Task: Functional-Dependency (source dataset: Auto-Relate)

Given a table represented shown below, your task is to determine functional dependencies that exist in the table. Recall that a functional dependency is a relationship where the value of one attribute (or a set of attributes), known as the determinant columns, can uniquely determines the value of another attribute, known as the dependent columns (e.g., student-ID => student-Name, car-model => car-make, employee-ID => department, etc.).

Instructions:
Analyze the provided table below, identify all functional dependencies that should semantically hold in the table. Do not produce spurious relationships that not semantically meaningful.

No explanation, format the output as a JSON object:{"Functional-Dependency": <list of functional dependencies>}. Each functional dependency should be represented as [<determinant column>, <dependent column>]

Input Table:
|   Game |   Roll |   P1D1 |   P1D2 |   P1Doubles_many |   P2D1 |   P2D2 |   P2Doubles_many |   P1.Multiplier_many |   P2.Multiplier_many |   P1.Multiplied |   P2.Multiplied |   P1.Multiplier |   P2.Multiplier |   P1Score |   P2Score |   P1Deduction |   P2Deduction |   P1Total |   P2Total | Win/Loss   |   P1Doubles |   P2Doubles |
|-------:|-------:|-------:|-------:|-----------------:|-------:|-------:|-----------------:|---------------------:|---------------------:|----------------:|----------------:|----------------:|----------------:|----------:|----------:|--------------:|--------------:|----------:|----------:|:-----------|------------:|------------:|
|    146 |      3 |      6 |      1 |                0 |      2 |      4 |                0 |                    3 |                    2 |               0 |               0 |               3 |               2 |        31 |        38 |             0 |            20 |        31 |        18 | Win        |           0 |           2 |
|    651 |      5 |      6 |      1 |                0 |      5 |      1 |                0 |                    2 |                    2 |               0 |               0 |               2 |               2 |        36 |        46 |             0 |            10 |        36 |        36 | Loss       |           0 |           1 |
|    505 |      3 |      2 |      2 |                0 |      1 |      6 |                0 |                    2 |                    2 |               0 |               0 |               2 |               2 |        35 |        50 |             0 |            10 |        35 |        40 | Loss       |           0 |           1 |
|    472 |      4 |      1 |      3 |                0 |      5 |      3 |                0 |                    1 |                    2 |               0 |               0 |               1 |               2 |        30 |        34 |             0 |             0 |        30 |        34 | Loss       |           2 |           0 |
|    712 |      4 |      5 |      2 |                0 |      1 |      2 |                0 |                    1 |                    1 |               0 |               0 |               1 |               1 |        35 |        32 |             0 |             0 |        35 |        32 | Win        |           0 |           0 |
|     33 |      4 |      5 |      3 |                0 |      4 |      5 |                0 |                    1 |                    3 |               0 |               0 |               1 |               3 |        34 |        40 |             0 |             0 |        34 |        40 | Loss       |           1 |           0 |
|    270 |      3 |      1 |      1 |                0 |      4 |      1 |                0 |                    1 |                    1 |               0 |               0 |               1 |               1 |        33 |        37 |             0 |             0 |        33 |        37 | Loss       |           0 |           0 |
|    997 |      4 |      3 |      3 |                0 |      2 |      4 |                0 |                    3 |                    3 |               0 |               0 |               3 |               3 |        67 |        33 |            30 |             0 |        37 |        33 | Win        |           2 |           0 |
|    356 |      1 |      4 |      4 |                0 |      3 |      4 |                0 |                    3 |                    3 |               0 |               0 |               3 |               3 |        32 |        40 |             0 |            15 |        32 |        25 | Win        |           0 |           1 |
|    324 |      4 |      1 |      4 |                0 |      3 |      1 |                0 |                    3 |                    2 |               0 |               0 |               3 |               2 |        31 |        32 |             0 |             0 |        31 |        32 | Loss       |           0 |           0 |
|    990 |      4 |      6 |      5 |                0 |      6 |      4 |                0 |                    3 |                    3 |               0 |               0 |               3 |               3 |        47 |        42 |             0 |             0 |        47 |        42 | Win        |           0 |           0 |
|    795 |      5 |      3 |      3 |                0 |      1 |      6 |                0 |                    1 |                    3 |               0 |               0 |               1 |               3 |        30 |        46 |             0 |            15 |        30 |        31 | Loss       |           0 |           1 |
|    493 |      2 |      4 |      3 |                0 |      4 |      2 |                0 |                    1 |                    2 |               0 |               0 |               1 |               2 |        38 |        34 |             0 |             0 |        38 |        34 | Win        |           0 |           0 |
|    979 |      2 |      1 |      1 |                0 |      3 |      5 |                0 |                    1 |                    1 |               0 |               0 |               1 |               1 |        28 |        27 |             0 |             0 |        28 |        27 | Win        |           0 |           0 |
|    382 |      5 |      3 |      5 |                0 |      1 |      3 |                0 |                    2 |                    1 |               0 |               0 |               2 |               1 |        42 |        27 |            10 |             0 |        32 |        27 | Win        |           1 |           1 |
|    251 |      3 |      2 |      3 |                0 |      6 |      6 |                1 |                    1 |                    2 |               0 |              24 |               1 |               2 |        31 |        73 |             0 |            20 |        31 |        53 | Loss       |           0 |           2 |
|    536 |      5 |      1 |      1 |                0 |      2 |      4 |                0 |                    1 |                    2 |               0 |               0 |               1 |               2 |        32 |        57 |             0 |            20 |        32 |        37 | Loss       |           2 |           2 |
|    710 |      4 |      2 |      4 |                0 |      5 |      1 |                0 |                    2 |                    3 |               0 |               0 |               2 |               3 |        50 |        39 |            10 |             0 |        40 |        39 | Win        |           1 |           0 |
|    132 |      4 |      6 |      4 |                0 |      3 |      4 |                0 |                    3 |                    1 |               0 |               0 |               3 |               1 |        35 |        33 |             0 |             0 |        35 |        33 | Win        |           0 |           0 |
|    568 |      3 |      1 |      6 |                0 |      2 |      3 |                0 |                    1 |                    2 |               0 |               0 |               1 |               2 |        39 |        39 |             0 |             0 |        39 |        39 | Loss       |           0 |           0 |
|    632 |      2 |      1 |      4 |                0 |      2 |      3 |                0 |                    1 |                    2 |               0 |               0 |               1 |               2 |        30 |        26 |             0 |             0 |        30 |        26 | Win        |           2 |           0 |
|    853 |      5 |      6 |      5 |                0 |      5 |      4 |                0 |                    2 |                    1 |               0 |               0 |               2 |               1 |        45 |        31 |             0 |             0 |        45 |        31 | Win        |           0 |           1 |
|    736 |      1 |      5 |      4 |                0 |      2 |      4 |                0 |                    1 |                    2 |               0 |               0 |               1 |               2 |        36 |        29 |             0 |             0 |        36 |        29 | Win        |           2 |           0 |
|    902 |      3 |      6 |      2 |                0 |      3 |      6 |                0 |                    1 |                    1 |               0 |               0 |               1 |               1 |        34 |        39 |             0 |             0 |        34 |        39 | Loss       |           0 |           3 |
|    400 |      2 |      2 |      5 |                1 |      4 |      1 |                0 |                    1 |                    1 |               7 |               0 |               1 |               1 |        34 |        31 |             0 |             0 |        34 |        31 | Win        |           1 |           0 |
|    275 |      5 |      1 |      6 |                0 |      5 |      3 |                0 |                    1 |                    1 |               0 |               0 |               1 |               1 |        39 |        37 |             0 |             0 |        39 |        37 | Win        |           1 |           2 |
|    189 |      4 |      6 |      5 |                0 |      6 |      6 |                1 |                    3 |                    1 |               0 |              12 |               3 |               1 |        39 |        43 |            15 |             0 |        24 |        43 | Loss       |           1 |           1 |
|    154 |      1 |      4 |      3 |                0 |      3 |      6 |                0 |                    1 |                    3 |               0 |               0 |               1 |               3 |        40 |        46 |             0 |            15 |        40 |        31 | Win        |           1 |           1 |
|    535 |      2 |      2 |      1 |                1 |      6 |      3 |                0 |                    1 |                    3 |               3 |               0 |               1 |               3 |        36 |        51 |             0 |            15 |        36 |        36 | Loss       |           2 |           1 |
|     66 |      5 |      6 |      6 |                0 |      1 |      4 |                0 |                    1 |                    1 |               0 |               0 |               1 |               1 |        43 |        26 |             0 |             0 |        43 |        26 | Win        |           0 |           1 |
|     21 |      5 |      6 |      6 |                0 |      1 |      3 |                0 |                    3 |                    1 |               0 |               0 |               3 |               1 |        71 |        37 |            30 |             0 |        41 |        37 | Win        |           2 |           0 |
|    941 |      1 |      3 |      5 |                0 |      3 |      6 |                0 |                    1 |                    3 |               0 |               0 |               1 |               3 |        50 |        42 |             0 |             0 |        50 |        42 | Win        |           0 |           0 |
|    343 |      2 |      2 |      5 |                1 |      6 |      4 |                0 |                    2 |                    1 |              14 |               0 |               2 |               1 |        49 |        37 |            10 |             0 |        39 |        37 | Win        |           1 |           1 |
|    639 |      1 |      1 |      1 |                1 |      4 |      2 |                0 |                    2 |                    1 |               4 |               0 |               2 |               1 |        33 |        32 |            10 |             0 |        23 |        32 | Loss       |           1 |           1 |
|    332 |      4 |      5 |      1 |                0 |      3 |      6 |                0 |                    3 |                    1 |               0 |               0 |               3 |               1 |        46 |        42 |            15 |             0 |        31 |        42 | Loss       |           1 |           1 |
|    741 |      4 |      5 |      6 |                0 |      1 |      6 |                0 |                    2 |                    2 |               0 |               0 |               2 |               2 |        38 |        34 |            10 |             0 |        28 |        34 | Loss       |           1 |           0 |
|    506 |      1 |      6 |      5 |                0 |      6 |      6 |                1 |                    1 |                    2 |               0 |              24 |               1 |               2 |        46 |        51 |             0 |            10 |        46 |        41 | Win        |           0 |           1 |
|    635 |      1 |      1 |      5 |                1 |      4 |      1 |                0 |                    1 |                    2 |               6 |               0 |               1 |               2 |        40 |        53 |             0 |            10 |        40 |        43 | Loss       |           1 |           1 |
|    667 |      2 |      2 |      4 |                1 |      4 |      3 |                0 |                    3 |                    1 |              18 |               0 |               3 |               1 |        59 |        42 |            30 |             0 |        29 |        42 | Loss       |           2 |           2 |
|    257 |      5 |      4 |      4 |                0 |      4 |      5 |                0 |                    3 |                    3 |               0 |               0 |               3 |               3 |        33 |        36 |             0 |             0 |        33 |        36 | Loss       |           0 |           0 |
|    569 |      3 |      2 |      4 |                0 |      5 |      5 |                1 |                    3 |                    2 |               0 |              20 |               3 |               2 |        33 |        48 |             0 |            10 |        33 |        38 | Loss       |           0 |           1 |
|    701 |      3 |      1 |      4 |                0 |      1 |      1 |                1 |                    3 |                    1 |               0 |               2 |               3 |               1 |        39 |        34 |             0 |             0 |        39 |        34 | Win        |           0 |           2 |
|    378 |      2 |      6 |      1 |                0 |      2 |      1 |                0 |                    1 |                    3 |               0 |               0 |               1 |               3 |        36 |        38 |             0 |            15 |        36 |        23 | Win        |           0 |           1 |
|    658 |      5 |      6 |      1 |                0 |      6 |      4 |                0 |                    1 |                    3 |               0 |               0 |               1 |               3 |        33 |        51 |             0 |            30 |        33 |        21 | Win        |           2 |           2 |
|    517 |      1 |      3 |      2 |                0 |      1 |      5 |                0 |                    1 |                    1 |               0 |               0 |               1 |               1 |        31 |        36 |             0 |             0 |        31 |        36 | Loss       |           1 |           0 |
|    152 |      1 |      4 |      4 |                0 |      5 |      5 |                1 |                    1 |                    1 |               0 |              10 |               1 |               1 |        38 |        37 |             0 |             0 |        38 |        37 | Win        |           0 |           3 |
|    418 |      2 |      3 |      1 |                0 |      5 |      5 |                1 |                    3 |                    3 |               0 |              30 |               3 |               3 |        54 |        75 |            15 |            30 |        39 |        45 | Loss       |           1 |           2 |
|    994 |      2 |      2 |      4 |                1 |      4 |      6 |                0 |                    3 |                    3 |              18 |               0 |               3 |               3 |        54 |        33 |            15 |             0 |        39 |        33 | Win        |           1 |           0 |
|    804 |      1 |      6 |      2 |                0 |      1 |      3 |                0 |                    2 |                    1 |               0 |               0 |               2 |               1 |        46 |        38 |            10 |             0 |        36 |        38 | Loss       |           1 |           0 |
|    675 |      3 |      4 |      4 |                0 |      4 |      1 |                0 |                    2 |                    2 |               0 |               0 |               2 |               2 |        38 |        30 |             0 |             0 |        38 |        30 | Win        |           0 |           0 |



label: Game
P1.Multiplier
Game
P2.Multiplier
Game
P1Score
Game
P2Score
Game
P1Deduction
Game
P2Deduction
Game
P1Total
Game
P2Total
Game
Win/Loss
Game
P1Doubles
Game
P2Doubles
output: {"Functional-Dependency": [["Game", "P1.Multiplier"], ["Game", "P2.Multiplier"], ["Game", "P1Score"], ["Game", "P2Score"], ["Game", "P1Deduction"], ["Game", "P2Deduction"], ["Game", "P1Total"], ["Game", "P2Total"], ["Game", "Win/Loss"], ["Game", "P1Doubles"], ["Game", "P2Doubles"]]}
labeled_fds: Game
P1.Multiplier
Game
P2.Multiplier
Game
P1Score
Game
P2Score
Game
P1Deduction
Game
P2Deduction
Game
P1Total
Game
P2Total
Game
Win/Loss
Game
P1Doubles
Game
P2Doubles
Game
P1Doubles_many
Game
P2.Multiplied
source: $MMTU_HOME/resources/Relationship-finding/FD/247934828