A Simple Formalization and Proof for the Mutilated Chess Board

Journal of The ACM, 1988

This paper presents the highlights of a formalization and proof of the Church-Rosser theorem that was carried out with the Boyer-Moore theorem prover.

Proceedings of the International Conference on Agents and Artificial Intelligence, 2015

We formalize and verify the superposition of rectangles in Isabelle/HOL. The superposition is associated with the arrangement of rectangular software windows while keeping some regions visible and other hidden. We adopt a qualitative spatial reasoning approach to represent these rectangles and the relations between their regions. The properties of the model are formally proved and show some characteristics of superposition operation. Although, this work is limited to 29 structures of rectangles, the superpositions produce hundreds of cases that are tedious to tackle in Isabelle/HOL. We also explain our strategy to optimize the proofs.

arXiv (Cornell University), 2024

Chess graphs encode the moves that a particular chess piece can make on an m × n chessboard. We study through these graphs through the lens of chip-firing games and graph gonality. We provide upper and lower bounds for the gonality of king's, bishop's, and knight's graphs, as well as for the toroidal versions of these graphs. We also prove that among all chess graphs, there exists an upper bound on gonality solely in terms of min{m, n}, except for queen's, toroidal queen's, rook's, and toroidal bishop's graphs.

The following statement is due to H. Steinhaus [Kalejdoskop matematyczny, PWN Warszawa 1954]. Consider a chessboard with some “mined” squares on. Assume that the king cannot go across the chessboard from the left edge to the right one without meeting a mined square. Then the rook can go across the chessboard from upper edge to the lower one moving exclusively on mined squares. According to W. Surówka several proofs of the Steinhaus Theorem seem to be incomplete or use induction on the size of the chessboard [W. Surówka, Pr. Nauk. Uniw. Slask. Katowicach Ann. Math. Silesianae 1399(7), 57-61 (1993; Zbl 0807.52018)]. In this note we generalize the Steinhaus Theorem assuming that the chessboard (= square) is divided into arbitrary polygons (not necessarily squares) and we show an algorithm allowing to find rook’s or king’s route between chosen opposite edges of the chessboard.

Malaya Journal of Matematik, 2020

A (p, q, r)-board that has pq + pr + qr squares consists of a (p, q)-, a (p, r)-, and a (q, r)-rectangle. Let S be the set of the squares. Consider a bijection f : S → [1, pq + pr + qr]. Firstly, for 1 ≤ i ≤ p, let x i be the sum of all the q + r integers in the i-th row of the (p, q + r)-rectangle. Secondly, for 1 ≤ j ≤ q, let y j be the sum of all the p + r integers in the j-th row of the (q, p + r)-rectangle. Finally, for 1 ≤ k ≤ r, let z k be the the sum of all the p + q integers in the k-th row of the (r, p + q)-rectangle. Such an assignment is called a (p, q, r)-design if {x i : 1 ≤ i ≤ p} = {c 1 } for some constant c 1 , {y j : 1 ≤ j ≤ q} = {c 2 } for some constant c 2 , and {z k : 1 ≤ k ≤ r} = {c 3 } for some constant c 3. A (p, q, r)-board that admits a (p, q, r)-design is called (1) Cartesian tri-magic if c 1 , c 2 and c 3 are all distinct; (2) Cartesian bi-magic if c 1 , c 2 and c 3 assume exactly 2 distinct values; (3) Cartesian magic if c 1 = c 2 = c 3 (which is equivalent to supermagic labeling of K(p, q, r)). Thus, Cartesian magicness is a generalization of magic rectangles into 3-dimensional space. In this paper, we study the Cartesian magicness of various (p, q, r)-board by matrix approach involving magic squares or rectangles. In Section 2, we obtained various sufficient conditions for (p, q, r)-boards to admit a Cartesian tri-magic design. In Sections 3 and 4, we obtained many necessary and (or) sufficient conditions for various (p, q, r)-boards to admit (or not admit) a Cartesian bi-magic and magic design. In particular, it is known that K(p, p, p) is supermagic and thus every (p, p, p)-board is Cartesian magic. We gave a short and simpler proof that every (p, p, p)-board is Cartesian magic. Keywords 3-dimensional boards, Cartesian tri-magic, Cartesian bi-magic, Cartesian magic.

A Isabelle/HOL formalisation of Gödel's two incompleteness theorems is presented. Aspects of the development are described in detail, including two separate treatments of variable binding: the nominal package [25] and de Bruijn indices [3]. The work followsŚwierczkowski's a detailed proof, using hereditarily finite set theory [23].

Information and Computation, 2003

History preserving bisimilarity (hp-bisimilarity) and hereditary history preserving bisimilarity (hhp-bisimilarity) are behavioural equivalences taking into account causal relationships between events of concurrent systems. Their prominent feature is that they are preserved under action refinement, an operation important for the top-down design of concurrent systems. It is shown that, in contrast to hp-bisimilarity, checking hhp-bisimilarity for finite labelled asynchronous transition systems is undecidable, by a reduction from the halting problem of 2-counter machines. To make the proof more transparent a novel intermediate problem of checking domino bisimilarity for origin constrained tiling systems is introduced and shown undecidable. It is also shown that the unlabelled domino bisimilarity problem is undecidable, which implies undecidability of hhp-bisimilarity for unlabelled finite asynchronous systems. Moreover, it is argued that the undecidability of hhp-bisimilarity holds for finite elementary net systems and 1-safe Petri nets.

Information and Computation, 1989

A restricted version of interactive proofs, denoted as Arthur-Merlin games, was defined by Babai in [B]. The difference is in that the verifier is restricted to" send messages Xj such that the concatenation x lX 2 ••• x, is the "secret" random string r. The verifier and prover in such protocols were denoted as Arthur (A) and Merlin (M) respectively, and the protocol is denoted as an AM protocol. AM [g (n)] and AM are dermed similarly to IP [g (n)] andIP.

We describe various computing techniques for tackling chessboard domination problems and apply these to the determination of domination and irredundance numbers for queens' and kings' graphs. In particular we show that γ(Q 15 ) = γ(Q 16 ) = 9, confirm that γ(Q 17 ) = γ(Q 18 ) = 9, show that γ(Q 19 ) = 10, show that i(Q18) = 10, improve the bound for i(Q 19 ) to 10 ≤ i(Q 19 ) ≤ 11, show that ir(Q n ) = γ(Q n ) for 1 ≤ n ≤ 13, show that IR(Q 9 ) = Γ(Q 9 ) = 13 and that IR(Q 10 ) = Γ(Q 10 ) = 15, show that γ(Q 4k+1 ) = 2k + 1 for 16 ≤ k ≤ 21, improve the bound for i(Q 22 ) to i(Q 22 ) ≤ 12, and show that IR(K 8 ) = 17, IR(K 9 ) = 25, IR(K 10 ) = 27, and IR(K 11 ) = 36.

i Preface Isabelle [10,] is a generic theorem prover. It has been instantiated to support reasoning in several object-logics: