Associative Unification and Symbolic Reasoning Modulo Associativity in Maude

Set constructors, finite sets, and logical semantics

Unification in a combination of arbitrary disjoint equational theories

Unification of infinite sets of terms schematized by primal grammars

The optimal repair property, which says that there is a finite set of optimal (i.e., entailment-maximal) repairs that covers all repairs, has turned out to be useful both in the context of ontology engineering and in belief change. We provide abstract order-theoretic conditions that guarantee the existence of finite sets of optimal repairs covering all repairs, and illustrate their use with abstract examples as well as with more practical examples from the realm of Description Logic (DL). The order-theoretic view on optimal repairs also reveals that there is a strong similarity between the optimal repair property and the existence of a finite complete set of unifiers for unification modulo equational theories. Applying Siekmann’s proposal to divide unification problems into the unification types unitary, finitary, infinitary, and zero to repair problems, we obtain a more fine-grained classification of repair problems. For the DL examples introduced in this paper, we observe that types unitary, finitary and zero can occur, but none of these examples provides us with an infinitary repair problem. However, we also show that unification problems can actually be viewed as repair problems in the abstract framework introduced in our previous work on contractions based on optimal repairs. Thus, within this framework, known results on unification types of certain equational theories provide us with examples of repair problems of these types.

We consider a time-optimal control problem on the plane with a circular vectogram of velocities and a non-convex target set with a boundary having a finite number of points of discontinuity of curvature. We study the problem of identifying and constructing scattering curves that form a singular set of the optimal result function in the case when the points of discontinuity of curvature have one-sided curvatures of different signs. It is shown that these points belong to pseudo-vertices that are characteristic points of the boundary of the target set, which are responsible for the generation of branches of a singular set. The structure of scattering curves and the optimal trajectories starting from them, which fall in the neighborhood of the pseudo-vertex, is investigated. A characteristic feature of the case under study is revealed, consisting in the fact that one pseudo-vertex can generate two different branches of a singular set. The equation of the tangent to the smoothness points of the scattering curve is derived. A scheme is proposed for constructing a singular set, based on the construction of integral curves for first-order differential equations in normal form, the right-hand sides of which are determined by the geometry of the boundary of the target in neighborhoods of the pseudo-vertices. The results obtained are illustrated by the example of solving the control problem when the target set is a one-dimensional manifold.

Lipschitz-continuous local isometric immersions: rigid maps and origami

A positive set constraint is of the form exp1\(\subseteq\)exp2, a negative set constraint is of the form exp1\(\subseteq\)exp2 where exp1 and exp2 are set expressions constructed using set variables, function symbols, and the set union, intersection and complement symbols. Decision algorithms for satisfiability of systems of positive and negative set constraints were given by Gilleron et al. [GTT93b], Aiken et al. [AKW93], and Charatonik and Pacholski [CP94]. In this paper, we study properties of the set of solutions of such systems and properties of solutions of interest for applications. The main decidability results are: for positive and negative set constraints, equivalence of systems is decidable, it is decidable whether or not a system has a unique solution; for positive set constraints, it is decidable whether or not a system has a least solution, it is decidable whether or not a system has a finite solution (i.e. the interpretation maps each set variable on a finite set).

J.-P. Doignon and J.-C. Falmagne have introduced a formal notion of knowledge space to develop a scientific approach to the assessment of knowledge. They define a knowledge space as a (finite) set Q of questions together with a collection K, of subsets of Q representing different knowledge states; in addition, they assume that K is closed under set unions. The theory of knowledge spaces can be effectively connected with formal concept analysis. The connection is established by the definition of a knowledge context as a triple (P, Q, I) where P is a finite set of persons, Q is a finite set of questions, and I is a binary relation between P and Q such that plq means: the person p cannot solve the question q. For each knowledge context there is a corresponding knowledge space which consists of the complements of all intents of this context; conversely, each knowledge space can be derived in this way. Via the described connection, methods of formal concept analysis can be successfully applied to the theory of knowledge spaces. In particular, the attribute exploration as a method of conceptual knowledge acquisition yields a new approach for building knowledge spaces.

Symbolic techniques usually use characteristic functions for representing sets of states. Boolean functional vectors provide an alternate set representation which is suitable for symbolic simulation. Their use in symbolic reachability analysis and model checking is limited, however, by the lack of algorithms for performing set operations. We present algorithms for set union, intersection and quantification that work with a canonical Boolean functional vector representation and show how this enables efficient symbolic simulation based reachability analysis. Our experimental results for reachability analysis indicate that the Boolean functional vector representation is often more compact than the corresponding characteristic function, thus giving significant performance improvements on some benchmarks.

In the paper [S1] I introduced a game, denoted by MG(J) (where J is a free ideal on some infinite set S) and called “the meager nowhere dense game for J”. The special case when J is the collection of finite subsets of the set S is called the countable-finite game on S. It proceeds as follows.First player ONE picks a countable set C1, then player TWO picks a finite set F1. Then in the second inning ONE picks a countable set C2 with C1 ⊂ C2 (unless explicitly indicated otherwise, “⊂” means “is a proper subset of”) and TWO responds with a finite set F2, and so on. The players construct a sequence (C1,F1,C2,F2,…,Ck,Fk,…) where for each positive integer k(i) Ck denotes ONE's countable set picked during the kth inning,(ii) Fk denotes TWO's finite set picked during the kth inning, and(iii) Ck ⊂ Ck + 1.Such a sequence is a play of the countable-finite game on S, and TWO wins this play if is contained in . The notion of a winning perfect information strategy is defined as usual (see, for example, [S1]). Zermelo-Fraenkel set theory together with the axiom of choice (denoted by ZFC; for a statement of the axioms see pp. xv–xvi of [K]) is a strong enough theory to build a winning perfect information strategy for player TWO in this game.Does TWO have a winning strategy requiring less than perfect information? Fix a positive integer k. A strategy of TWO which requires knowledge of only at the most the k most recent moves of ONE is said to be a k-tactic. For the countable-finite game on an infinite set S the following facts about the existence of winning k-tactics for TWO are proved in [S1]:1) TWO does not have a winning 1-tactic (Theorem 1 of [S1]).2) If the cardinality of S is less than ℵ2 then TWO has a winning 2-tactic (Corollary 4 of [S1]).3) If TWO has a winning k-tactic in the countable-finite game on an infinite set S, then TWO has a winning 3-tactic (Proposition 15 of [SI]).

In this paper we study notions of a “meager subset” of a hyper-finite set. We work within an ω-saturated nonstandard universe and fix a hyperfinite natural number Є *N∖N. We shall consider subsets of the set = {1, 2, …,H}.By analogy with the meager subsets of the real interval [0, 1], a notion of meager subset of should have the following properties.1. Finite sets, countable unions of meager sets, subsets of meager sets, and translates of meager sets should be meager.2. The Baire Category Theorem should hold; that is, should not be meager.3. The internal analogue of the Cantor set should be meager.4. The notion of a meager set should be with respect to a natural topology on .5. There should exist meager subsets of of Loeb measure one.6. Sierpiński and Lusin sets should have hyperfinite counterparts with properties similar to the classical case.Property 5 is desirable so that, as with Lebesgue measure and Baire category on [0, 1], the topological and measure-theoretic notions of “large” and “small” sets are incomparable.Property 6, while not as necessary as the other five, is desirable because of the strong interplay between measure and category in the classical results about Lusin and Sierpinski sets. Our objective is to find notions of meager set which have a relationship to Loeb measure similar to the classical relationship between meager sets and Lebesgue measure.

The authors describe an efficient Boolean algebraic algorithm to compute the probability of a union of nondisjoint sets as applied to symbolic reliability analysis. Coherent networks and fault-trees with statistically-independent components characterized by their minimal pathsets or cutsets are used as examples for generating the nondisjoint sets. The algorithm uses the concept of multiple variable inversion originally proposed by A. Grnarov et al. (1979). The authors illustrate improvements in the use of the multiple variable inversion technique for this problem using two examples. The algorithm is extended to compute the reliability importance of a given component (sensitivity of system reliability to the component reliability). A computer program implementing the modified algorithm is used to solve and obtain measured time complexities for a large set of network and fault tree models. >

Ramsey's theorem [5] asserts that every infinite set X has the following partition property (RP): For every partition of the set [X]2 of two-element subsets of X into two pieces, there is an infinite subset Y of X such that [Y]2 is included in one of the pieces. Ramsey explicitly indicated that his proof of this theorem used the axiom of choice. Kleinberg [3] showed that every proof of Ramsey's theorem must use the axiom of choice, although rather weak forms of this axiom suffice. J. Dawson has raised the question of the position of Ramsey's theorem in the hierarchy of weak axioms of choice.In this paper, we prove or refute the provability of each of the possible implications between Ramsey's theorem and the weak axioms of choice mentioned in Appendix A.3 of Jech's book [2]. Our results, along with some known facts which we include for completeness, may be summarized as follows (the notation being as in [2]):A. The following principles do not (even jointly) imply Ramsey's theorem, nor does Ramsey's theorem imply any of them:the Boolean prime ideal theorem,the selection principle,the order extension principle,the ordering principle,choice from wellordered sets (ACW),choice from finite sets,choice from pairs (C2).B. Each of the following principles implies Ramsey's theorem, but none of them follows from Ramsey's theorem:the axiom of choice,wellordered choice (∀kACk),dependent choice of any infinite length k (DCk),countable choice (ACN0),nonexistence of infinite Dedekind-finite sets (WN0).

The purpose of this chapter is to explore the possibilities of the maximum principle (MP) approach for the class of min–max control problems dealing with construction of the optimal control strategies for a class of uncertain models given by a system of ordinary differential equations with unknown parameters from a given finite set. The problem under consideration belongs to the class of optimization problems of the min–max type and consists in the design of a control providing a “good” behavior if applied to all models from a given class. Here a version of the robust maximum principle applied to the min–max Bolza problem with a terminal set is presented. The cost function contains a terminal term as well as an integral one. A fixed horizon is considered. The main result deals with finite parametric uncertain sets involved in a model description. The min–max LQ control problem is considered in detail.

Intérêt d’une indication topographique élective de l’exérèse-suture avec greffe de peau totale du triangle excédentaire adjacent : la région temporofrontale