postgres=# SELECT version();
 PostgreSQL 10.6 (Ubuntu 10.6-0ubuntu0.18.04.1) on x86_64-pc-linux-gnu, compiled by gcc (Ubuntu 7.3.0-16ubuntu3) 7.3.0, 64-bit

CREATE TABLE State(    -- TM states
  sid INTEGER PRIMARY KEY,    -- 0 is always the initial state
  isFinal BOOLEAN NOT NULL,
  sname TEXT UNIQUE NOT NULL -- just for show
);

CREATE TABLE Symbol( -- TM symbols
  cid INTEGER PRIMARY KEY,  -- 0 is always the blank symbol
  cname TEXT UNIQUE NOT NULL
);

CREATE TABLE Transition( -- TM transition function
  sid INTEGER NOT NULL REFERENCES State,     -- initial state
  symbol INTEGER NOT NULL REFERENCES Symbol, -- & symbol
  UNIQUE(sid, symbol),
  new_state INTEGER NOT NULL REFERENCES State,
  new_symbol INTEGER NOT NULL REFERENCES Symbol,
  move INTEGER NOT NULL CHECK(move=-1 OR move=1)
);

CREATE TABLE Tape( -- TM initial tape contents
  tid INTEGER PRIMARY KEY,
  symbol INTEGER REFERENCES Symbol
);

WITH RECURSIVE running(rid, sid, tape, pos) AS (
  -- first store initial tape contents
  SELECT 0, 0, ARRAY(SELECT symbol FROM Tape ORDER BY tid), 1
  UNION
  -- then proceed to compute iterations
  SELECT p.rid+1, t.new_state,
         -- build updated tape as an array
         p.tape[1:p.pos-1] ||                      -- prefix
           t.new_symbol ||                         -- updated cell
           p.tape[p.pos+1:array_length(p.tape,1)], -- suffix
         -- move cursor position
         p.pos+t.move
  FROM running AS p
  -- get state details, to know whether to stop
  JOIN State AS s ON (p.sid=s.sid)
  -- get corresponding state transition
  JOIN Transition AS t ON (t.sid=p.sid AND
                           -- coalesce defaults to blank
                           t.symbol=COALESCE(p.tape[p.pos],0))
  WHERE -- stop on a final state
        NOT s.isFinal
)
-- just store the computed table
INSERT INTO Run
  SELECT * FROM running;

ERROR:  insert or update on table "run" violates foreign key constraint "run_sid_fkey"
DETAIL:  Key (sid)=(0) is not present in table "state".

{state, color} -> {next_state, next_color, offset(header position)}

{1, 2} -> {1, 1, -1},
{1, 1} -> {1, 2, -1},
{1, 0} -> {2, 1, 1 },
{2, 2} -> {1, 0, 1 },
{2, 1} -> {2, 2, 1 },
{2, 0} -> {1, 2, -1}

INSERT INTO State VALUES(0,FALSE,1),(1,FALSE,2); 
INSERT INTO Symbol VALUES(0,'0'),(1,'1'),(2,'2');
INSERT INTO Transition VALUES
(0,2,0,1,-1),
(0,1,0,2,-1),
(0,0,1,1, 1),
(1,2,0,0, 1),
(1,1,1,2, 1),
(1,0,0,2,-1)

INSERT INTO Tape VALUES(0,0);

WITH RECURSIVE running(rid, sid, tape, pos) AS (
  -- first store initial tape contents
  SELECT 0, 0, ARRAY(SELECT symbol FROM Tape ORDER BY tid), 1
  UNION
  -- then proceed to compute iterations
  SELECT p.rid+1, t.new_state,
         -- build updated tape as an array
         p.tape[1:p.pos-1] ||                      -- prefix
           t.new_symbol ||                         -- updated cell
           p.tape[p.pos+1:array_length(p.tape,1)], -- suffix
         -- move cursor position
         p.pos+t.move
  FROM running AS p
  -- get state details, to know whether to stop
  JOIN State AS s ON (p.sid=s.sid)
  -- get corresponding state transition
  JOIN Transition AS t ON (t.sid=p.sid AND
                           -- coalesce defaults to blank
                           t.symbol=COALESCE(p.tape[p.pos],0))
  WHERE -- stop on a final state
        --NOT s.isFinal -- There is no halt state.
        NOT p.rid = 100 -- By changing this number 
                        -- you can set the number of generations of cellular automata on tape.
)
-- just store the computed table
INSERT INTO Run
  SELECT * FROM running;

2 -> orange
1 -> yellowfins
0 -> white

{{1, 2} -> {2, 2, 1}, {1, 1} -> {1, 1, 1}, {1, 0} -> {3, 1, -1}, {2, 2} -> {2,1, 1}, {2, 1} -> {1, 2, 1}, {3, 2} -> {3, 2, -1}, {3, 1} -> {3, 1, -1}, {3, 0} -> {1, 0, 1}}