,
it writes symbol 18, m oves left, and remains in state A. This
is represented by a p hasing Rotlung Reanimator under Bob’s
control saying “Whenever Rotlung Reanimator or another
Aetherborn dies, c reate a 2/2 white Sliver creature token.”
By contrast, when the q
1
program sees symbol 11, it
writes symbol 12, m oves left, and changes to state B. This is
represented by a phasing Xathrid Necromancer under Bob’s
control saying “Whe never Xathrid Necromancer or another
Kavu creature you control dies, create a tapped 2/2 white
Leviathan creature token.”
In both cases this token is cr eated under Bob’s control on
turn T1, but Alice’s Illusory Gains triggers and grants her
control of it. In the case where it’s tappe d, that means at
the beginning of turn T2, it will untap. This causes Alice’s
Mesmeric Orb’s trigger to be put on the stack at the same time
as Bob’s Wild Evocation’s trigger ( sin ce no pla yer receives
priority during the untap step). Alice is the active pla yer, so
Alice’s tr igger is put on the stack first and then Bob’s [16]; so
the Wild Evocation trigger resolves, forcing Alice to ca st and
resolve Cleansing Beam, before the Mesmeric Orb trigger
resolves.
When the Mesmeric Orb trigger does reso lve, it tries to
put the Coalition Victory from the top of Alice’s library into
her graveyard. But Wheel of Sun and Moon modifies this
event to put Coalition Victory onto the bottom of her library,
just underneath the Cleansing Beam that’s just resolved.
Once all these triggers are resolved, Alice proceeds to her
draw step. When the state is not c hanging, she will draw
Coalition Victory at this point, but when the state is changing,
that card is skipped and she moves on to draw Soul Snuffers
in turn T2’s draw step, so she will cast it on turn T3.
The net result of this is that the computation step is 3
turns long for each player when the state is changing, but
4 turns long f or each player when the state is not changing. In
the normal 4-turn operation, Bob’s p hasing Reanima tors and
Necromancers will phase in twice and ph ase out twice , and be
in the same state on one cycle’s turn T1 as they were in the
previous cycle’s turn T1. But when changing state, they will
have changed phase by the next cycle’s turn T1, switching the
Turing machine’s state.
E. Out of Tape
The Turing tape can be initialised to a ny desired length
before starting processing. But it is preferable to allow the
machine to run on a simulated infinite tape: in other words,
to assume that any uninitialised tape space contains symbol
3 (the blank symbol in the (2, 18 ) UTM), represented by
creature type Cephalid. This is accomplished by having the
ends of the currently-initialised tape marked by two special
tokens, one green Lhurgoyf and one white Rat. Su ppose we’ve
exhausted all the initialised tape to the left. This means that
the casting of Infest on turn T1 kills the Lhurgoyf rather than
one of the normal tape types. This does not directly trigg er
any of the normal Reanimators/Necromancers. Instead, Bob
has another Rotlung Re animator hacked to read “Wh e never
Rotlung Reanimator or another Lhurgoyf dies, create a 2/2
green Lhurgoyf creature token”, and A lice has a Rotlung
Reanimator hacked to read “Whenever Rotlung Reanimator
or another Lhurgoyf dies, create a 2/2 black Cephalid creature
token.” Bob’s trigger will resolve first, then Alice’s.
First, Bob’s Reanimator trigger creates a new Lhurgoyf just
to the left of the current head. (Alice’s Illusory Gains triggers
and gives her control of this new Lhurgoyf, but that will
soon change.) We have one c opy of Shared Tr iumph set to
Lhurgoyf (“Creatures of th e chosen type get +1/+1”) so this
token arrives as a 3/3.
Second, Alice’s Reanimator trigger now creates a 2/2 black
Cephalid under Alice’s control. The same two copies of Dread
of Night as before are giving all black cre atures
−
2/
−
2, so the
black Cephalid w ill arrive as a 0/0 and immediately die.
The death of this Cephalid triggers one of th e regular
phasing Reanimators o f Bob’s just as if a tape cell containing
symbol 3 had been read: a new 2/2 token is created and
Illusory Gains moves to that new token. The green Lhurgoyf
token ser ving as an end-of-tape marker has be en recreated one
step over to th e left.
The situation for the white Rat representing the right-hand
end o f the tape is exactly equivalent. Bob has a Rotlung
Reanimator hacked to read “Whenever Rotlung Reanimator
or another Rat dies, crea te a 2/2 white Rat creature token”;
Alice has a Rotlung Reanimator hacked to read “Whenever
Rotlung Reanim ator or another Rat dies, create a 2/2 black
Cephalid creature token”; and we have ano ther Shared Tri-
umph set to Rat.
6
(This algorithm would be a little more complex if reading
symbol 3 could cause a state change in the (2, 18) UTM , but
thankfully it ca nnot.)
F. Halting
We choose to encode halting as making Alice win th e game.
When the Turing machine doesn’t change state, Alice casts
the card Coalition Victory on her third turn. It reads “You win
the game if you control a lan d of eac h basic land type and a
creature of e ach color.” This normally accomplishes nothing
because she controls no blue cr e atures (Prismatic Lace has
been used to give her creatures of all the other colours). She
does, however, control one land, and also controls Prismatic
Omen, which re ads “L ands you control are eve ry basic land
type in addition to their other types.” Recall that Choke is in
play, preventing he r from activating the mana ability of this
land.
When the halt symbol is read (symbol 17 in state A), the
appropriate phasing Reanimator of Bob’s read s “Whenever
Rotlung Reanimator or another Rhino dies, create a 2/2 blue
Assassin creature token.” Alice’s Illusory Gains takes control
of this Assassin token in th e usual way in turn T1. She now
meets the condition for Coalition Victory when she casts it
on turn T 3, and wins the game.
If the encoded machine does not in fact h a lt then th e game
has entered an unbreakable deterministic in finite loop , which
is specified as a draw by r ule 104.4b [16].
V. DISCUSSION
A. Consequen ces for Computational The ories of Ga m e s
This construction establishes that Magic: The Gathering is
the most computationally complex real-world game known in
the literature. I n addition to showing that optimal strategic
play in Magic is non-computable, it also shows that merely
evaluatin g the deterministic consequences of past moves in
Magic is non-computable. The full complexity of optimal
strategic play remains an open qu estion, as do ma ny other
computational aspects of Magic. For example, a player ap-
pears to have infinitely many moves available to them from
some board states of Magic. Whether or not there exists a
real-world game o f Magic in which a player has infinitely
many meaningfully different moves available to them has the
potentially to highly impact the way we understand and model
games as a form of computation.
Indeed , this result raises several interesting philosophical
questions about games as a form of computa tion. Some
authors, such as Demaine and Hearn [9], have sought a formal
framework for modelling games that is strictly sub-Turing.
Unlike the open-world, non-strategic game s in which Turing
machines have been constructed be fore, Magic: The Gathering
is unam biguously a two-playe r strategic gam e like such models
attempt to represen t. Therefore this result shows th a t any sub-
Turing model is n ecessarily inadequate to capture all gam es.
Quite the opposite: it seems likely that a super-Turing model
of game s would be necessary to exp la in Magic. The na¨ıve
extension of Demaine and Hea rn’s Constraint Logic to allow
for unbounded memory appears to be meaningless, although
it’s possible that a clever appr oach would bring success.
Open Problem 3: Does there exist a generalisation of
Constraint Logic that explains the computational complexity
of Magic: The G athering?
Although our construction is reducible to the Halting Prob-
lem, the fact that even evaluating a board is non-computable
is strongly suggestive that th e complexity of strategic play is
greater than that. We be lieve there is strong evidence that the
true computation al complexity is far highe r. In particular, we
conjecture:
Conjecture 4: Playing Magic: The Gathering optimally is
at least as hard as 0
(ω)
.
Whether or not it is possible for there to be a real-world
game whose computational complexity is strictly harder than
0
(ω)
is an interesting p hilosophical question. If not, then this
conjecture would imply that Magic: The Gathering is as har d
as it is possible for a real-world game to be.
B. Real-world Playab ility a nd Legality
While ther e are practical difficulties invo lved with correctly
setting up the necessary board state, such as running out of
space on y our table, a sufficiently tena cious player could set
up and execute this construction in a re al-world tournament
TABLE III
60-CARD DECKLIS T TO PLAY THE TURING MACHI NE I N A LEGACY TOURNAMENT
Card
Purpose Card Purpose Card Purpose
4 Ancient Tom b Bootstrap 1 Rotlung Reanimator Logic processing 1 Xathrid Necromancer Change state
4 Lotus P etal
Bootstrap 1 Cloak of Invisibility Logic processing 1 Mesmeric Orb Change state
4 Grim Monolith Infinite mana device 1 Infest Logic processing 1 Coalition Victory Halting device
4 Power Artifact Infinite mana device 1 Cleansing Beam Logic processing 1 Prismatic Omen Halting device
4 Gemstone Array
Infinite mana device 1 Soul Snuffers Logic processing 1 Choke Halting device
4 Staff of Domination Draw rest of deck 1 Illusory Gains Logic processing 1 Recycle Remove choices
1 Memnarch Make token copies 1 Privileged Position Logic processing 1 Blazing Archon Remove choices
1 Stolen Identity Make token copies 1 Steely Resolve Logic processing 1 Djinn Illuminatus Simplify setup
1 Artificial Evolution
Edit cards 1 Vigor Logic processing 1 Reito Lantern Simplify setup
1 Olivia Voldaren Edit cards 1 Fungus Sliver Logic processing 1 Claws of Gix Simplify setup
1 Glamerdye Edit cards 1 Dread of Night Logic processing 1 Riptide Replicator Set up tape
1 Prismatic Lace
Edit cards 1 Wild Evocation Forced play device 1 Capsize Set up tape
1 Donate
Edit card control 1 Wheel of Sun and Moon Forced play device 1 Karn Liberated Cleanup after setup
1 Reality Ripple Edit card phase 1 Shared Triumph Infinite tape device 1 Fathom Feeder Cleanup after setup
7
game of Magic: The Gathering. An example 60-card deck that
is ca pable of executing this construction on the first turn of
the game and which is legal in the competitive Legacy format
can be seen in Table III.
With the correct d raw, the deck uses Ancient Tomb and
three Lotus Petals to play Grim Monolith a nd Power Arti-
fact and generate unlimited colour less mana, at which point
Staff of Domination draws the rest of the deck and Ge m-
stone A rray gene rates unlimited coloured mana. T he deck
casts most of the permanents immediately, and uses Stolen
Identity to make to ken copies of them (using Memnarch
first on the en chantments like Cloak of Invisibility). The
tape is initialised with Riptide Replicator and Capsize. Dj inn
Illuminatus or Reito Lantern allow repeated casting of the
text-modification card s, as well as Reality Ripple whic h sets
the phase of the Rotlung Reanimators and Donate which
gives most permanents to Bob. Once everything is set up,
Steely Resolve is cast, and then Karn Liberated and Capsize
are used to exile all setup permanen ts and all cards from
Bob’s han d, eventually exiling Capsize and Karn Liberate d
themselves. Now no player has any remainin g choices except
to let the Turing machine execute.
In addition to the Comprehensive Rules [16], play at sanc-
tioned Magic: The Gathering tourna ments is also governed by
the Tournament Rules [17]. Some of these rules, most notably
the ones involving slow play, may effect an individual’s ability
to successfully execute the combo due to concerns a bout the
sheer amount of time it would take to manually move the
tokens ar ound to simulate a computation on a Tu ring machine.
This would not be a concern for two agents with sufficiently
high computational power, a s the Tournament Rules also
provide a mechanism called “shortcuts” for players to skip
carrying out laborious loops if both players agree on the game
state at the beginning and the end of the shortcut.
VI. CONCLUSION
We have presented a methodology for embeddin g Ro-
gozhin’s (2, 18) universal Turing machine in a two-player
game of Magic: The Gathering. Consequently, we have shown
that identifying the outcome of a game of Magic in which all
moves are forced for the r e st of the gam e is undecidable. In
addition to solving a decade-old outstanding open problem, in
the process of arriving at our resu lt we showed that Magic:
The Gathering d oes not fit assumptions commonly made by
computer scientists while modelling games. We conjec ture that
optimal play in Magic is far harder than this result alon e
implies, and leave the true complexity of Magic and the
reconciliation of Magic with existing theories of games fo r
future research.
ACKNOWLEDGEMENTS
We are grateful to C-Y. Howe for help simplifying our Tur-
ing machine constructio n considerab ly and to Adam Yedidia
for conversations about the design and construction of Turing
machines.
REFERENCES
[1] David Auger and Oliver Teytaud. The frontier of decidability in partially
observable recursive games. International Journal of Foundations of
Computer Science, 2012.
[2] Stella Biderman and Bjørn Kjos-Hanssen. Non-comparable natu-
ral numbers. Theoretical Computer Science Stack Exchange, 2018.
https://cstheory.stackexchange.com/q/41384(version:2018-08-16).
[3] Krishnendu Chatterjee and Rasmus Ibsen-Jensen. T he complexity of
deciding legality of a single step of Magic: The Gathering. In 22nd
European Conference on Artificial Intelligence, 2016.
[4] Alex Churchill. Magic: The Gathering is Turing complete v5, 2012.
https://www.toothycat.net/
∼
hologram/Turing/.
[5] Alex Churchill et al. Magic is Turing complete (the Turing machine
combo), 2014. http://tinyurl.com/pv3n2lg.
[6] Peter I. Cowling Colin D. Ward and Edward J. Powley. Ensemble deter-
minization in Monte Carlo tree search for the imperfect information card
game Magic: The Gathering. In IEEE Transactions on Computational
Intelligence and AI in Games, volume 4, 2012.
[7] Michael J. Coulombe and Jayson Lynch. Cooperating in video games?
Impossible! Undecidability of team multiplayer games. In 9th Interna-
tional Conference on Fun with Algorithms, 2018.
[8] Erik D. Demaine and Robert A. Hearn. Playing games with algorithms:
algorithmic combinatorial game theory. In 26th Symp. on Mathematical
Foundations in Computer Science, pages 18–32, 2001.
[9] Erik D. Demaine and Robert A. Hearn. Constraint logic: A uniform
framework for modeling computation as games. In 2008 23rd Annual
IEEE Conference on Computational Complexity, pages 149–162, 2008.
[10] Erik D. Demaine and Robert A. H earn. Games, Puzzles, and Computa-
tion. CRC Press, 2009.
[11] Alexander Esche. Mathematical Programming and Magic: The Gather-
ing. PhD thesis, Northern Illinois University, 2018.
[12] Eugenio Fortanely. Personal communication, 2018.
[13] H. G. Rice. Classes of recursively enumerable sets and their decision
problems. Trans. Amer. Math. Soc., 74:358366, 1953.
[14] Yurii Rogozhin. Sm all universal Turing machines. Theoretical Computer
Science, 168(2):215–240, 1996.
[15] Colin D. Ward and Peter I. Cowling. Monte Carlo search applied to
card selection in Magic: The Gathering. In CIG’09 Proceedings of the
5th international conference on Computational Intelligence and Games,
pages 9–16, 2009.
[16] Wizards of the Coast. Magic: The G ath-
ering comprehensive rules, Aug 2018.
https://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules.
[17] Wizards of the Coast. Magic: The
Gathering tournament rules, Aug 2018.
https://wpn.wizards.com/sites/wpn/files/attachements/mtg
mtr 21jan19 en.pdf.
8
#### What is a Creature Token?
Creature tokens are effectively markers that represent a creature card with specific properties. Creature tokens can be created as the result of a spell for instance. Creature Tokens are not actual Magic cards. They cannot exist in your hand or deck. While playing the game, players can use whatever is more player friendly for them to represent a creature token with specific properties (e.g. die, piece of torn paper, etc).
Caulombe and Lynch were able to reason about those video games by structuring them into an abstract game on graphs that serves as a simplification of the Team Computation Game they mention (which is undecidable).
Here is an example of how they built those graphs leveraging elements of a game (Super Smash Bros in this case):
If you are curious to learn more about it, here is the [paper](https://drops.dagstuhl.de/opus/volltexte/2018/8805/pdf/LIPIcs-FUN-2018-14.pdf) by Caulombe and Lynch.
Unlike games like Chess and Go, researchers still haven't been able to develop an AI capable playing magic at the level of the best human Magic players.
Rengo Kriegspiel is a "blindfolded", team variant of Go. There are two teams with two players each, a referee and five Go sets. The players move alternatingly. Each player keeps track of their own moves on their own board; they are not informed about teammates' or opponents' moves. The referee keeps track of the complete game and informs a player if their move was illegal, in which case they can try again. The referee removes captured stones from all affected boards.
*A game of Rengo Kriegspiel*
Rice's Theorem states that all nontrivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior for a given set of inputs (e.g. does the program terminate for all inputs), unlike a syntactic property (e.g. does the program contain a go-to statement)
Rogozhin's (2,18) Turing Machine is a Turing machine with 2 states and 18 different symbols (that can be written to the "tape")
### Rogozhin Turing Machines
When Alan Turing came up with the idea of a universal machine he had in mind the simplest computing model powerful enough to calculate all possible functions that can be calculated.
Turing Machines are generally defined with a set of symbols (which can be written to the "tape") and a set of states that the Turing Machine can be in. The natural question is: What is the smallest possible Turing Machine? Claude Shannon showed that two symbols were sufficient so long as enough states were used (or vice versa), and that it was always possible to exchange states for symbols.
Yurii Rogozhin worked on finding the smallest universal Turing Machines. Generally, the syntax for Universal Turing Machines is $UTM(m; n)$ - which denotes the class of universal Turing machines with $m$ states and $n$ symbols.
As the authors state, there are a number of real world games that show non-trivial computational complexity:
- **Tetris** - If the sequence of pieces is given, it is NP-complete to determine whether it is possible to keep the stack below a finite height
- **Crossword puzzle construction** - NP-complete
- **Dots-and-boxes** - NP-hard
- **Mahjong Solitaire** - NP-complete
- **Minesweeper** - coNP-Hard
- **FreeCell** - NP-Complete
- **Jigsaw puzzles** - Jigsaw puzzles can be formalized in a variety of ways, different edge matching rules, different boundary conditions, etc. Many versions are NP-Complete
- **Lemmings** - NP-Complete
Cards mentioned by the authors:
Zero-sum games are 2-player games where the sum of the payoffs of the players is 0 - these are games of pure competition between the players. Zero-sum stochastic games are dynamic zero-sum games played in discrete time with probabilistic transitions between states. In games with imperfect information (like Poker) there might be certain parts of hidden state that are not visible to one or more players
Example of a Turing Machine built by a player in Minecraft
## Playing Magic
The full rules of Magic are extensive - you can find the official document [here (https://media.wizards.com/2021/downloads/MagicCompRules%20202109224.pdf), which is over 250 pages long.
#### Overview of game
A typical game of Magic usually involves two players. Each player has their own deck of cards, either one previously constructed or made from a limited pool of cards for the event. A player starts the game with a "life total" of twenty and loses the game when their life total is reduced to zero. A player's life total is typically reduced via an attack by an unblocked creature or by direct damage caused by spells. A player can also lose if they must draw from an empty deck. Some cards specify other ways to win or lose the game. Additionally, the "Golden Rule of Magic" states that "whenever a card's text directly contradicts the rules, the card takes precedence".
If you wish to learn how to play Magic, [this video](https://www.youtube.com/watch?v=RZyXU1L3JXk) is a good resource.
### Magic The Gathering
Magic: The Gathering is a trading card game created by the American mathematician and inventor Richard Garfield (great-grandson of U.S. President James A. Garfield) and released in 1993. Garfield was influenced by his passion for the roleplaying game Dungeons & Dragons.
Magic is considered to be the first trading card game. It successfully combined elements of board games and card collecting. During a game of Magic, two players compete by leveraging their homebuilt deck of cards to defeat their opponent by spending "mana" to play creatures, spells and other items. Players can customize their deck before each battle, choosing cards from their own library. More than 20,000 different cards have been released for Magic: The Gathering since its release and there are an estimated 20 million players all over the world.
*Two players playing Magic: the Gathering*