Unification Theory - An Introduction

Aus der Einleitung: „Equational unification is a generalization of syntactic unification in which semantic properties of function symbols are taken into account. For example, assume that the function symbol '+' is known to be commutative. Given the unication problem x + y ≐ a + b (where x and y are variables, and a and b are constants), an algorithm for syntactic unification would return the substitution {x ↦ a; y ↦ b} as the only (and most general) unifier: to make x + y and a + b syntactically equal, one must replace the variable x by a and y by b. However, commutativity of '+' implies that {x ↦ b; y ↦ b} also is a unifier in the sense that the terms obtained by its application, namely b + a and a + b, are equal modulo commutativity of '+'. More generally, equational unification is concerned with the problem of how to make terms equal modulo a given equational theory, which specifies semantic properties of the function symbols that occur in the terms to be unified.

Unification in the union of disjoint equational theories : combining decision procedures

Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms which compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictions--a slight generalization of unification problems with constants--is decidable for these theories. As a consequence of this new method, we can for example show that general A-unifiability, i.e., solvability of A-unification problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol. Our method can also be used to combine algorithms which compute finite complete sets of unifiers. Manfred Schmidt-Schauß\u27 combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also get the new result that unification in the union of disjoint equational theories is finitary, if general unification--i.e., unification of terms with additional free function symbols--is finitary in the single theories

E-unification for subsystems of S4

This paper is concerned with the unification problem in the path logics associated by the optimised functional translation method with the propositional modal logics \textit{K}, \textit{KD}, \textit{KT}, \textit{KD4}, \textit{S4} and \textit{S5}. It presents improved unification algorithms for certain forms of the right identity and associativity laws. The algorithms employ mutation rules, which have the advantage that terms are worked off from the outside inward, making paramodulating into terms superfluous

String unification is essentially infinitary

A unifier of two terms s and t is a substitution sigma such that ssigma=tsigma and for first-order terms there exists a most general unifier sigma in the sense that any other unifier delta can be composed from sigma with some substitution lambda, i.e. delta=sigmacirclambda. This notion can be generalised to E-unificationwhere E is an equational theory, =_{E} is equality under E andsigmaa is an E-unifier if ssigma =_{E}tsigma. Depending on the equational theory E, the set of most general unifiers is always a singleton (as above), or it may have more than one, 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\u27s problem, Markov\u27s problem, unification of word equations or Makanin\u27s problem as it is often called in the literature) is the E-unification problem, where E = {f(x,f(y,z))=f(f(x,y),z)}, i.e. unification under associativity or string unification once we drop the fs and the brackets. It is well known that this problem is infinitary and decidable. Essential unifiers, as introduced by Hoche and Szabo, generalise the notion of a most general unifier and have a dramatically pleasant effect on the set of most general unifiers: 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 an essentially finitary theory. The most dramatic reduction known so far is obtained for idempotent semigroups or bands as they are called in computer science: bands are of type nullary, i.e. there exist two unifiable terms s and t, but the set of most general unifiers is not enumerable. This is in stark contrast to essential unification: the set of essential unifiers for bands always exists and is finite. We show in this paper that the early hope for a similar reduction of unification under associativity is not justified: string unification is essentially infinitary. But we give an enumeration algorithm for essential unifiers. And beyond, this algorithm terminates when the considered problem is finitary

Inklusion von Patternsprachen und verwandte Probleme

A pattern is a word that consists of variables and terminal symbols. The pattern language that is generated by a pattern A is the set of all terminal words that can be obtained from A by uniform replacement of variables with terminal words. For example, the pattern A = a x y a x (where x and y are variables, and the letter a is a terminal symbol) generates the set of all words that have some word a x both as prefix and suffix (where these two occurrences of a x do not overlap). Due to their simple definition, pattern languages have various connections to a wide range of other areas in theoretical computer science and mathematics. Among these areas are combinatorics on words, logic, and the theory of free semigroups. On the other hand, many of the canonical questions in formal language theory are surprisingly difficult. The present thesis discusses various aspects of the inclusion problem of pattern languages. It can be divide in two parts. The first one examines the decidability of pattern languages with a limited number of variables and fixed terminal alphabets. In addition to this, the minimizability of regular expressions with repetition operators is studied. The second part deals with descriptive patterns, the smallest generalizations of arbitrary languages through pattern languages ("smallest" with respect to the inclusion relation). Main questions are the existence and the discoverability of descriptive patterns for arbitrary languages.Ein Pattern ist ein Wort aus Variablen und Terminalsymbolen. Die von einem Pattern A erzeugte Patternsprache ist die Menge aller Terminalwörter, die durch eine uniforme Ersetzung der Variablen in A durch Terminalwörter erzeugt werden können. So beschreibt das Pattern A = a x y a x (wobei x und y Variablen sind und a ein Terminal ist) die Menge aller Wörter, die ein Wort der Form a x sowohl als Präfix, als auch als Suffix haben (ohne dass sich diese beiden Vorkommen von a x überlappen). Wegen ihrer einfachen Definition besitzen Patternsprachen eine Vielzahl von Verbindungen zu verschiedenen anderen Gebieten der theoretischen Informatik und Mathematik, unter anderem zur Wortkombinatorik, Logik und der Theorie freier Halbgruppen. Andererseits führen viele der üblichen sprachtheoretischen Fragestellungen bei Patternsprachen zu kombinatorischen Problemen von überraschender Schwierigkeit. Die vorliegende Dissertation widmet sich verschiedenen Aspekten des Inklusionsproblems von Patternsprachen und kann in zwei Teile unterteilt werden. Der erste Teil untersucht die Entscheidbarkeit des Inklusionsproblems für Sprachen, die von Pattern mit beschränkter Variablenzahl über Terminalalphabeten von beschränkter Größe erzeugt werden. Darüber hinaus werden verschiedene Aspekte der Minimierbarkeit von regulären Ausdrücken mit Rückreferenzen betrachtet. Der zweite Teil der Dissertation handelt von deskriptiven Pattern; d.h. denjenigen Pattern, die die (hinsichtlich der Inklusion) kleinsten Verallgemeinerungen einer gegebenen Sprache erzeugen. Hauptfragen sind hierbei die Existenz und die Auffindbarkeit deskriptiver Pattern für beliebige Sprachen

An interpretation of paradigmatic morphology

The thesis has as its goal the extension of current approaches in the description of natural languages, based on logics of partial information, to the area of morphology. I review work in a number of areas which may inform the study of morphology. I define a system for the representation of lexical and morphological information similar in descriptive aims to the system of Word and Paradigm (WP) morphology developed by Matthews, although somewhat different in technical details. I show that this system has a simple mathematical structure and indicate how it is related to current proposals in the field of feature value logics for linguistic description. The descriptive use of the system is demonstrated by an analysis of verbal paradigms from Latin.The attested shortcomings of WP are reanalysed in the light of the formalization developed above, and I show that, contrary to previous claims, the structures developed for the formalization of WP may be both adequate for describing the morphology of non-inflecting languages and concise in so doing. These assertions are supported by sample analyses of the morphology of Turkish, taken as an exemplary agglutinating language, and of Semitic