Michael Soltys

Unshuffling a square is NP-hard

A shuffle of two strings is formed by interleaving the characters into a new string, keeping the characters of each string in order. A string is a square if it is a shuffle of two identical strings. There is a known polynomial time dynamic programming algorithm to determine if a given string z is the shuffle of two given strings x, y; however, it has been an open question whether there is a polynomial time algorithm to determine if a given string z is a square. We resolve this by proving that this problem is NP-complete via a many-one reduction from 3-Partition. The problem of recognizing a square shuffle is NP-complete.It remains complete over finite alphabets of size 7.The proof is based on a many-one reduction from 3-Partition.

On normalization of inconsistency indicators in pairwise comparisons

Abstract In this study, we provide mathematical and practice-driven justification for using [ 0 , 1 ] normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, is presented. A new type of paradox of infinity is described.

The proof complexity of linear algebra

We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley-Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities.

On the Complexity of Computing Winning Strategies for Finite Poset Games

This paper is concerned with the complexity of computing winning strategies for poset games. While it is reasonably clear that such strategies can be computed in PSPACE, we give a simple proof of this fact by a reduction to the game of geography. We also show how to formalize the reasoning about poset games in Skelley’s theory

An Introduction to the Analysis of Algorithms

\mathbf{W}_{1}^{1}

Feasible proofs of matrix properties with csanky's algorithm

for PSPACE reasoning. We conclude that

Matrix identities and the pigeonhole principle

\mathbf{W}_{1}^{1}

Constructing an indeterminate string from its associated graph

can use the “strategy stealing argument” to prove that in poset games with a supremum the first player always has a winning strategy.

Weak theories of linear algebra

This textbook covers the mathematical foundations of the analysis of algorithms. The gist of the book is how to argue, without the burden of excessive formalism, that a given algorithm does what it is supposed to do. The two key ideas of the proof of correctness, induction and invariance, are employed in the framework of pre/post-conditions and loop invariants. The algorithms considered are the basic and traditional algorithms of computer science, such as Greedy, Dynamic and Divide & Conquer. In addition, two classes of algorithms that rarely make it into introductory textbooks are discussed. Randomized algorithms, which are now ubiquitous because of their applications to cryptography; and Online algorithms, which are essential in fields as diverse as operating systems (caching, in particular) and stock-market predictions. This self-contained book is intended for undergraduate students in computer science and mathematics.

Complex Ranking Procedures

We show that Csankys fast parallel algorithm for computing the characteristic polynomial of a matrix can be formalized in the logical theory LAP, and can be proved correct in LAP from the principle of linear independence. LAP is a natural theory for reasoning about linear algebra introduced in [8]. Further, we show that several principles of matrix algebra, such as linear independence or the Cayley-Hamilton Theorem, can be shown equivalent in the logical theory QLA. Applying the separation between complexity classes