More Modal Semantics without Possible Worlds

Daniel Skurt, Hitoshi Omori
2016 IfColog journal of logics and their applications (FLAP)  
A unifier of two terms s and t is a substitution σ such that sσ " tσ and for first-order terms there exists a most general unifier σ in the sense that any other unifier δ can be composed from σ with some substitution λ, i.e. δ " σ ˝λ. For many practical applications it turned out to be useful to generalize this notion to E-unification, where E is an equational theory, " E is equality under E and σ is an E-unifier if sσ " E tσ. Depending on the equational theory E, the set of most general
more » ... s is always a singleton (as above) or it may have more than one unifier, either finitely or infinitely many unifiers and for some theories it may not even exist, in which case we call the theory of type nullary. String unification (or Löb's problem, Markov's problem, unification of word equations or Makanin's problem as it is often called in the literature) is the Eunification problem, where E " tf px, f py, zqq " f pf px, yq, zqu, i.e. unification under associativity or string unification once we drop the f s and the brackets. It is well known that this problem is infinitary and decidable. Essential unifiers, as introduced by Hoche and Szabo, generalize the notion of a most general unifier and have a dramatically pleasant effect in the sense that the set of essential unifiers is often much smaller than the set of most general unifiers. Essential unification may even reduce an infinitary theory to We would like to thank our first reviewer at the unification workshop in 2008 and the interesting discussion there and afterwards with several participants of the workshop. All of this led to a complete reformulation of our basic definitions and greatly simplified the proofs and the general presentation, finally leading to our more general framework based on the encompassment order as presented here and in [67] . We also acknowledge the very critical and competent later reviews of this paper. We are also indebted to Artur Jez' substantial contribution to paragraph 3.1, where he pointed to a serious flaw in our first version of this paper (the unitary, finitary result)
dblp:journals/flap/SkurtO16 fatcat:otzojve45vdzjderdms77skfsm