A knowledge system S describing a part of real-world does, in general, not contain complete information. Reasoning with incomplete information is prone to errors since any belief derived from S may be false in the present state of the world. A false belief may suggest wrong decisions and lead to harmful actions. So, an important goal is to make false beliefs as unlikely as possible. This work introduces the notions of typical atoms and typical models, and shows that reasoning with typical models minimises the expected number of false beliefs over all ways of using incomplete information. Various properties of typical models are studied, in particular, correctness and stability of beliefs suggested by typical models, and their connection to oblivious reasoning.

This study is motivated by the question whether there exists a characteristic feature common to the phenomenon of phase transition of properties taking place in the course of evolution of various sets of objects of different nature. A novel measure model of phase transition is presented, providing an insight into the process underlying different phase transitions. First, it is shown to conform to several famous and extensively investigated cases of phase transition. Then a few close consequences of applying the measure model are presented. It has been proved that monotone properties undergo a sharp phase transition. The measure model provides a sufficient condition of a sharp phase transition common to monotone and non-monotone properties as well. The measure model reveals that if a property Q undergoes a phase transition in the course of evolution of objects of a certain type, then there exist other secondary properties related to the measure of Q that also undergo a phase transition during the evolution.

A large Knowledge System operating for a long time almost inevitably becomes ‘polluted’ by wrong data that make the system inconsistent. Despite this fact, a sizeable part of the system remains unpolluted, and retains useful information. It is widely adopted that a maximally consistent subset of a system (mc-subset) contains a significant portion of unpolluted data. So, determining mc-subsets is a necessary step towards reasoning with inconsistent knowledge. We consider extensions of the MAX-SAT problem, investigate characteristic features of mc-subsets, present algorithms for computing all or major mc-subsets of inconsistent sets of clauses, and, report results of experiments evaluating parameters of mc-subsets.

Given a system description and its observed behaviour, there usually exist several possible explanations of the situation. Different explanations may contradict one another and suggest conflicting actions. So, in order to take proper actions sanctioned by the real situation, we need to single out an explanation that is most adequate to the reality. Existing methods for determining of such a best explanation require an extensive statistical knowledge about the system behaviour, which often is either only partially available or not satisfactorily reliable. Trying to overcome this lack of data, we introduce the notion of evidence as a measure of credibility of beliefs based on the semantic information contained in the system and supplied by the observations. Reasoning by evidence is applied to the process of determining a best abductive explanation. The notion of conjecture is presented as a generalization of explanation.Then reasoning by evidence is applied to testing and refining of diagnoses.

As was shown recently, many important AI problems require counting the number of models of propositional formulas. The problem of counting models of such formulas is, according to present knowledge, computationally intractable in a worst case. Based on the Davis-Putnam procedure, we present an algorithm, CDP, that computes the exact number of models of a propositional CNF or DNF formula F. Let m and n be the number of clauses and variables of F, respectively, and let p denote the probability that a literal l of F occurs in a clause C of F, then the average running time of CDP is shown to be O(nm^d), where d=-1/log(1-p). The practical performance of CDP has been estimated in a series of experiments on a wide variety of CNF formulas.

The complexity of logical inference in knowledge systems grows with the system size, determined, in particular, by the number of its clauses. This growth is so rapid, that reasoning in sizeable systems becomes intractable.However, given a system S and a query Q, should we be able to detect a subset of S informative enough to provide a correct answer to Q, and small enough to fit into the range of tractable computation,the answer to Q could be produced efficiently. This paper presents a way of implementing this idea.

It is well known that a parsimonious semantics (along with its elegance) may lead to counterintuitive beliefs. This paper presents a semantics of majority of models, SMJ, that suggests an affirmative answer to the question in the title. Numerous observations show a common-sense plausibility of SMJ and provide it with a probabilistic justification. Computational aspects of SMJ are considered.

A logic system S describing a world W contains certain semantic information about W. We define the quantity of this information, and the information value of a formula F relative to S. A close relationship is revealed between the information of S and the cardinality of the set of its models. Then the evidence of a formula F given by S is introduced providing basis for reasoning by evidence as a way of non-monotonic reasoning. The latter is presented in the form of plausible world assumption, PWA possessing certain advantages over the known approaches to non-monotonic reasoning: PWA computes a plausible model for any first-order logic system, and provides a high degree of relative monotonicity to its beliefs. The most important feature of PWA is its semantic plausibility demonstrated by a number of examples.

We extend the query optimization techniques known as algebric manipulations with relational expressions [48] to work with deductive databases. In particular, we propose a method for moving data-independent selections and projections into recursive axioms, which extends all other known techniques for performing that task [2, 3, 9, 18, 20]. We also show that, in a well-defined sense, our algorithm is optimal among the algorithms that propagate data-independent selections through recursion.

The authors present a logic, called RI (reasoning with inconsistency), that treats any set of clauses, either consistent or not, in a uniform way. In this logic, consequences of a contradiction are not nearly as damaging as in the standard predicate calculus, and meaningful information can still be extracted from an inconsistent set of formulas. RI has a resolution-based sound and complete proof procedure. It is a much richer logic than the predicate calculus, and the latter can be imitated within RI in several different ways (depending on the intended meaning of the predicate calculus formulas). The authors also introduce a novel notion of epistemic entailment and show its importance for investigating inconsistency in the predicate calculus.>

Answering queries in indefinite systems is a difficult problem both computationally, since it involves non-Horn clauses and factoring, and conceptually, concerning producing beliefs for formulas not derivable from the system. to provide a basis for reasonable beliefs, we propose new criteria as an alternative to the Full Information Principle. Then an approach to producing stable beliefs, called Plausible World Assumption (PWA), is introduced. It is shown how a set of non-Horn clauses can be transformed into a set of so called singleton-head-rules such that evaluation of a given query is reduced to processing of a set of Horn clauses relevant to the query. Finally, algorithms are presented for computing facts and beliefs for atomic queries in accord with the PWA. This method is shown to be more efficient than the known techniques for query evaluation in indefinite systems.

It is shown how Horn logic programs can be implemented using database techniques, namely, mostly bottom-up in combination with certain top-down elements (as opposed to the top-down implementations of logic programs prevailing so far). The proposed method is sound and complete. It easily lends itself to a parallel implementation and is free of nonlogical features like backtracking. As an extension to the common approach to deductive databases, function symbols are allowed to appear in programs, and it is shown that much of database query optimization can be applied to optimize logic programs. An important advantage of present approach is its ability to evaluate successfully many programs that terminate under neither pure top-down nor bottom-up evaluation strategies.>

We show how Horn logic programs can be implemented using database techniques, namely, mostly bottom-up in combination with certain top-down elements (as opposed to the top-down implementations of logic programs prevailing so far). The proposed method is sound and complete. It easily lends itself to a parallel implementation, and is free of nonlogical features, like backtracking. In extension to the common approach to deductive databases, we allow function symbols to appear in programs. An important advantage of this method is that it terminates in many cases in which PROLOG and SLD-resolution do not.

Recently developed inferential database systems face some common problems: a very fast growth of search space and difficulties in recognizing inference termination (especially for recursive axioms). These shortcomings stem mainly from the fact that the inference process is usually separated from database operations. A problem-oriented inferential system i8 described which refers to the database prior to query (or subquery) processing, so that the inference from the very beginning is directed by data relevant to the query. A multiprocessor implementation of the system is presented based on a computer network conforming to database relations and axioms. The system provides an efficient indication of query termination, and is complete in the sense that it produces all correct answers to a query in a finite time.

Using a nonprocedural language for query formulation requires certain automatization of a query answering process. Given a query for creation of a new relation, the problem is to find an efficient procedure which produces this relation from a given relational database. We concentrate upon sequences of join operations which losslessly produce a relation required by a query. A new property of such sequences is analyzed which provides a basis for the presented algorithms that construct an efficient join procedure. The algorithms have polynomial complexity. A modified AND/OR graph is used for the display of a given set of dependencies and a collection of relations representing a database.