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

Using Automated Reasoning Techniques for Deductive Databasis

This report presents a proposal for a deduction component that supports the query mechanism of relational databases. The query-subquery (QSQ) paradigm is currently very popular in the database community since it focuses the deduction process on the relevant data. We show how to extend the QSQ paradigm from Horn clauses to arbitrary predicate logic formulae such that disjunctions in the consequent of an implication, negation in its logical meaning and arbitrary recursive predicates can be handled without restrictions. Various techniques to improve the search behaviour, such as lemma generation, query generalization etc. can be incorporated. Furthermore we show how to use clause graphs for compile time optimizations in the presence of recursive clauses and to support the run time processing

Holonic multi-agent systems

A holonic multi-agent paradigm is proposed, where agents give up parts of their autonomy and merge into a super-agent"(a holon), that acts - when seen from the outside - just as a single agent again. We explore the spectrum of this new paradigm, ranging from definitorial issues over classification of possible application domains, an algebraic characterization of the merge operation, to implementational aspects: We propose algorithms for holon formation and on-line re-configuration. Based on some general criteria for the distinction between holonic and non-holonic domains, we examine domains suitable for holonic agents and sketch the implementation of holonic agents in these scenarios. Finally, a case study of a holonic agent system is presented in detail: TELETRUCK system is a fleet management system in the transportation domain

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

Computer supported mathematics with Ωmega

AbstractClassical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spaces—often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited.The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive, search based as well as proof planning techniques.The Ωmega system is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for a working mathematician as well as a software engineer when employing formal methods for quality assurance. In particular, Ωmega supports proof development at a human-oriented abstract level of proof granularity. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ωmega has many characteristics in common with systems like NuPrL, CoQ, Hol, Pvs, and Isabelle. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λClam at Edinburgh

Concept logics

Concept languages (as used in BACK, KL-ONE, KRYPTON, LOOM) are employed as knowledge representation formalisms in Artificial Intelligence. Their main purpose is to represent the generic concepts and the taxonomical hierarchies of the domain to be modeled. This paper addresses the combination of the fast taxonomical reasoning algorithms (e.g. subsumption, the classifier etc.) that come with these languages and reasoning in first order predicate logic. The interface between these two different modes of reasoning is accomplished by a new rule of inference, called constrained resolution. Correctness, completeness as well as the decidability of the constraints (in a restricted constraint language) are shown

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

Isolation, Cloning and Structural Characterisation of Boophilin, a Multifunctional Kunitz-Type Proteinase Inhibitor from the Cattle Tick

Inhibitors of coagulation factors from blood-feeding animals display a wide variety of structural motifs and inhibition mechanisms. We have isolated a novel inhibitor from the cattle tick Boophilus microplus, one of the most widespread parasites of farm animals. The inhibitor, which we have termed boophilin, has been cloned and overexpressed in Escherichia coli. Mature boophilin is composed of two canonical Kunitz-type domains, and inhibits not only the major procoagulant enzyme, thrombin, but in addition, and by contrast to all other previously characterised natural thrombin inhibitors, significantly interferes with the proteolytic activity of other serine proteinases such as trypsin and plasmin. The crystal structure of the bovine α-thrombin·boophilin complex, refined at 2.35 Å resolution reveals a non-canonical binding mode to the proteinase. The N-terminal region of the mature inhibitor, Q16-R17-N18, binds in a parallel manner across the active site of the proteinase, with the guanidinium group of R17 anchored in the S1 pocket, while the C-terminal Kunitz domain is negatively charged and docks into the basic exosite I of thrombin. This binding mode resembles the previously characterised thrombin inhibitor, ornithodorin which, unlike boophilin, is composed of two distorted Kunitz modules. Unexpectedly, both boophilin domains adopt markedly different orientations when compared to those of ornithodorin, in its complex with thrombin. The N-terminal boophilin domain rotates 9° and is displaced by 6 Å, while the C-terminal domain rotates almost 6° accompanied by a 3 Å displacement. The reactive-site loop of the N-terminal Kunitz domain of boophilin with its P1 residue, K31, is fully solvent exposed and could thus bind a second trypsin-like proteinase without sterical restraints. This finding explains the formation of a ternary thrombin·boophilin·trypsin complex, and suggests a mechanism for prothrombinase inhibition in vivo

Unification of sets and multisets

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. It is also shown that completeness and minimality is closely related to the notion of a basis for the linear solution space of these equations . Terms under associativity (A) and commutativity (C) closely resemble the datastructure multi sets (sets which may contain multiple occurrences of the same element) , which is used in the matching of patterns (pattern directed invocation) in many Al-languages. The problem was first investigated in [ 40], this paper presents an alternative solution, which is an improvement over [ 40]. Unification under associativity, commutativity and idempotence is shown to behave in an exactly parallel manner to unification under {A, C} with the proviso that the arithmetic equations are replaced by Boolean equations. Terms under associativity, commutativity and idempotence closely resemble the datastructure set, which is also used in the matching of patterns in many Al-languages. The combination of these axioms occur in automatic theorem proving, e.g. set theoretic Intersection and union have these properties