Relating semantic and proof-theoretic concepts for polynomial time decidability of uniform word problems

Compares three approaches to polynomial-time decidability for uniform word problems for quasi-varieties. Two of the approaches, by T. Evans (1951) and S. Burris (1995), respectively, are semantic, referring to certain embeddability and axiomatizability properties. The third approach is more proof-theoretic in nature, inspired by D. McAllester´s (1993) concept of local inference. We define two closely related notions of locality for equational Horn theories and show that both of the criteria of Evans and Burris lie in between these two concepts. In particular, the variant we call “stable locality” is shown to subsume both Evans´ and Burris´s methods