Exploring Thue's 1914 paper on the transformation of strings according to given rules

Axel Thue's paper of 1914 on string rewriting was made famous by Emil Post when, in 1947, he proved the word problem for Thue systems to be undecidable. Yet, only the first two pages of Thue's paper are directly relevant to Post's work in 1947, and the remaining 30 pages seem to have been cast into the shade. Based on a recently completed translation of this paper, I hope to shed some light on the remaining part of this work, and to advocate its relevance for the history of computing. Thue's paper has been "passed by reference" into the history of computing, based mainly on a small section of that work. A closer study of the remaining parts of that paper highlight a number of important themes in the history of computing: the transition from algebra to formal language theory, the analysis of the "computational power" (in a pre-1936 sense) of rules, and the development of algorithms to generate rule-sets

Exploring Thue's 1914 paper on the transformation of strings according to given rules

Axel Thue's paper of 1914 on string rewriting was made famous by Emil Post when, in 1947, he proved the word problem for Thue systems to be undecidable. Yet, only the first two pages of Thue's paper are directly relevant to Post's work in 1947, and the remaining 30 pages seem to have been cast into the shade. Based on a recently completed translation of this paper, I hope to shed some light on the remaining part of this work, and to advocate its relevance for the history of computing. Thue's paper has been "passed by reference" into the history of computing, based mainly on a small section of that work. A closer study of the remaining parts of that paper highlight a number of important themes in the history of computing: the transition from algebra to formal language theory, the analysis of the "computational power" (in a pre-1936 sense) of rules, and the development of algorithms to generate rule-sets

Solving Commutative Relaxations of Word Problems

We present an algebraic characterization of the standard commutative relaxation of the word problem in terms of a polynomial equality. We then consider a variant of the commutative word problem, referred to as the “Zero-to-All reachability” problem. We show that this problem is equivalent to a finite number of commutative word problems, and we use this insight to derive necessary conditions for Zero-to-All reachability. We conclude with a set of illustrative examples

P is not equal to NP

SAT is not in P, is true and provable in a simply consistent extension B' of a first order theory B of computing, with a single finite axiom characterizing a universal Turing machine. Therefore, P is not equal to NP, is true and provable in a simply consistent extension B" of B.Comment: In the 2nd printing the proof, in the 1st printing, of theorem 1 is divided into three parts a new lemma 4, a new corollary 8, and the remaining part of the original proof. The 2nd printing contains some simplifications, more explanations, but no error has been correcte

Rigidity is undecidable

We show that the problem `whether a finite set of regular-linear axioms defines a rigid theory' is undecidable.Comment: 8 page