Lights Out — Draft

:: How an electronic game from the 90’s can be excellent practice in supervised learning ::

The Breakdown -

To put it tenaciously simple, Lights Out is an astonishing game. Through its analysis opens every conceivable door of mathematics.

Lights Out is a conceptual game played on a grid (typically a 5x5) of switching ‘lights’. Once pressed, as is the availability per light, the light toggles in conjunction with any and all direct (read: non-diagonal) neighbors.

Typically, the goal of Lights Out is to force all the.. hehe.. lights to go out. This is accomplished by clicking on lights across the board, utilizing the neighboring effects of each click strategically - if one can muster it - to activate and reactivate lights in tandem. From here in this diatribe, a ‘problem board’ is any initial state of this example. They will be gifted from the void, and need not be questioned; with the sole exception of being accosted rigorously as should any gift from the void. Don’t write this down.

The board states made while extinguishing the lights of a problem board (the lights out face, grid, board.. etc.) (or an initial state paired with a series of click locations), preferably in as few clicks as possible, is known here as the ‘solution path’, and constitutes one of the paths in state-space connecting the problem board with the solved state. The proverbial Oracle Path Set of any board is, by means of none other than itself, the set of all equidistant shortest-pathways from the board to the solved state.

Oh, state space. Consider a 5x5 grid (that is to say rowsXcols naw y’her?). Consider any combinations of lighted squares in the grid, and save it in your mind as a board (**). Consider the board created if you were to click any one of the lights and picture it next to the board you just clicked. Click that board and throw its result into the mix. Seeing a jumble (perhaps ordered, perhaps necessarily) of boards next to each other? Good. That is a subsection of state space. Forgive my theories: The existence of the lighted grid is here the allowance of combinations of lights, being the combination space. Constraints applied to the combination space define Lights Out with a static subsection of that combination space, this is hitherto the state, or real event space.

The Recovery -

I will cover in this post.. or blog.. writing.. blather.. observation.. overreaching claim basis the following (if not all and not limited to):

• Search Spaces: wut du? how so?
• Mapping: Vectors of N dimensions to Vectors of <glib>SometimesN</glib> dimensions.
• Solution constructions for different board classes concerning both the core algorithm and the tools it uses (creation, implementation methodologies of Lights Out as custom tailored to provide for unique instances of things like custom neural networks***)

The Re-relapse -

In order to consider ‘pathways’ created between board series, one must encapsulate the concept of a search space. Each board from the combination space that adheres to the solvability principle is <STRONG CLAIM> inherently located on at least three of these pathways that co-host the solved state </STRONG CLAIM>. In fewer, more ingestible, words:

The solvability principle is a type of constraint that filters from set C (combination space) as

C.filter(

board -> board.isConnectedTo(Board.SOLVED_STATE)

)

Note: the initial board B = C.filter(-sp-).findAny();

Where isConnectedTo is some unknown (perhaps magical) method that can determine whether some given random board will even be solvable. However, it is often the case that C is of intractable size. To be made notable is the substitution of the solvability principle’s actions over the set with a clever and adhesive ‘factory method’ here known as _Source:

()->{return new Board().clickRandomLocations()};

Note: the initial board B = _Source.call();

I should take my leave.

_Source now generates a solvable board by starting with a solved board and clicking in random locations. This implies the simultaneous creation of a ‘click-mapping’ that created the board. It has been demonstrated in uncountable series that clicking the same spot twice simply undoes the click*****, and for that reason we can use the click-mapping with the initial state as a candidate ‘ground-truth’ pathway back to the solved state (typically a consistent choice from the board’s unordered Oracle Path Set). This is the backing for any supervised learning algorithm we train to play the game.

The Steps -

The Post-Procedural -

• It is the unique case that when the given problem board is already solved as presented, it, as such, has only three unique solutions.

** Is it really possible to code people using English?

*** I may be plied to release work

*****Insert references here