Understanding the QuickXPlain Algorithm: Simple Explanation and Formal Proof

In his seminal paper of 2004, Ulrich Junker proposed the QuickXPlain algorithm, which provides a divide-and-conquer computation strategy to find within a given set an irreducible subset with a particular (monotone) property. Beside its original application in the domain of constraint satisfaction problems, the algorithm has since then found widespread adoption in areas as different as model-based diagnosis, recommender systems, verification, or the Semantic Web. This popularity is due to the frequent occurrence of the problem of finding irreducible subsets on the one hand, and to QuickXPlain's general applicability and favorable computational complexity on the other hand. However, although (we regularly experience) people are having a hard time understanding QuickXPlain and seeing why it works correctly, a proof of correctness of the algorithm has never been published. This is what we account for in this work, by explaining QuickXPlain in a novel tried and tested way and by presenting an intelligible formal proof of it. Apart from showing the correctness of the algorithm and excluding the later detection of errors (proof and trust effect), the added value of the availability of a formal proof is, e.g., (i) that the workings of the algorithm often become completely clear only after studying, verifying and comprehending the proof (didactic effect), (ii) the shown proof methodology can be used as a guidance for proving other recursive algorithms (transfer effect), and (iii) the possibility of providing "gapless" correctness proofs of systems that rely on (results computed by) QuickXPlain, such as numerous model-based debuggers (completeness effect)

Polynomial-time algorithms for generation of prime implicants

AbstractA notion of a neighborhood cube of a term of a Boolean function represented in the canonical disjunctive normal form is introduced. A relation between neighborhood cubes and prime implicants of a Boolean function is established. Various aspects of the problem of prime implicants generation are identified and neighborhood cube-based algorithms for their solution are developed. The correctness of algorithms is proven and their time complexity is analyzed. It is shown that all presented algorithms are polynomial in the number of minterms occurring in the canonical disjunctive normal form representation of a Boolean function. A summary of the known approaches to the solution of the problem of the generation of prime implicants is also included

Fast subsumption checks using anti-links

The concept of "anti-link" is defined, and useful equivalence-preserving operations based on anti-links are introduced.These operations eliminate a potentially large number of subsumed paths in a negation normal form formula.Those anti-links that directly indicate the presence of subsumed paths are characterized. The operations have linear time complexity in the size of that part of the formula containing the anti-link. The problem of removing all subsumed paths in an NNF formula is shown to be NP-hard, even though such formulas may be small relative to the size of their path sets. The general problem of determining whether there exists a pair of subsumed paths associated with an arbitrary anti-link is shown to be NP-complete. Additional techniques based on "strictly pure full blocks" are introduced and are also shown to eliminate redundant subsumption checks. The effectiveness of these techniques is examined with respect to some benchmark examples from the literature

A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises all that can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology

文字法によるブール関数の最小化について

This paper describes a method for deriving minimal sums of Boolean functions. Themethod, unlike most others, does not assign numerical values to minterms. Instead it dealswith minterms and terms directly by means of literals.The use of literals lends itself to analgebraic approach. The speed and efficiency of the calculations make the method suitablefor handling functions of a relatively high number of literals

Méthodes permettant la prédominance de connaissances subsumées

Cette thèse s inscrit dans le domaine de l Intelligence Artificielle symbolique. Elle y traite d une question fondamentale liée à la représentation des connaissances et des raisonnements à base de logique. Plus précisément, elle s intéresse au problème pouvant se produire lors de l insertion dans un ensemble de connaissances d une information qui peut déjà en être déduite. Comment faire en sorte que cette nouvelle information vienne préempter les informations qui permettent son inférence ? Supposons par exemple qu un ensemble de prémisses contienne l information Si l interrupteur est enclenché alors la pièce est éclairée . Il est naturel d espérer que l ajout d une règle additionnelle, en un sens plus précise que la première, et qui exprime que Si l interrupteur est enclenché et si l ampoule n est pas cassée alors la pièce est éclairée , puisse venir la préempter. En effet, il ne doit plus être suffisant de savoir que L interrupteur soit enclenché pour en conclure que La pièce est éclairée : il faut aussi que L ampoule ne soit pas cassée . Remarquons que la seconde règle est consistante avec la première et que les cadres de logiques non monotones et de révision de croyances ou de mise à jour ne traitent pas a priori de ce problème. Nous adressons d abord cette question dans le cadre de la logique classique et ensuite dans un cadre plus général de représentation à base de logiques non monotones, et particulièrement de celles permettant la représentation de règles avec exceptions reposant sur des tests de consistance.This thesis is in line with the symbolic Artificial Intelligence domain. It deals with a fondamental issue of the logic-based knowledge and reasoning representation. Most particulary, this thesis is interested in the issue occurring when a piece of information is added to a knowledge set which already entails it. How to make sure that this new piece of information prevails the ones that allow its inference? Suppose for instance that a premisses set contains the piece of information If the switch is on then the room is lighted . It is natural to expect that adding an additional rule, in a way more precise than the first one, and which assert that If the switch is on and if the lamp bulb is not broken then the room is lighted , could prevail it. Indeed, it will not be sufficient to know that The switch is on to conclude that The room is lighted : it is now necessary that The lamp bulb is not broken . Let us note that the second rule is consistent with the first one and that the non monotonic logics and beliefs revision or update frameworks do not handle this issue in principle. First of all, we adress this issue in the classical logic framework, and secondly in a most general framework of non monotonic logic based representation, and particulary the ones that allow the representation of rules with exceptions subject of consistency test.ARRAS-Bib.electronique (620419901) / SudocSudocFranceF