Exploring Tetris as a Transformation Semigroup

Abstract

Tetris is a popular puzzle video game, invented in 1984. We formulate two versions of the game as a transformation semigroup and use this formulation to view the game through the lens of Krohn-Rhodes theory. In a variation of the game upon which it restarts if the player loses, we find permutation group structures, including the symmetric group $S_5$ which contains a non-abelian simple group as a subgroup. This implies, at least in a simple case, that iterated Tetris is finitarily computationally universal.

Appendix: Krohn-Rhodes Theory and the Holonomy Decomposition

The Krohn-Rhodes (KR) theorem describes a general decomposition of transformation semigroups in terms of wreath products of the finite simple groups and the flip-flop monoid. A visualization of the flip-flop monoid is shown in Fig. 6.

Fig. 6

The flip-flop monoid on $X = \{1,2\}$ is given by the set of transformations $S = \{A,B,I\}$

Theorem 2

(Krohn-Rhodes decomposition [12]) A finite transformation semigroup (X, S), with states X and semigroup S acting on states by transformations, has a decomposition

$$(X,S) \text { divides } H_1 \wr H_2 \wr H_3 \ldots \wr H_n$$

with components $H_1, H_2, \ldots , H_n$, such that each $H_i$ is a finite simple group dividing S or the flip-flop monoid.

The decomposition given by this theorem tends to be far from optimal in practice. Therefore, most practical implementations of semigroup decomposition use the holonomy method described in [7].

We will reproduce the relevant definitions and theorems here. Define the set Q as

$$Q = \{\{X \cdot s\} | s \in S\} \cup \{X\} \cup \{\{a\} | a \in X \}$$

then we can define a relation on Q called subduction.

Definition 2

(Subduction) Let $S^I$ denote S with a new identity element appended. Given, $A,B \in Q$, we define an reflexive, transitive relation on Q,

$$A \le B \iff \exists s \in S^I, A \subseteq B \cdot s$$

Furthermore, let $A < B$ if $A \le B$ but not $B \le A$. This relation, which we will call subduction, induces an equivalence relation on Q: $A \equiv B \iff A \le B$ and $B \le A$. For each equivalence class $A /\!\!\equiv$ in $Q /\!\!\equiv$, let $\bar{A}$ be a unique representative.

Definition 3

(Tiles) Define A to be a tile of B if $A \subsetneq B$ and

$$\forall Z \in Q, (A \le Z \le B \implies Z = A \text { or } Z = B)$$

If $A \in Q$ with $|A|>1$, the set of tiles of A is $\Theta _A \subset Q$

Definition 4

(Holonomy group) The holonomy group, written $H_A$, of A is the set of permutations of $\Theta _A$ induced by the elements of $S^I$. If we let $H_A$ act on $\Theta _A$, then $(\Theta _A,H_A)$ is the holonomy permutation group of A.

Definition 5

(Height of an Image Set) The height of $A\in Q$ is h(A), where h(A) is the length of the longest strict subduction chain up to A.

We are now able to state the holonomy decomposition theorem, which asserts that the semigroup (X, S) divides a cascade product, from which the Krohn-Rhodes (KR) decomposition (Theorem 2) can be derived. The holonomy theorem describes the transformation semigroup (X, S) in terms of symmetries in the way transformations in S act on the set of tiles of the $\bar{A} \in Q$.

Theorem 3

(Holonomy decomposition [7]) Let (X, S) be a finite transformation semigroup, with $h = h(X)$ the height of X. For each i with $1\le i \le h$, let

$$(\Phi _i, \mathfrak {H_i}) = \prod _{\{A\in Q: h(A)=i,\, \bar{A}=A\}} (\Theta _{\bar{A}}, H_{\bar{A}})$$

$(\Phi _i, \mathfrak {H_i})$ is a permutation group and $(\Phi _i, \overline{\mathfrak {H_i}})$ is the permutation-reset transformation semigroup obtained by appending all constant maps to $\mathfrak {H_i}$. Then

$$(X,S) \text { divides } (\Phi _1, \overline{\mathfrak {H_1}}) \wr (\Phi _2, \overline{\mathfrak {H_2}}) \wr \cdots \wr (\Phi _h, \overline{\mathfrak {H_h}}).$$