MACE 2.0 Reference Manual and Guide

MACE is a program that searches for finite models of first-order statements. The statement to be modeled is first translated to clauses, then to relational clauses; finally for the given domain size, the ground instances are constructed. A Davis-Putnam-Loveland-Logeman procedure decides the propositional problem, and any models found are translated to first-order models. MACE is a useful complement to the theorem prover Otter, with Otter searching for proofs and MACE looking for countermodels.Comment: 10 page

Algebarsko modeliranje kvantno-mehaničkih jednadžbi u konačno i beskonačno dimenzionalnim Hilbertovim prostorima

The Hilbert space of quantum mechanics has a dual representation in lattice theory, called the Hilbert lattice. In addition to offering the potential for new insights, the lattice-theoretical approach may be computationally efficient for certain kinds of quantum mechanics problems, particularly if, in the future, we are able to exploit what may be a “natural” fit with quantum computation. The equations that hold in the Hilbert space lattice representation are not completely known and are poorly understood, although much progress has been made in the last several years. This work contributes to the development of these equations, with special attention to the so-called generalized orthoarguesian equations. Many new results that do not appear in the literature are given, along with their detailed proofs. In addition, possible approaches for work towards answering some remaining open questions are discussed.Prošireni sažetak. Pozadina. Stanja u kvantnoj mehanici mogu se modelirati kao vektori u Hilbertovom prostoru. Skup zatvorenih podprostora konačno ili beskonačno dimenzionalnog Hilbertova prostora član je klase čestica koje se zovu Hilbertove rešetke (Hilbert lattice, HL). (Rešetka je djelomično uređen skup u kojemu svaka dva člana imaju najmanju gornju i najveću donju granicu. Ovaj i svi drugi ovdje korišteni termini formalno su definirani u disertaciji). Obratno, moguće je izvesti Hilbertov prostor polazeći od HL. Zbog ovog dvostrukog odnosa razumijevanje svojstava HL-a može dovesti do boljeg razumijevanja svojstava Hilbertova prostora. Osim što nudi mogućnost novih uvida, teorijski pristup rešetki može biti računski efikasan za neke vrste kvantno mehaničkih problema, naročito ako, u budućnosti, budemo mogli koristiti ono što bi mogao biti “prirodno” odgovarajući dio za kvantno računanje. Jednadžbe koje u Hilbertovu prostoru podržavaju prikaz rešetke nisu u potpunosti poznate i nedovoljno ih se razumije, premda je tijekom nekoliko posljednjih godina učinjen veliki napredak. Familija svih HL-ova definirna je (aksiomatizirana) skupomuvjeta prvoga reda koji uključuju (egzistencijalne) kvantifikatore. Za određeni broj uvjeta nultog reda odnosno jednadžbi bez kvantifikatora, može se pokazati da vrijede u svakom HL-u. Najočigledniji od njih su jednadžbe koje definiraju bilo koju rešetku (te posebno svaku orto-rešetku), koje su dio skupa aksioma. Godine 1937. Husimi je otkrio ortomodularni zakon (koji je sada takoder dio HL definicije), koji je bio intenzivno obrađen u literaturi o klasi ortomodularnih rešetki (OML), kojih je HL podklasa. Za razliku od uvjeta prvoga reda, jednadžbe nam omogućavaju da direktno baratamo objektima u podprostoru Hilbertova prostora i dobijemo vrstu računske “algebre” za rad s tim objektima. Jednadžbe su posebno prikladne za efikasne računske tehnike. Klasa rešetki definirana samo jednadžbama, kao što je OML, naziva se jednadžbenim varijetetom. Klasa HL-a sama po sebi nije jednadžbeni varijetet (za što je dokaz naveden u disertaciji). Usprkos tome, klasa rešetki koju je generirao (tj. koja zadovoljava) skup jednadžbi koje vrijede u HL-u može se proučavati odvojeno kao superklasa od HL-a i svi rezultati su automatski primjenjivi na HL kao poseban slučaj. Jedan važan neriješen problem je pronaći sve moguće jednadžbene zakone koji vrijede u HL-u. S jačim jednadžbama moguće je proučiti više karakteristika HL-a korištenjem samo jednadžbenih varijeteta. Ovdje je kratki pregled napretka postignutog do sada. Trebalo je nekoliko desetljeća nakon Husimieva OML zakona da se pronade drugi, a to je bio ortoarguesiev zakon kojega je otkrio Alan Day 1975. 1981. g. Godowski je otkrio nezavisnu beskonačnu familiju HL jednadžbi, baziranu na kvantnim probabilističkim stanjima. Te jednadžbe nazivano “Godowski-eve jednadžbe” ili n-Gos. Godine 1986. Mayet je našao algoritam za generiranje većeg skupa jednadžbi (nazvan MGEs), koji je također utemeljen na stanjima, čiji su podskup bile Godowski-eve jednadžbe), premda se na početku nije znalo da li je ikoja od njih nezavisna od n-Go jednadžbi. Od 2006. do 2009., Megill i Pavičić pronašli su nove jednadžbe utemeljene na Mayet-ovu algoritmu za koje se pokazalo da se ne daju izvesti iz Godowski-evih. U 2000. g.Megill i Pavičić otkrili su novu familiju jednadžbi koje vrijede u HL-u—generalizirane ortoarguesieve jednadžbe, nazvane nOA zakoni (n ≥ 3). OA zakon Alan-a Day-a je drugi član ove serije, zakon 4OA, a Greechie/Godowski-eve jednadžbe izvedene iz OA su ekvivalentne prvome članu, zakonu 3OA. Dok je otvoren problem da li se obitelj nOA sastoji od uzastopno jačih jednadžbi, mi smo dokazali (obimnim kompjuterskim traženjem protuprimjera) da su zakoni 3OA, 4OA, 5OA i 6OA uzastopno jači. 2011. g. uspjeli smo dokazati da je zakon 7OA jači od zakona 6OA. Godine 1995. Maria Soler je dokazala da je dodavanjem dva dodatna HL aksioma, moguće iz HL-a izvesti Hilbertov prostor čije je polje jedno od “klasičnih” polja kvantne mehanike (realno, kompleksno ili kvaternionsko). Soler-in teorem upotpunio je dugo neostvareni cilj da se Hilbertov prostor kvantne mehanike izvede iz nekog HL-a pokazujući da su oni dualni. Godine 2006. Mayet je opisao novu obitelj jednadžbi, nazvanu E jednadžbe, utemeljenu na jednom svojstvu Hilbertova prostora koje se naziva vektorski-valuiranim stanjem. Važno je reći da te jednadžbe ne vrijede za svako moguće polje koje se može dovesti u vezu s Hilbertovim prostorom već samo za ona polja s karakteristikom 0, koja uključuju klasična polja kvantne mehanike. To nam daje jednadžbeni uvjet koji je u stvari ovisan o (te ih tako djelomično i opisuje) Soler-inim dodatnim uvjetima (prvoga reda) dodanim HL-u. Ovdje ćemo ukratko sumirati ključne teme pokrivene u disertaciji koje se odnose na traženje novih HL jednadžbi. Ortomodularne rešetke. Veliki broj uvjeta koji vrijede u OML-u prikupljen je u poglavlju 3, za kasniju uporabu. Oni koji se nisu ranije pojavili u literaturi popraćeni su detaljnim dokazima. Određeni broj novih rezultata naveden je za takozvanu Sasaki hook operaciju, koja postaje koristan alat u kasnijim poglavljima. Ortoarguesieve jednadžbe. Poglavlje 4 predstavlja ekstenzivnu studiju generaliziranih ortoarguesievih rešetki (jednadžbeni varijeteti nOA). Prezentiran je revidirani dokaz tih zakona i razmotreni su poznati rezultati neovisnosti (sve do 7OA). Nekoliko sustava označavanja, korisnih u različitim situacijama, uvedeno je kako bi se kompaktno eprezentirale te jednadžbe. Mnoge jednadžbe koje su ekvivalentne i koje su posljedice zakona nOA, koje su gotovo sve nove, izvedene su uz detaljne dokaze. Važan neriješen, otvoren problem je “pretpostavka ortoarguesievog identiteta,” koja propituje da li je uvjet poznat kao zakon ortoarguesievog identiteta ekvivalentan ortoarguesianskom zakonu. Ako ova pretpostavka vrijedi, bila bi moćan alat za dokazivanje teorema. Jedna ekstenzivna studija koja je posljedica ove pretpostavke, jednako kao i drugih pretpostavki koje je impliciraju, predstavlja središnji dio posljednjeg odjeljka poglavlja 3. Ostale jednadžbe Hilbertove rešetke. Poglavlje 5 razmatra druge gore spomenute jednadžbene varijetete. Posebno je predstavljeno 16 novih Mayet-Godowski-evih jednadžbi (MGEs), otkrivenih kao dio ove disertacije. Poglavlje 6 istražuje svojstva superpozicije prvoga reda i modularnu simetriju, od čega niti jedno do sada nije dovelo do nove jednadžbe. Prezentirana je pretpostavljena jednadžba izvedena iz modularne simetrije, ali je otvoreni problem da li njen izvod (počevši od modularne simetrije) vrijedi u svim OML-ovima. Jednadžbe rešetke za konačno dimenzionalne Hilbertove prostore. Konačno dimenzionalni Hilbertovi prostori važni su za mnoge probleme u kvantnoj mehanici, uključujući većinu eksperimenata koji uključuju čestična stanja i većinu pristupa kvantnom računanju. Poglavlje 7 razmatra modularni zakon i Arguesiev zakon koji vrijedi u zatvorenim podprostorima konačno dimenzionalnih Hilbertovih prostora. Izvedena je nova serija Arguesievih zakona višeg reda. Prodiskutirane su moguće primjene Pappusova zakona projektivne geometrije

Applying automated deduction to natural language understanding

AbstractVery few natural language understanding applications employ methods from automated deduction. This is mainly because (i) a high level of interdisciplinary knowledge is required, (ii) there is a huge gap between formal semantic theory and practical implementation, and (iii) statistical rather than symbolic approaches dominate the current trends in natural language processing. Moreover, abduction rather than deduction is generally viewed as a promising way to apply reasoning in natural language understanding. We describe three applications where we show how first-order theorem proving and finite model construction can efficiently be employed in language understanding.The first is a text understanding system building semantic representations of texts, developed in the late 1990s. Theorem provers are here used to signal inconsistent interpretations and to check whether new contributions to the discourse are informative or not. This application shows that it is feasible to use general-purpose theorem provers for first-order logic, and that it pays off to use a battery of different inference engines as in practice they complement each other in terms of performance.The second application is a spoken-dialogue interface to a mobile robot and an automated home. We use the first-order theorem prover spass for checking inconsistencies and newness of information, but the inference tasks are complemented with the finite model builder mace used in parallel to the prover. The model builder is used to check for satisfiability of the input; in addition, the produced finite and minimal models are used to determine the actions that the robot or automated house has to execute. When the semantic representation of the dialogue as well as the number of objects in the context are kept fairly small, response times are acceptable to human users.The third demonstration of successful use of first-order inference engines comes from the task of recognising entailment between two (short) texts. We run a robust parser producing semantic representations for both texts, and use the theorem prover vampire to check whether one text entails the other. For many examples it is hard to compute the appropriate background knowledge in order to produce a proof, and the model builders mace and paradox are used to estimate the likelihood of an entailment

Relational Structure Theory: A Localisation Theory for Algebraic Structures

This thesis extends a localisation theory for finite algebras to certain classes of infinite structures. Based on ideas and constructions originally stemming from Tame Congruence Theory, algebras are studied via local restrictions of their relational counterpart (Relational Structure Theory). In this respect, first those subsets are identified that are suitable for such a localisation process, i. e. that are compatible with the relational clone structure of the counterpart of an algebra. It is then studied which properties of the global algebra can be transferred to its localisations, called neighbourhoods. Thereafter, it is discussed how this process can be reversed, leading to the concept of covers. These are collections of neighbourhoods that allow information retrieval about the global structure from knowledge about the local restrictions. Subsequently, covers are characterised in terms of a decomposition equation, and connections to categorical equivalences of algebras are explored. In the second half of the thesis, a refinement concept for covers is introduced in order to find optimal, non-refinable covers, eventually leading to practical algorithms for their determination. Finally, the text establishes further theoretical foundations, e. g. several irreducibility notions, in order to ensure existence of non-refinable covers via an intrinsic characterisation, and to prove under some conditions that they are uniquely determined in a canonical sense. At last, the applicability of the developed techniques is demonstrated using two clear expository examples.:1 Introduction 2 Preliminaries and Notation 2.1 Functions, operations and relations 2.2 Algebras and relational structures 2.3 Clones 3 Relational Structure Theory 3.1 Finding suitable subsets for localisation 3.2 Neighbourhoods 3.3 The restricted algebra A|U 3.4 Covers 3.5 Refinement 3.6 Irreducibility notions 3.7 Intrinsic description of non-refinable covers 3.8 Elaborated example 4 Problems and Prospects for Future Research Acknowledgements Index of Notation Index of Terms BibliographyDiese Dissertation erweitert eine Lokalisierungstheorie für endliche Algebren auf gewisse Klassen unendlicher Strukturen. Basierend auf Ideen und Konstruktionen, die ursprünglich der Tame Congruence Theory entstammen, werden Algebren über lokale Einschränkungen ihres relationalen Gegenstücks untersucht (Relationale Strukturtheorie). In diesem Zusammenhang werden zunächst diejenigen Teilmengen identifiziert, welche für einen solchen Lokalisierungsprozeß geeignet sind, d. h., die mit der Relationenklonstruktur auf dem Gegenstück einer Algebra kompatibel sind. Es wird dann untersucht, welche Eigenschaften der globalen Algebra auf ihre Lokalisierungen, genannt Umgebungen, übertragen werden können. Nachfolgend wird diskutiert, wie dieser Vorgang umgekehrt werden kann, was zum Begriff der Überdeckungen führt. Dies sind Systeme von Umgebungen, welche die Rückgewinnung von Informationen über die globale Struktur aus Kenntnis ihrer lokalen Einschränkungen erlauben. Sodann werden Überdeckungen durch eine Zerlegungsgleichung charakterisiert und Bezüge zu kategoriellen Äquivalenzen von Algebren hergestellt. In der zweiten Hälfte der Arbeit wird ein Verfeinerungsbegriff für Überdeckungen eingeführt, um optimale, nichtverfeinerbare Überdeckungen zu finden, was letztlich zu praktischen Algorithmen zu ihrer Bestimmung führt. Schließlich erarbeitet der Text weitere theoretische Grundlagen, beispielsweise mehrere Irreduzibilitätsbegriffe, um die Existenz nichtverfeinerbarer Überdeckungen vermöge einer intrinsischen Charakterisierung sicherzustellen und, unter gewissen Bedingungen, zu beweisen, daß sie in kanonischer Weise eindeutig bestimmt sind. Schlußendlich wird die Anwendbarkeit der entwickelten Methoden an zwei übersichtlichen Beispielen demonstriert.:1 Introduction 2 Preliminaries and Notation 2.1 Functions, operations and relations 2.2 Algebras and relational structures 2.3 Clones 3 Relational Structure Theory 3.1 Finding suitable subsets for localisation 3.2 Neighbourhoods 3.3 The restricted algebra A|U 3.4 Covers 3.5 Refinement 3.6 Irreducibility notions 3.7 Intrinsic description of non-refinable covers 3.8 Elaborated example 4 Problems and Prospects for Future Research Acknowledgements Index of Notation Index of Terms Bibliograph

Automatic Proofs and Counterexamples for Some Ortholattice Identities

This note answers questions on whether three identities known to hold for orthomodular lattices are true also for ortholattices. One identity is shown to fail by MACE, a program that searches for counterexamples, an the other two are proved to hold by EQP, an equational theorem prover. The problems, from work in quantum logic, were given to us by Norman Megill. Keywords: Automatic theorem proving, ortholattice, quantum logic, theory of computation. 1 Introduction An ortholattice is an algebra with a binary operation (join) and a unary operation 0 (complement) satisfying the following (independent) set of identities. x y = (x 0 y 0 ) 0 (definition of meet) x y = y x (x y) z = x (y z) x (x y) = x x 00 = x x (y y 0 ) = y y 0 Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. From these identities one can..

A global workspace framework for combined reasoning

Artificial Intelligence research has produced many effective techniques for solving a wide range of problems. Practitioners tend to concentrate their efforts in one particular problem solving paradigm and, in the main, AI research describes new methods for solving particular types of problems or improvements in existing approaches. By contrast, much less research has considered how to fruitfully combine different problem solving techniques. Numerous studies have demonstrated how a combination of reasoning approaches can improve the effectiveness of one of those methods. Others have demonstrated how, by using several different reasoning techniques, a system or method can be developed to accomplish a novel task, that none of the individual techniques could perform. Combined reasoning systems, i.e., systems which apply disparate reasoning techniques in concert, can be more than the sum of their parts. In addition, they gain leverage from advances in the individual methods they encompass. However, the benefits of combined reasoning systems are not easily accessible, and systems have been hand-crafted to very specific tasks in certain domains. This approach means those systems often suffer from a lack of clarity of design and are inflexible to extension. In order for the field of combined reasoning to advance, we need to determine best practice and identify effective general approaches. By developing useful frameworks, we can empower researchers to explore the potential of combined reasoning, and AI in general. We present here a framework for developing combined reasoning systems, based upon Baars’ Global Workspace Theory. The architecture describes a collection of processes, embodying individual reasoning techniques, which communicate via a global workspace. We present, also, a software toolkit which allows users to implement systems according to the framework. We describe how, despite the restrictions of the framework, we have used it to create systems to perform a number of combined reasoning tasks. As well as being as effective as previous implementations, the simplicity of the underlying framework means they are structured in a straightforward and comprehensible manner. It also makes the systems easy to extend to new capabilities, which we demonstrate in a number of case studies. Furthermore, the framework and toolkit we describe allow developers to harness the parallel nature of the underlying theory by enabling them to readily convert their implementations into distributed systems. We have experimented with the framework in a number of application domains and, through these applications, we have contributed to constraint satisfaction problem solving and automated theory formation