Undecidability Bounds for Integer Matrices Using Claus Instances

Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More

We study the decidability of three well-known problems related to integer matrix multiplication: Mortality (M), Zero in the Left-Upper Corner (Z), and Zero in the Right-Upper Corner (R). Let d and k be positive integers. Define M(k) as the following special case of the Mortality problem: given a set X of d-by-d integer matrices such that the cardinality of X is not greater than k, decide whethe...

On Markov’s Undecidability Theorem for Integer Matrices

We study a problem considered originally by A. Markov in 1947: Given two matrix semigroups, determine whether or not they contain a common element. This problem was proved undecidable by Markov for 4×4 matrices, even in the very restrict form, and for 3× 3 matrices by Krom in 1981. Here we give a new proof in the 3× 3 case which gives undecidability in an almost as restricted form as the result...

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

Undecidability in binary tag systems and the Post correspondence problem for four pairs of words

Since Cocke and Minsky proved 2-tag systems universal, they have been extensively used to prove the universality of numerous computational models. Unfortunately, all known algorithms give universal 2-tag systems that have a large number of symbols. In this work, tag systems with only 2 symbols (the minimum possible) are proved universal via an intricate construction showing that they simulate c...

A linear formulation with O(n) variables for the quadratic assignment problem

We present an integer linear formulation that uses the so-called “distance variables” to solve the quadratic assignment problem (QAP). The model involves O(n) variables. Valid equalities and inequalities are additionally proposed. We further improved the model by using metric properties as well as an algebraic characterization of the Manhattan distance matrices that Mittelman and Peng [28] rece...