Unification in Abelian Semigroups

Unification in equational theories, i.e. solving of equations in varieties, is a basic operation in Computational Logic, in Artificial Intelligence (AI) and in many applications of Computer Science. In particular the unification of terms in the presence of an associative and commutative f unction, i.e. solving of equations in Abelian Semigroups, turned out to be of practical relevance for Term Rewriting Systems, Automated Theorem Provers and many AI-programming languages. The observation that unification under associativity and commutativity reduces to the solution of certain linear diophantine equations is the basis for a complete and minimal unification algorithm. The set of most general unifiers is closely related to the notion of a basis for the linear solution space of these equations. These results are extended to unification in free term algebras combined with Abelian Semigroups

Algebraic geometry over algebraic structures II: Foundations

In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.Comment: 55 page

Homotopy Type Theory in Lean

We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201

Opening the AC-Unification Race

This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems

Brane and string field structure of elementary particles

The two quantizations of QFT,as well as the attempt of unifying it with general relativity,lead us to consider that the internal structure of an elementary fermion must be twofold and composed of three embedded internal (bi)structures which are vacuum and mass (physical) bosonic fields decomposing into packets of pairs of strings behaving like harmonic oscillators characterized by integers mu corresponding to normal modes at mu (algebraic) quanta.Comment: 50 page

An Analytic Method for SS-Expansion involving Resonance and Reduction

In this paper we describe an analytic method able to give the multiplication table(s) of the set(s) involved in an $S$-expansion process (with either resonance or $0_S$-resonant-reduction) for reaching a target Lie (super)algebra from a starting one, after having properly chosen the partitions over subspaces of the considered (super)algebras. This analytic method gives us a simple set of expressions to find the partitions over the set(s) involved in the process. Then, we use the information coming from both the initial (super)algebra and the target one for reaching the multiplication table(s) of the mentioned set(s). Finally, we check associativity with an auxiliary computational algorithm, in order to understand whether the obtained set(s) can describe semigroup(s) or just abelian set(s) connecting two (super)algebras. We also give some interesting examples of application, which check and corroborate our analytic procedure and also generalize some result already presented in the literature.Comment: v3, 47 pages, misprints corrected in Fortschritte der Physik, Published online 7 November 201