# On Primitive Recursive Characteristics of Chess

## Abstract

**:**

## 1. Introduction

`X`or

`O`) whose turn it is to play on b within M moves where M is a positive integer [6] and returns 0 if it is not winnable. As shown in this investigation, the characteristic of stalemate for a chess game is decidable by a decision procedure that determines if the end board of the game is a stalemate for the player whose turn it is to play on it. We refer to a feature of a D2PBG as a decision characteristic if there exists a decision procedure to decide it.

‘‘[w]e thus require that a computationterminate after some finite numberof steps; we do not insist on an apriori ability to estimate this number.’’

## 2. Preliminaries

`min`is similar to Kleene’s operator $\mu $ [3]. If $P({x}_{1},\dots ,{x}_{n})$ is a n-ary predicate, we occasionally write $P({x}_{1},\dots ,{x}_{n})$ to abbreviate $P({x}_{1},\dots ,{x}_{n})=1$ and write $\neg P({x}_{1},\dots ,{x}_{n})$ to abbreviate $P({x}_{1},\dots ,{x}_{n})=0$.

## 3. G-Number Operators

#### 3.1. Assignment

#### 3.2. Count

#### 3.3. Append

#### 3.4. Concatenation

_{3}(x, t + 1) = (x)

_{s(t)}⊗

_{l}F3(x, t).

#### 3.5. G-Number Generator

#### 3.6. Subsequence

#### 3.7. Removal

#### 3.8. Predicate Mapping

#### 3.9. Position

#### 3.10. Association

## 4. Chess

#### 4.1. Boards

#### 4.2. Pawn Metamorphosis

#### 4.3. Valid Pieces and Boards

#### 4.4. Diagonals, Rows, Columns

#### 4.5. Potentially Reachable Positions

#### 4.6. Actual Reachability for Bishop 15

**Lemma**

**1.**

#### 4.7. Actual Reachability

**Lemma**

**2.**

**Lemma**

**3.**

#### 4.8. Checks, Mates, and Stalemates

#### 4.9. Dead Position

**1**) b contains only 14 (white king) and 30 (black king); (

**2**) b contains only 14, 30 and a bishop; (

**3**) b contains only 14, 30 and a knight; (

**4**) b contains only 14, 30, one white bishop, and one black bishop such that one bishop is light-colored and the other one-dark-colored.

**2a**) b contains only 14, 30, 12; (

**2b**) b contains only 14, 30, 15; (

**2c**) b contains only 14, 30, 28; (

**2d**) b contains only 14, 30, 31; (

**2e**) b contains only 14, 30, and exactly one metamorphic bishop.

**3**) b contains only 14, 30, 11; (

**3b**) b contains only 14, 30, 16; (

**3c**) b contains only 14, 30, 27; (

**3d**) b contains only 14, 30, 32. (

**3e**) b contains only 14, 30, and exactly one metamorphic knight.

**4a**) b contains 14, 30, 15, 31; (

**4b**) b contains 14, 30, 12, 28; (

**4c**) b contains 14, 30, and two metamorphic bishops one of which is light-colored and the other dark-colored. Let the primitive recursive predicate

#### 4.10. Moves

#### 4.11. History

#### 4.12. Games

**Lemma**

**4.**

**Lemma**

**5.**

#### 4.13. Repetition and 50-Move Rule

#### 4.14. Classification of Games

**Lemma**

**6.**

**Lemma**

**7.**

**Lemma**

**8.**

## 5. Procedures

**Theorem**

**1.**

**Corollary**

**1.**

## 6. Discussion

‘‘[a]t each turn, the rules define both what moves are legaland the effect that each possible move will have; there isno element of chance. In contrast to card games in which theplayers’ hands are hidden, each player has complete informationabout his opponent’s position, including the choices open tohim and the moves he has made. The game begins from a specifiedstate, often a configuration of men on a board. It ends in awin for one player and a loss for the other, or possibly ina draw.’’

- Is chess a D2PBG with a finite history? In other words, is there $t>0$ such that $(\forall {t}^{\prime})\{{t}^{\prime}\le t\vee {E}_{{t}^{\prime}}=0\}$? (See Equation (79)).
- Are there computable decision procedures for player x that are not primitive recursive and that guarantee for x to win or draw an unfinished game if $\overline{x}$ uses only primitive recursive decision procedures?
- Are there games that x cannot lose when x uses only primitive recursive decision procedures while $\overline{x}$ uses computable decision procedures that are not primitive recursive?

## 7. Summary

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bolon, T. How to Never Lose at Tic Tac Toe; Book Country: New York, NY, USA, 2013. [Google Scholar]
- Daly, W., Jr. Computer Strategies for the Game of Qubic. Master’s Thesis, Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA, 1961. [Google Scholar]
- Kleene, S.C. Introduction to Metamathematics; D. Van Nostrand: New York, NY, USA, 1952. [Google Scholar]
- Rogers, H., Jr. Theory of Recursive Functions and Effective Computability; The MIT Press: Cambridge, MA, USA, 1988. [Google Scholar]
- Davis, M.; Sigal, R.; Weyuker, E. Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science, 2nd ed.; Harcourt, Brace & Company: Boston, MA, USA, 1994. [Google Scholar]
- Kulyukin, V. On Primitive Recursiveness of Tic Tac Toe. In Proceedings of the International Conference on Foundations of Computer Science (FCS’19), Las Vegas, NV, USA, 29 July–1 August 2019; pp. 9–15. [Google Scholar]
- Meyer, M.; Ritchie, D. The Complexity of Loop Programs. In Proceedings of the ACM National Meeting, Washington, DC, USA; 1967; p. 465. Available online: https://people.csail.mit.edu/meyer/meyer-ritchie.pdf (accessed on 11 February 2022).
- The Longest Tournament Chess Game. Available online: https://www.chess.com/blog/ThummimS/the-longest-tournament-chess-game (accessed on 21 January 2022).
- Barr, A.; Feigenbaum, E. The Handbook of Artificial Intelligence; Addison-Wesley: Reading, MA, USA, 1982; Volume 1. [Google Scholar]
- Russell, S.; Norvig, P. Artificial Intelligence: A Modern Approach; Pearson: Hoboken, NJ, USA, 1995. [Google Scholar]
- Krizhevsky, A.; Sutskever, I.; Hinton, G.E. ImageNet classification with deep convolutional neural networks. In Proceedings of the 25th International Conference on Neural Information Processing Systems, Lake Tahoe, NV, USA, 3–6 December 2012; pp. 1097–1105. [Google Scholar]

**Figure 3.**Board ${b}^{\prime}$ after two moves (i.e., (1) e4 e5 and (2) Kf3 Kf6) by each player on b in Figure 1.

**Figure 4.**Light-colored positions (bolded) on valid board b (See Figure 2 for all board position numbers).

**Figure 5.**Dark-colored positions (bolded) on valid board b (See Figure 2 for all board position numbers).

**Table 1.**Metamorphic positions for the white and black pawns encoded as G-numbers and the symbols that denote the G-numbers.

Pawn | Metamorphic Positions |
---|---|

2 | [1, 2, 3, 4, 5, 6, 7] |

3 | [1, 2, 3, 4, 5, 6, 7, 8] |

4 | [1, 2, 3, 4, 5, 6, 7, 8] |

5 | [1, 2, 3, 4, 5, 6, 7, 8] |

6 | [1, 2, 3, 4, 5, 6, 7, 8] |

7 | [1, 2, 3, 4, 5, 6, 7, 8] |

8 | [1, 2, 3, 4, 5, 6, 7, 8] |

9 | [2, 3, 4, 5, 6, 7, 8] |

18 | [57, 58, 59, 60, 61, 62, 63] |

19 | [57, 58, 59, 60, 61, 62, 63, 64] |

20 | [57, 58, 59, 60, 61, 62, 63, 64] |

21 | [57, 58, 59, 60, 61, 62, 63, 64] |

22 | [57, 58, 59, 60, 61, 62, 63, 64] |

23 | [57, 58, 59, 60, 61, 62, 63, 64] |

24 | [57, 58, 59, 60, 61, 62, 63, 64] |

25 | [58, 59, 60, 61, 62, 63, 64] |

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**MDPI and ACS Style**

Kulyukin, V. On Primitive Recursive Characteristics of Chess. *Mathematics* **2022**, *10*, 1016.
https://doi.org/10.3390/math10071016

**AMA Style**

Kulyukin V. On Primitive Recursive Characteristics of Chess. *Mathematics*. 2022; 10(7):1016.
https://doi.org/10.3390/math10071016

**Chicago/Turabian Style**

Kulyukin, Vladimir. 2022. "On Primitive Recursive Characteristics of Chess" *Mathematics* 10, no. 7: 1016.
https://doi.org/10.3390/math10071016