NotAwfulTech

Way too many lines follow (but gives the option to finding swaps "manually"):

#!/usr/bin/env jq -n -crR -f

( # If solving manually input need --arg swaps
  # Expected format --arg swaps 'n01-n02,n03-n04'
  # Trigger start with --arg swaps '0-0'
  if $ARGS.named.swaps then $ARGS.named.swaps |
    split(",") | map(split("-") | {(.[0]):.[1]}, {(.[1]):.[0]}) | add
  else {} end
) as $swaps |

[ inputs | select(test("->")) / " " | del(.[3]) ] as $gates |

[ # Defining Target Adder Circuit #
  def pad: "0\(.)"[-2:];
  (
    [ "x00", "AND", "y00", "c00" ],
    [ "x00", "XOR", "y00", "z00" ],
    (
      (range(1;45)|pad) as $i |
      [ "x\($i)", "AND", "y\($i)", "c\($i)" ],
      [ "x\($i)", "XOR", "y\($i)", "a\($i)" ]
    )
  ),
  (
    ["a01", "AND", "c00", "e01"],
    ["a01", "XOR", "c00", "z01"],
    (
      (range(2;45) | [. , . -1 | pad]) as [$i,$j] |
      ["a\($i)", "AND", "s\($j)", "e\($i)"],
      ["a\($i)", "XOR", "s\($j)", "z\($i)"]
    )
  ),
  (
    (
      (range(1;44)|pad) as $i |
      ["c\($i)", "OR", "e\($i)", "s\($i)"]
    ),
    ["c44", "OR", "e44", "z45"]
  )
] as $target_circuit |

( #        Re-order xi XOR yi wires so that xi comes first        #
  $gates | map(if .[0][0:1] == "y" then  [.[2],.[1],.[0],.[3]] end)
) as $gates |

#  Find swaps, mode=0 is automatic, mode>0 is manual  #
def find_swaps($gates; $swaps; $mode): $gates as $old |
  #                   Swap output wires                #
  ( $gates | map(.[3] |= ($swaps[.] // .)) ) as $gates |

  # First level: 'x0i AND y0i -> c0i' and 'x0i XOR y0i -> a0i' #
  #      Get candidate wire dict F, with reverse dict R        #
  ( [ $gates[]
      | select(.[0][0:1] == "x" )
      | select(.[0:2] != ["x00", "XOR"] )
      | if .[1] == "AND" then { "\(.[3])": "c\(.[0][1:])"  }
      elif .[1] == "XOR" then { "\(.[3])": "a\(.[0][1:])"  }
      else "Unexpected firt level op" | halt_error end
    ] | add
  ) as $F | ($F | with_entries({key:.value,value:.key})) as $R |

  #       Replace input and output wires with candidates      #
  ( [ $gates[]  | map($F[.] // .)
      | if .[2] | test("c\\d") then [ .[2],.[1],.[0],.[3] ] end
      | if .[2] | test("a\\d") then [ .[2],.[1],.[0],.[3] ] end
    ] # Makes sure that when possible a0i comes 1st, then c0i #
  ) as $gates |

  # Second level:   use info rich 'c0i OR e0i -> s0i' gates   #
  #      Get candidate wire dict S, with reverse dict T       #
  ( [ $gates[]
      | select((.[0] | test("c\\d")) and .[1] == "OR" )
      | {"\(.[2])": "e\(.[0][1:])"}, {"\(.[3])": "s\(.[0][1:])"}
    ] | add | with_entries(select(.key[0:1] != "z"))
  ) as $S | ($S | with_entries({key:.value,value:.key})) as $T |

  ( #      Replace input and output wires with candidates     #
    [ $gates[] | map($S[.] // .) ] | sort_by(.[0][0:1]!="x",.)
  ) as $gates  | #                   Ensure "canonical" order #

  [ # Diff - our input gates only
    $gates - $target_circuit
    | .[] | [ . , map($R[.] // $T[.] // .) ]
  ] as $g |
  [ # Diff +  target circuit only
    $target_circuit - $gates
    | .[] | [ . , map($R[.] // $T[.] // .) ]
  ] as $c |

  if $mode > 0 then
    #    Manual mode print current difference    #
    debug("gates", $g[], "target_circuit", $c[]) |

    if $gates == $target_circuit then
      $swaps | keys | join(",") #   Output successful swaps  #
    else
      "Difference remaining with target circuit!" | halt_error
    end
  else
    # Automatic mode, recursion end #
    if $gates == $target_circuit then
      $swaps | keys | join(",") #   Output successful swaps  #
    else
      [
        first(
          # First case when only output wire is different
          first(
            [$g,$c|map(last)]
            | combinations
            | select(first[0:3] == last[0:3])
            | map(last)
            | select(all(.[]; test("e\\d")|not))
            | select(.[0] != .[1])
            | { (.[0]): .[1], (.[1]): .[0] }
          ),
          # "Only" case where candidate a0i and c0i are in an
          # incorrect input location.
          # Might be more than one for other inputs.
          first(
            [
              $g[] | select(
                ((.[0][0]  | test("a\\d")) and .[0][1] == "OR") or
                ((.[0][0]  | test("c\\d")) and .[0][1] == "XOR")
              ) | map(first)
            ]
            | if length != 2 then
                "More a0i-c0i swaps required" | halt_error
              end
            | map(last)
            | select(.[0] != .[1])
            | { (.[0]): .[1], (.[1]): .[0] }
          )
        )
      ] as [$pair] |
      if $pair | not then
        "Unexpected pair match failure!" | halt_error
      else
        find_swaps($old; $pair+$swaps; 0)
      end
    end
  end
;

find_swaps($gates;$swaps;$swaps|length)

What helped was looking at the penultimate robot's button choices when moving the keypad robot. After the first one or two levels, the transitions settle into the table in the appendix. I will not explain the code.

  (P(0, 1), P(0, 1)): [],
  (P(0, 1), P(0, 2)): [btn.r],
  (P(0, 1), P(1, 0)): [btn.d, btn.l],
  (P(0, 1), P(1, 1)): [btn.d],
  (P(0, 1), P(1, 2)): [btn.d, btn.r],
  (P(0, 2), P(0, 1)): [btn.l],
  (P(0, 2), P(0, 2)): [],
  (P(0, 2), P(1, 0)): [btn.d, btn.l, btn.l],
  (P(0, 2), P(1, 1)): [btn.l, btn.d],
  (P(0, 2), P(1, 2)): [btn.d],
  (P(1, 0), P(0, 1)): [btn.r, btn.u],
  (P(1, 0), P(0, 2)): [btn.r, btn.r, btn.u],
  (P(1, 0), P(1, 0)): [],
  (P(1, 0), P(1, 1)): [btn.r],
  (P(1, 0), P(1, 2)): [btn.r, btn.r],
  (P(1, 1), P(0, 1)): [btn.u],
  (P(1, 1), P(0, 2)): [btn.u, btn.r],
  (P(1, 1), P(1, 0)): [btn.l],
  (P(1, 1), P(1, 1)): [],
  (P(1, 1), P(1, 2)): [btn.r],
  (P(1, 2), P(0, 1)): [btn.l, btn.u],
  (P(1, 2), P(0, 2)): [btn.u],
  (P(1, 2), P(1, 0)): [btn.l, btn.l],
  (P(1, 2), P(1, 1)): [btn.l],
  (P(1, 2), P(1, 2)): [],

Ended up reading wikipedia to lift one the Bron-Kerbosch methods:

#!/usr/bin/env jq -n -rR -f

reduce (
  inputs / "-" #         Build connections dictionary         #
) as [$a,$b] ({}; .[$a] += [$b] | .[$b] += [$a]) | . as $conn |


#  Allow Loose max clique check #
if $ARGS.named.loose == true then

# Only works if there is at least one pair in the max clique #
# That only have the clique members in common.               #

[
  #               For pairs of connected nodes                   #
  ( $conn | keys[] ) as $a | $conn[$a][] as $b | select($a < $b) |
  #             Get the list of nodes in common                  #
      [$a,$b] + ($conn[$a] - ($conn[$a]-$conn[$b])) | unique
]

# From largest size find the first where all the nodes in common #
#    are interconnected -> all(connections ⋂ shared == shared)   #
| sort_by(-length)
| first (
  .[] | select( . as $cb |
    [
        $cb[] as $c
      | ( [$c] + $conn[$c] | sort )
      | ( . - ( . - $cb) ) | length
    ] | unique | length == 1
  )
)

else # Do strict max clique check #

# Example of loose failure:
# 0-1 0-2 0-3 0-4 0-5 1-2 1-3 1-4 1-5
# 2-3 2-4 2-5 3-4 3-5 4-5 a-0 a-1 a-2
# a-3 b-2 b-3 b-4 b-5 c-0 c-1 c-4 c-5

def bron_kerbosch1($R; $P; $X; $cliques):
  if ($P|length) == 0 and ($X|length) == 0 then
    if ($R|length) > 2 then
      {cliques: ($cliques + [$R|sort])}
    end
  else
    reduce $P[] as $v ({$R,$P,$X,$cliques};
      .cliques = bron_kerbosch1(
        .R - [$v] + [$v]     ; # R ∪ {v}
        .P - (.P - $conn[$v]); # P ∩ neighbours(v)
        .X - (.X - $conn[$v]); # X ∩ neighbours(v)
           .cliques
      )    .cliques    |
      .P = (.P - [$v]) |       # P ∖ {v}
      .X = (.X - [$v] + [$v])  # X ∪ {v}
    )
  end
;

bron_kerbosch1([];$conn|keys;[];[]).cliques | max_by(length)

end

| join(",") # Output password