Nonmonotonic reasoning, preferential models and cumulative logics

Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen. Both proof-theoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay, while the latter has been advocated by Y. Shoham in. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams. The "preferential" models proposed here are a much stronger tool than Adams' probabilistic semantics. The basic language used in this paper is that of propositional logic. The extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper.

Nonmonotonic Reasoning, Preferential Models and Cumulative Logics

(1958). The Mathematics of Sentence Structure. The American Mathematical Monthly: Vol. 65, No. 3, pp. 154-170.

The Mathematics of Sentence Structure

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Implementing Mathematics with The Nuprl Proof Development System

This paper proposes a modular ACTOR architecture and definitional method for artificial intelligence that is conceptually based on a single kind of object: actors [or, if you will, virtual processors, activation frames, or streams]. The formalism makes no presuppositions about the representation of primitive data structures and control structures. Such structures can be programmed, micro-coded, or hard wired in a uniform modular fashion. In fact it is impossible to determine whether a given object is "really" represented as a list, a vector, a hash table, a function, or a process. The architecture will efficiently run the coming generation of PLANNER-like artificial intelligence languages including those requiring a high degree of parallelism. The efficiency is gained without loss of programming generality because it only makes certain actors more efficient; it does not change their behavioral characteristics. The architecture is general with respect to control structure and does not have or need goto, interrupt, or semaphore primitives. The formalism achieves the goals that the disallowed constructs are intended to achieve by other more structured methods. PLANNER Progress "Programs should not only work, but they should appear to work as well." PDP-1X Dogma The PLANNER project is continuing research in natural and effective means for embedding knowledge in procedures. In the course of this work we have succeeded in unifying the formalism around one_ fundamental concept: the ACTOR. Intuitively, an ACTOR is an active agent which plays a role on cue according to a script. We use the ACTOR metaphor to emphasize the inseparability of control and data flow in our model. Data structures, functions, semaphores, monitors, ports, descriptions, Quillian nets, logical formulae, numbers, identifiers, demons, processes, contexts, and data bases can all be shown to be special cases of actors. All of the above are objects with certain useful modes of behavior. Our formalism shows how all of the modes of behavior can be defined in terms of one kind of behavior: sending messages to actors. An actor is always invoked uniformly in exactly the same way regardless of whether it behaves as a recursive function, data structure, or process. "It is vain to multiply Entities beyond need." William of Occam "Monotheism is the Answer." The unification and simplification of the formalisms for the procedural embedding of knowledge has a great many benefits for us: FOUNDATIONS: The concept puts procedural semantics [the theory of how things operate] on a firmer basis. It will now be possible to do cleaner theoretical studies of the relation between procedural semantics and set-theoretic semantics such as model theories of the quantificational calculus and the lambda calculus. LOGICAL CALCULAE: A procedural semantics is developed for the quantificational calculus. The logical constants FOR-ALL, THERE-EXISTS, AND, OR, NOT, and IMPLIES are defined as actors. KNOWLEDGE BASED PROGRAMMING is programming in an environment which has a substantial knowledge base in the application area for which the programs are intended. The actor formalism aids knowledge based programming in the following ways: PROCEDURAL EMBEDDING of KNOWLEDGE, TRACING BEHAVIORAL DEPENDENCIES, and SUBSTANTIATING that ACTORS SATISFY their INTENTIONS. INTENTIONS: Furthermore the confirmation of properties of procedures is made easier and more uniform. Every actor has an INTENTION which checks that the prerequisites and the context of the actor being sent the message are satisfied. The intention is the CONTRACT that the actor has with the outside world. How an actor fullfills its contract is its own business. By a SIMPLE BUG we mean an actor which does not satisfy its intention. We would like to eliminate simple debugging of actors by the META-EVALUATION of actors to show that they satisfy their intentions. Suppose that there is an external audience of actors E which satisfy the intentions.of the actors to which they send messages. Intuitively, the principle of ACTOR INDUCTION states that the intentions of all actions caused by E are in turn satisfied provided that the following condition holds: If for each actor A the' intention of A is satisfied => that the intentions of all actors sent messages by A are satisfied. Computational induction [Manna], structural induction [Burstall], and Peano induction are all special cases of ACTOR induction. Actor based intentions have the following advantages: The intention is decoupled from the actors it describes. Intentions of concurrent actions are more easily disentangled. We can more elegantly write intentions The intentions are written in the same formalism as the Because for dialogues between actors. procedures they describe. Thus for example intentions can have intentions, protection is an intrinsic property of actors, we hope to be able to deal with protection issues in the same straight forward manner as more conventional intentions. Intentions of data structures are handled by the same machinery as for all other actors. COMPARATIVE SCHEMATOLOGY: The theory of comparative power of control structures is I

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A universal modular ACTOR formalism for artificial intelligence

The present invention relates to a process for sizing cellulose fibers or cellulose fiber containing materials and to a composition for carrying out the process. More particularly the invention relates to a process for sizing according to which cellulose fibers or cellulose fiber containing materials in a manner known per se are brought into contact with compounds having the general formula WHEREIN R1 is an organic, hydrophobic group having 8 to 40 carbon atoms and R2 is an alkyl group having 1 to 7 carbon atoms or has the same meaning as R1.

Gerhard Gentzen

Papers

Untersuchungen über das logische Schließen. II

The collected papers of Gerhard Gentzen

Investigations into Logical Deduction

Die Widerspruchsfreiheit der reinen Zahlentheorie

Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie