In this paper we propose a metatheory, MT which represents the computation which implements its object theory, OT, and, in particular, the computation which implements deduction in OT. To emphasize this fact we say that MT is a metatheory of a mechanized object theory. MT has some "unusual" properties, e.g. it explicitly represents failure in the application of inference rules, and the fact that large amounts of the code implementing OT are partial, i.e. they work only for a limited class of inputs. These properties allow us to use MT to express and prove tactics, i.e. expressions which specify how to compose possibly failing applications of inference rules, to interpret them procedurally to assert theorems in OT, to compile them into the system implementation code, and, finally, to generate MT automatically from the system code. The definition of MT is part of a larger project which aims at the implementation of self-reflective systems, i.e. systems which are able to intros...

This paper describes a reasoning system, called GETFOL, able to introspect (the code implementing) its own deductive machinery, to reason deductively about it in a declarative metatheory and to produce new executable code which can then be pushed back into the underlying implementation. In this paper we discuss the general architecture of GETFOL and the problems related to its implementation.

In this article we formally describe a declarative approach for encoding plan operators in proof planning, the so-called methods. The notion of method evolves from the much studied concept tactic and was first used by Bundy. While significant deductive power has been achieved with the planning approach towards automated deduction, the procedural character of the tactic part of methods, however, hinders mechanical modification. Although the strength of a proof planning system largely depends on powerful general procedures which solve a large class of problems, mechanical or even automated modification of methods is nevertheless necessary for at least two reasons. Firstly methods designed for a specific type of problem will never be general enough. For instance, it is very difficult to encode a general method which solves all problems a human mathematician might intuitively consider as a case of homomorphy. Secondly the cognitive ability of adapting existing methods to suit novel situa...

Many real world applications require systems with both reasoning and sensing/acting capabilities. However, most often, these systems do not work properly, i.e. they fail to execute actions and rarely perceive the external world correctly. No action, even if apparently simple, is guaranteed to succeed and, therefore, no planning can be "sound" (with respect to the real world) without taking into account failure. In this paper, we present a theory of planning that provides (1) a language that allows us to express failure; (2) a declarative formal semantics for this language; (3) a logic for reasoning about (possibly failing) plans. 1 Failure Many real world applications require systems with both reasoning and sensing/acting capabilities. Reasoning allows systems to achieve high level goals. Acting and sensing allows them to work in a complex and unpredictable external environment. Most often, systems sensing and acting in the world do not work properly, i.e. they fail to exe...

We propose multi-context systems (MC systems) as a formal framework for the specification of complex reasoning. MC systems provide the ability to structure the specification of "global" reasoning in terms of "local" reasoning sub-patterns. Each sub-pattern is modeled as a deduction in a context, formally defined as an axiomatic formal system. The global reasoning pattern is modeled as a concatenation of contextual deductions via bridge rules, i.e. inference rules that infer a fact in one context from facts asserted in other contexts. Besides the formal framework, in this paper we propose a three layer architecture designed to specify and automatize complex reasoning. At the first level we have object-level contexts (called s-contexts) for domain specifications. Problem solving principles and, more in general, meta-level knowledge about the application domain is specified in a distinct context, called Problem Solving Context (PSC). On top of s-contexts and PSC, we have a further context, called MT , where it is possible to specify strategies to control multi-context reasoning spanning through s-contexts and PSC. We show how GETFOL can be used as a computer tool for the implementation of MC systems and for the automatization of multi-context deductions.

The reasoning power of human-oriented plan-based reasoning systems is primarily derived from their domain-specific problem solving knowledge. Such knowledge is, however, intrinsically incomplete. In order to model the human ability of adapting existing methods to new situations we present in this work a declarative approach for representing methods, which can be adapted by so-called meta-methods. Since apparently the success of this approach relies on the existence of general and strong meta-methods, we describe several meta-methods of general interest in detail by presenting the problem solving process of two familiar classes of mathematical problems. These examples should illustrate our philosophy of proof planning as well: besides planning with the current repertoire of methods, the repertoire of methods evolves with experience in that new ones are created by meta-methods which modify existing ones.

ion, Reformulation, and Approximation, SARA-95, Ville d'Esterel, Canada, 1995, forthcoming. Adapting the Diagonalization Method by Reformulations Xiaorong Huang Manfred Kerber Lassaad Cheikhrouhou Fachbereich Informatik Universitat des Saarlandes D-66041 Saarbrucken Germany fhuang---kerber---lassaadg@cs.uni-sb.de Abstract Extending the plan-based paradigm for automated theorem proving, we developed in previous work a declarative approach towards representing methods in a proof planning framework to support their mechanical modification. This paper presents a detailed study of a class of particular methods, embodying variations of a mathematical technique called diagonalization. The purpose of this paper is mainly twofold. First we demonstrate that typical mathematical methods can be represented in our framework in a natural way. Second we illustrate our philosophy of proof planning: besides planning with a fixed repertoire of methods, meta-methods create new methods by modifying e...

REX School/Workshop on Foundations of Object Oriented Languages, Lecture Notes in Computer Science, May 1990. [Yon91] A. Yonezawa. A Reflective Object Oriented Concurrent Language. Lecture Notes in Computer Science, 441:254--256, 1991. 17 [Giu92] F. Giunchiglia. The GETFOL Manual - GETFOL version 1. Technical Report 9204-01, DIST - University of Genova, Genoa, Italy, 1992. Forthcoming IRST-Technical Report. [GMMW77] M.J. Gordon, R. Milner, L. Morris, and C. Wadsworth. A Metalanguage for Interactive Proof in LCF. CSR report series CSR-16-77, Department of Artificial Intelligence, Dept. of Computer Science, University of Edinburgh, 1977. [GMW79] M.J. Gordon, A.J. Milner, and C.P. Wadsworth. Edinburgh LCF - A mechanized logic of computation, volume 78 of Lecture Notes in Computer Science. Springer Verlag, 1979. [GT91] F. Giunchiglia and P. Traverso. Reflective reasoning with and between a declarative metatheory and the implementation code. In Proc. of the 12th International Joint C

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A lot of the human ability to prove hard mathematical theorems can be ascribed to a problem-specific problem solving know-how. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their know-how to new related problems. These two aspects, having rich experience and extending it by need, can be simulated in a proof planning framework: the problem-specific reasoning knowledge is represented in form of declarative planning operators, called methods; since these are declarative, they can be mechanically adapted to new situations by so-called metamethods. In this contribution we apply this framework to two prominent proofs in theorem proving, first, we present methods for proving the ground completeness of binary resolution, which essentially correspond to key lemmata, and then, we show how these methods can be reused for the proof of the ground completeness of lock resolution.