It has been shown that the beliefs of an autoepistemic reasoner can be captured within a classical belief logic if we add the concept of “this is all I know” to the logic, where “this” can be understood as the knowledge base of the agent in question. In this paper we introduce a new construct, which allows us to express that “this is all I know about a certain subject matter.“ This relativized all-I-know concept turns out to be sufficient to model autoepistemic reasoning.
In this paper, we study how several patterns of deductive generalization from positive and negative examples can be relaxed to handle forms of defeasible reasoning, using default logic as a case study. We compare the resulting paradigms and establish the logical conditions under which they can take place.
Two different types of updates of information states are distinguished corresponding to two different kinds of incoming information: information about the actual state of domain; and information about changes made to the state of that domain. Gärdenfors [1986] proves a theorem showing that the first kind of update function can have certain intuitively appealing properties only on pain of triviality. Here a similar trivialisation result is proved for the second kind of update function.
Monotonic and non-monotonic reasoning is introduced into inductive inference. In inductive inference. which is a mathematical theory of algorithmic learning from possibly incomplete information. monotonicity means to construct hypotheses somehow incrementally. whereas the necessity of non-monotonic reasoning indicates that during hypothesis formation considerable belief revisions may be required. Therefore, it is of a particular interest to find areas of inductive inference where monotonic construction of hypotheses is always possible. It turned out that in the area of inductive inference of total recursive functions monotonicity can rarely be guaranteed. These results are compared to the problem of inductively inferring text patterns from finite samples. For this area. there is a universal weakly monotonic inductive inference algorithm. The computability of a stronger algorithm which is developed depends on the decidability of the inclusion problem for pattern languages. This problems remains open. Unfortunately. the latter algorithm turns out to be inconsistent. i.e. it sometimes generates hypotheses not able to reflect the information they are build upon. Consistency and monotonicity can hardly be achieved simultaneously. It arises the question under which circumstances an inductive inference algorithm for learning text patterns can be both consistent and monotonic. This problem class is characterized by closedness under intersection.
The goal of this work is to develop a formal logical foundation of the representation and the retrieval of cases in case-based reasoning. An adequate basis therefor provides the default logic with priorities. We present transformations which construct defaults from the memory of cases such that the retrieval of knowledge in case-based reasoning corresponds roughly to the preferred subtheory obtained by the defaults.
The purpose of this paper is to discuss various notions of nonmonotonic derivability from a conceptual point of view and to give a classification of some important forms of nonmonotonic derivability.
Inductive inference is the theory of identifying recursive functions from examples. In [26], [27], [30] the following thesis was stated: Any class of recursive functions which is identifiable at all can always be identified by an enumeratively working strategy. Moreover, the identification can always be realized with respect to a suitable nonstandard (i.e. non-Gödel) numbering. We review some of the results which have led us to state this thesis. New results are presented concerning monotonic identification and corroborating the thesis. Some of the consequences of the thesis are discussed involving the development of the theory of inductive inference during the last decade. Problems of further investigation as well as further applications of non-Gödel numberings in inductive inference are summarized.
The present paper surveys results and presents open problems concerning the limiting-effective synthesis of optimal programs for recursive functions given by input-output examples.Five different formalizations of the intuitive notion ''optimal program'' are given. In particular, it is studied under what conditions the knowledge that every function from a function class does possess an ''optimal program'' is sufficient to infer such an ''optimal program'' in the limit for each function contained in the class.
In the present paper strong-monotonic, monotonie and weak-monotonic reasoning is studied in the context of algorithmic language learning theory from positive as well as from positive and negative data. Strong-monotonicity describes the requirement to only produce better and better generalizations when more and more data are fed to the inference device. Monotonic learning reflects the eventual interplay between generalization and restriction during the process of inferring a language. However, it is demanded that for any two hypotheses the one output later has to be at least as good as the previously produced one with respect to the language to be learnt. Weakmonotonicity is the analogue of cumulativity in learning theory. We relate all these notions one to the other as well as to previously studied modes of identification, thereby in particular obtaining a strong hierarchy.
The present paper is concerned with a unifying approach to non-monotonic reasoning in clause logic programming. An outline of a general theory of non-monotonic reasoning is given that is aimed at a foundation of this area. Typical results are presented in the frame of this theory and problems related to logic programming are discussed.
In this paper we consider supernormal defaults [Poo88] with a strict partial order defining their priorities [Bre91]. We investigate their relation to minimal or preferential entailment and show that the semantics given in [Bre91] has to be modified in order to be equivalent to a preferential model approach. Concering the multiple extension problem, we introduce the careful view as an alternative to the credulous and skeptical one, which is needed to handle the generalized closed world assumption [Min82] within this framework. Given this “declaritive semantics” of such default theories, we will present a deduction algorithm for query answering. Compared to other approaches, the algorithm is quite efficient and general. Especially, it is able to generate disjunctive answers, to support the credulous, skeptical and careful view; and to cut fruitless search paths early. In order to check the applicability of defaults as soon as possible, we introduce the notion of a partial extension.
Predicate synthesis from examples (PreSE) becomes nowadays an acknowledged topic in Machine Learning (ML). Less known in ML (and in Artificial Intelligence as well), however, is predicate synthesis from formal specifications (PreS). The importance of PreS was pointed out by logicians (Skolem, Péter, ...) interested in recursive functions. It became clear that a false first order formula F of the form ∀x A(x) may specify a predicate P such that ∀x {P(x) ⇒ A(x)} is true, i.e., P describes the set S of all x for which A(x) is true. We say that F is a formal specification of P. Until now, automated construction of a definition of P for F has been partially tackled in program synthesis from incomplete specifications. However, as we illustrate in the paper, very often this approach succeeds to find a proper subset S' of S only. Therefore, a new method is necessary for PreS. In this paper we describe an algorithm and we show that this algorithm together with inductive theorem proving (ITP), i.e., proving theorems using mathematical induction principle can be considered as a tool for PreS, because it provides a (recursive) definition of P. We will present also an application of PreS to simplifying proofs of implications, as well as an application of PreS to discovery of recursive calls which lead to synthesizing efficient programs.
Several well-known inductive inference strategies change the actual hypothesis only when they discover that it “provably misclassifies” an example seen so far. This notion is made mathematically precise and its general power is characterized. In spite of its strength it is shown that this approach is not of “universal” power. Consequently, then hypotheses are considered which “unprovably misclassify” examples and the properties of this approach are studied. Among others it turns out that this type is of the same power as monotonic identification. Finally, it is shown that “universal” power can be achieved only when an unbounded number of alternations of these dual types of hypotheses is allowed.
This paper makes three main points. We observe first that the inference relation induced by a set of JTMS justification rules under the grounded model semantics (or equivalently, by a logic program with negation under the Gelfond-Lifschitz semantics) is not in general cumulative: the addition to a set of assumptions of some of the derivable conclusions may lead to a loss of others.We then show how cumulativity may be restored by adapting,a technique recently applied by Brewka to default logic. The basic idea is to upgrade the universe of discourse: replace the elementary propositions, between which inference customarily takes place, by more complex items consisting of elementary propositions indexed by certain of the ''reasons'' that lead to their acceptance.However, as we finally show, the indexed JTMS still has a shortcoming: it does not give an adequate treatment of the phenomenon of ''floating conclusions''. The problem of finding an alternative aproach that handles floating conclusions adequately without losing cumulativity again, remains open.
In this paper we continue investigations of proof theory of default logic. It turns out that, similarly to classical logic, default theories can be represented in normal forms.
We consider probabilistic inductive inference of Gödel numbers of total recursive functions when the set of possible errors is allowed to be infinite, but with bounded density. We have obtained hierarchies of classes of functions identifiable with different probabilities up to sets with fixed density. The obtained hierarchies turn out to be different from those which we have in the case of exact identification.
Given several input/output examples of some function we can state the problem: what is the “simplest” function which complies with these examples. This problem is well studied and is known to be very hard in the general case. In this paper we address a special case of the problem, when the target function can be expressed as a simple composition of known functions. We propose a new inductive synthesis algorithm for this case and show that it is efficient enough to synthesize complex geometry formulas.
A tree pattern is a structured pattern known as a term in formal logic, and a tree pattern language is the set of trees which are the ground instances of a tree pattern. In this paper, we deal with the class of tree languages whose language is defined as a union of at most k tree pattern languages, where k is an arbitrary fixed positive number. In particular, We present a polynomial time algorithm that, given a finite set of trees, to find a set of tree patterns that defines a minimal union of at most k tree pattern languages containing the given set. The algorithm can be considered as a natural extension of Plotkin's anti-unification algorithm, which finds a minimal single tree pattern language containing the given set. By using the algorithm, we can realize a consistent and conservative polynomial time inference machine that identifies the class of unions of k tree pattern languages in the limit from positive data for every k > 0.
Recently, in the context of learning probability distributions or stochastic rules, learning strategies which take into account both the simplicity of the hypothesized model and how well it explains the data have been shown to be effective. There are several strategies which fall into this general category, such as the minimum description length (MDL) principle and Occam's Razor. In this paper, we give au intuitive account of the reason why hypotheses obtained by such strategies may exhibit fast convergence to the true or optimal model as the sample size increases. We do so using the notion of ‘uniform convergence.’ We then investigate how we might apply the ‘uniform convergence method’ to estimate the convergence rates of these strategies, using the well-known Kullback-Leibler divergence as the distance measure between probabilistic information sources. In the process of doing so, we show that in fact for proving fast convergence with respect to the Kullback-Leibler divergence by the uniform convergence method, it is convenient to modify the MDL principle. We thus propose a new principle of statistical estimation, which we call ‘NIC(a new information criterion),’ motivated primarily by the goal of proving fast convergence to the true model.
A pattern is a finite string of constants and variables. The language of a pattern is the set of strings which can be óbtained by substituting non-null strings for the variables in the pattern. We consider the problem of learning pattern languages from positive example. We show that, for every k, the whole family of k-variable pattern languages can be identified in the limit by a consistent polynomial-time strategy.
