Tag Archives: tibit

Tibit game with two players: trickster and webster

Here is a version of the tibit game with two players, called “trickster” and “webster”.

The webster is the first player. The trickster is the second player.

The webster manipulates (i.e. modifies) a “web”, which is a

  • trivalent graph
  • with oriented arrows
  • with a cyclic orientation of arrows around any node (i.e. locally planar)
  • it may have free arrows, i.e. ones with free tail or free tip.

Tokens.   The webster  has one type of token, called a “termination token”.  The trickster has two types of tokens, one yellow, another magenta.

Moves.  When his turn comes,  any player can do one of the moves listed further, or he may choose to pass.

tibbit_5

Some of the webster moves are “reversible”, meaning that the webster may do them in both direction. Let’s say that the “+” direction is the one from left to right in the graphical moves and also means that the webster may put a termination token (but not take one). The “-” direction is from right to left in the graphical moves and also means that the webster may take a termination token (but not put one).

tibbit_6

The loop rule.  It is possible that, after a move by one of the players, the web is no longer a trivalent graph, because a loop (an arrow which closes itself, without any node or token on it) appears. In such a case the loop is erased before the next move.

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This is a collaborative game.

There are two webs, with tokens on them, the first called the “datum” and the second called the “goal”.

The players have to modify the datum into the goal, by playing collaboratively, in the following way.

The game has two parts.

Preparation.     The players start from a given web with given tokens placed on it (called the “datum”).  Further,   the webster builds a web which contains the datum and the trickster places tokens on it, but nowhere in the datum .

Eventually they obtain a larger web which contains the datum.

Alternatively, the players may  choose an initial   web, with tokens on it, which contains the datum.

Play.   Now the webster can do only the “+” moves and the trickster can’t put any token on the web.  The players try to obtain a web, with tokens on it, which contains the goal by using their other moves.

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This one looks like a bit, but wait, what’s the other one, a hobbit?

I shall use chemlambda in this post. To make it more readable for those who are used with GLC, I shall change the notations  from chemlambda, as described in the first figure.

propagator_1

The node \phi  is the fan-in node, which replaces the dilation node from GLC.

There is a particular small graph which behaves like a bit, a bit. Look:

propagator_2

The (composite) move PROP has been used before. Let me recall that PROP is needed for making the graph of the combinator W to self-replicate. There is a whole discussion about whether is reasonable to have all moves reversible. The conclusion, for chemlambda, is that if we want chemlambda to be Turing universal, then we need only the “+” moves, supplemented by the PROP+ move. Of course that chemlambda is TURING universal if we can use all moves, but using PROP+ instead of using FAN-IN-  seems like a reasonable idea.

In this second figure we see that the “bit” from the left of the first graph, when grafted to the “in” arrow of a fan-out node, comes out by the out arrows of the fan-out node. Behaves like a bit, only that there is only an appearance of a “signal” which circulates through the “wires” (i.e. arrows).

There is another reason to call it a bit, namely that the same pair fan-out node — application node appears in the Church encoding of the naturals, as seen in GLC or chemlambda. More precisely, the natural 3, for example, has 3 such pairs, the natural n (when n is not 0) has n such pairs.  (see Figure 22, p. 14, arXiv:1312.4333)

This “bit” circulates as well through an application node, as explained in the following figure:

propagator_5

A strange thing happens if we graft it to the right out arrow of a lambda abstraction node.

propagator_3

Wait! What’s that at the left out arrow … of the fan-out node? A co-bit? A hobbit?

First, let us notice that this time we used only “+” moves, that’s very good. Then, it looks like that we lost the lambda node, which transformed magically into a fan-out node, and the bit transformed into the hobbit.

Hobbit, or better co-bit? The next figure show that a hobbit and a bit annihilate and produce a loop.

propagator_4Is the hobbit a co-bit?

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