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Multilanguage Hierarchical Logics (or: How We Can Do Without Modal Logics)
, 1994
"... MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to ..."
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Cited by 178 (47 self)
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MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to modal logics. The motivations of our proposal are technical, epistemological and implementational. From a technical point of view, we prove, among other things, that the set of theorems of the most common modal logics can be embedded (under the obvious bijective mapping between a modal and a first order language) into that of the corresponding ML systems. Moreover, we show that ML systems have properties not holding for modal logics and argue that these properties are justified by our intuitions. This claim is motivated by the study of how ML systems can be used in the representation of beliefs (more generally, propositional attitudes) and provability, two areas where modal logics have been extensively used. Finally, from an implementation point of view, we argue that ML systems resemble closely the current practice in the computer representation of propositional attitudes and metatheoretic theorem proving.
Reflective Reasoning With and Between a Declarative Metatheory and the Implementation Code
, 1994
"... The goal of this paper is to present a theorem prover where the underlying code has been written to behave as the procedural metalevel of the object logic. We have then defined a logical declarative metatheory MT which can be put in a onetoone relation with the code and automatically generated ..."
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Cited by 23 (16 self)
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The goal of this paper is to present a theorem prover where the underlying code has been written to behave as the procedural metalevel of the object logic. We have then defined a logical declarative metatheory MT which can be put in a onetoone relation with the code and automatically generated from it. MT is proved correct and complete in the sense that, for any object level deduction, the wff representing it is a theorem of MT, and viceversa. Such theorems can be translated back in the underlying code. This opens up the possibility of deriving control strategies automatically by metatheoretic theorem proving, of mapping them into the code and thus of extending and modifying the system itself. This seems a first step towards "really" selfreflective systems, ie. systems able to reason deductively about and modify their underlying computation mechanisms. We show that the usual logical reflection rules (so called reflection up and down) are derived inference rules of the system.
A Metatheory of a Mechanized Object Theory
, 1994
"... 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" prope ..."
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Cited by 22 (10 self)
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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 selfreflective systems, i.e. systems which are able to intros...
Hierarchical MetaLogics: Intuitions, Proof Theory and Semantics
, 1991
"... The goal of this paper is to provide a possible foundation for metareasoning in the fields of artificial intelligence and computer science. We first investigate the relationship that we want to hold between metatheory and objecttheory. We then outline a methodology in which reflection rules serve ..."
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Cited by 13 (9 self)
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The goal of this paper is to provide a possible foundation for metareasoning in the fields of artificial intelligence and computer science. We first investigate the relationship that we want to hold between metatheory and objecttheory. We then outline a methodology in which reflection rules serve to deductively generate a metatheory from its object theory. Finally, we apply this methodology and define a hierarchical metalogic, namely a formal system generating an entire metahierarchy, which is sound and complete with respect to a semantics formalising the desired meta/object relationship.
Automating MetaTheory Creation and System Extension
 In AI*IA91 (Italian Association for Artificial Intelligence
, 1991
"... In this paper we describe a first experiment with a new approach for building theorem provers that can formalize themselves, reason about themselves, and safely extend themselves with new inference procedures. Within the GETFOL system we have built a pair of functions that operate between the system ..."
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Cited by 6 (3 self)
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In this paper we describe a first experiment with a new approach for building theorem provers that can formalize themselves, reason about themselves, and safely extend themselves with new inference procedures. Within the GETFOL system we have built a pair of functions that operate between the system's implementation and a theory about this implementation. The first function lifts the actual inference rules to axioms that comprise a theory of GETFOL's inference capabilities. This allows us to turn the prover upon itself whereby we may formally reason about its inference rules and derive new rules. The second function flattens new rules back into the underlying system. This provides a novel means of safe system selfextension and an efficient way of executing derived rules. 1 Introduction Theorem proving systems are generally viewed as static systems or black boxes. One cares not how the underlying rules are implemented, but only that they mimic the inference rules of some desired logic...
A System for MultiLevel Mathematical Reasoning
, 1990
"... We present a system, called GETFOL, where, for any given mathematical object theory, it is possible to define a provably correct and complete metatheory MT. Theorem proving in MT can be used to build metatheoretic representations of object level proofs. Within GETFOL, these representations can be ..."
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Cited by 1 (1 self)
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We present a system, called GETFOL, where, for any given mathematical object theory, it is possible to define a provably correct and complete metatheory MT. Theorem proving in MT can be used to build metatheoretic representations of object level proofs. Within GETFOL, these representations can be executed to prove object level theorems. Mathematical proofs can thus be built by intermixing reasoning in the object theory and reasoning in the metatheory. This provides a very flexible way to mechanize mathematical reasoning.
Metatheories About Provability and Metatheories About Proofs in a Multicontext System
"... modify MT. MT has the good properties to do what it has been defined for. In this note, we take as a significative example the problem to reason about proof structures, possibly proofs of the same theorem. For this purpose we define a second metatheory, MP. Then, at the end the multicontext system c ..."
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modify MT. MT has the good properties to do what it has been defined for. In this note, we take as a significative example the problem to reason about proof structures, possibly proofs of the same theorem. For this purpose we define a second metatheory, MP. Then, at the end the multicontext system comprises OT, MT and MP. We claim that having two separated metatheories is natural and keeps both the metatheories significantly simpler than a single metatheory, with straightforward advantages, both for theorem proving and for representing knowledge, both intellectual and computational. The aim of this note is to define formally MP, to compare MP with MT and to show how the two different metatheories can be used to perform different kinds of metareasoning, and how they can be integrated in a multilangage system. 1 2 Two different metatheories for two different goals Here we describe the crucial features of the two metatheories. MT: the metatheory MT has b
La Deduzione Automatica
"... Scopo di questo articolo e` dare una panoramica introduttiva alla deduzione automatica, mettendo in evidenza obiettivi, differenze e similitudini di alcuni fra i piu` importanti approcci al problema. ..."
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Scopo di questo articolo e` dare una panoramica introduttiva alla deduzione automatica, mettendo in evidenza obiettivi, differenze e similitudini di alcuni fra i piu` importanti approcci al problema.
Valid Extensions of Introspective Systems: A Foundation for Reflective Theorem Provers
, 1994
"... Introspective systems have been proved useful in several applications, especially in the area of automated reasoning. In this paper we propose to use structured algebraic specifications to describe the embedded account of introspective systems. Our main result is that extending such an introspective ..."
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Introspective systems have been proved useful in several applications, especially in the area of automated reasoning. In this paper we propose to use structured algebraic specifications to describe the embedded account of introspective systems. Our main result is that extending such an introspective system in a valid manner can be reduced to development of correct software. Since sound extension of automated reasoning systems again can be reduced to valid extension of introspective systems, our work can be seen as a foundation for extensible introspective reasoning systems, and in particular for reflective provers. We prove correctness of our mechanism and report on first experiences we have made with its realization in the KIV system (Karlsruhe Interactive Verifier).