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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.
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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Cited by 53 (2 self)
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
MetaProgramming in Logic Programming
 Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... data types are facilitated in Godel by its type and module systems. Thus, in order to describe the metaprogramming facilities of Godel, a brief account of these systems is given. Each constant, function, predicate, and proposition in a Godel program must be specified by a language declaration. The ..."
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Cited by 46 (3 self)
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data types are facilitated in Godel by its type and module systems. Thus, in order to describe the metaprogramming facilities of Godel, a brief account of these systems is given. Each constant, function, predicate, and proposition in a Godel program must be specified by a language declaration. The type of a variable is not declared but inferred from its context within a particular program statement. To illustrate the type system, we give the language declarations that would be required for the program in Figure 1. BASE Name. CONSTANT Tom, Jerry : Name. PREDICATE Chase : Name * Name; Cat, Mouse : Name. Note that the declaration beginning BASE indicates that Name is a base type. In the statement Chase(x,y) ! Cat(x) & Mouse(y). the variables x and y are inferred to be of type Name. Polymorphic types can also be defined in Godel. They are constructed from the base types, type variables called parameters, and type constructors. Each constructor has an arity 1 attached to it. As an...
Reflection in logic, functional and objectoriented programming: a short comparative study
 Proc. of the IJCAI’95 Workshop on Reflection and Metalevel Architectures andtheir Applications in AI,1995
"... Département d’informatique et de recherche opérationnelle ..."
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Cited by 38 (1 self)
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Département d’informatique et de recherche opérationnelle
(ML)²: A formal language for KADS models of expertise
, 1993
"... This paper reports on an investigation into a formal language for specifying kads models of expertise. After arguing the need for and the use of such formal representations, we discuss each of the layers of a kads model of expertise in the subsequent sections, and define the formal constructions tha ..."
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Cited by 35 (9 self)
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This paper reports on an investigation into a formal language for specifying kads models of expertise. After arguing the need for and the use of such formal representations, we discuss each of the layers of a kads model of expertise in the subsequent sections, and define the formal constructions that we use to represent the kads entities at every layer: ordersorted logic at the domain layer, metalogic at the inference layer, and dynamiclogic at the task layer. All these constructions together make up (ml) 2 , the language that we use to represent models of expertise. We illustrate the use of (ml) 2 in a small example model. We conclude by describing our experience to date with constructing such formal models in (ml) 2 , and by discussing some open problems that remain for future work. 1 Introduction One of the central concerns of "knowledge engineering" is the construction of a model of some problem solving behaviour. This model should eventually lead to the construction of a...
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 23 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
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...
The Logic of Provability
, 1997
"... Contents 1. Introduction, Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2. Modal logic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3. Proof of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 4. Fixed poi ..."
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Cited by 21 (2 self)
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Contents 1. Introduction, Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2. Modal logic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3. Proof of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 4. Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 5. Propositional theories and Magarialgebras . . . . . . . . . . . . . . . . . . . . . 368 6. The extent of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 369 7. Classification of provability logics . . . . . . . . . . . . . . . . . . . . . . . . . . 371 8. Bimodal and polymodal provability logics . . . . . . . . . . . . . . . . . . . . . 374 9. Rosser orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.
Metaprogramming in Logic
 Encyclopedia of Computer Science and Technology
, 1994
"... In this review of metaprogramming in logic we pay equal attention to theoretical and practical issues: the contents range from mathematical and logical preliminaries to implementation and applications in, e.g., software engineering and knowledge representation. The area is one in rapid development b ..."
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Cited by 17 (0 self)
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In this review of metaprogramming in logic we pay equal attention to theoretical and practical issues: the contents range from mathematical and logical preliminaries to implementation and applications in, e.g., software engineering and knowledge representation. The area is one in rapid development but we have emphasized such issues that are likely to be important for future metaprogramming languages and methodologies. 1 Introduction The term `metaprogramming' relates to `programming' as `metalanguage' relates to `language' and `metalogic' to `logic': programming where the data represent programs. It should be no surprise that metaprogramming with logic programming languages takes advantage of many results from metalogic. In the most general interpretation we would say that `metaprogramming ' refers to any kind of computer programming where the input or output represents programs. We will refer to a program of this kind as a metaprogram and to its data as object programs. Analogousl...
Multilanguage First Order Theories of Propositional Attitudes
, 1991
"... The goal of this paper is to present a new family of formal systems, so called multilanguage systems (MLsystems), which allow the use of multiple distinct first order languages and inference rules whose premises and consequences need not belong to the same language. MLsystems are argued to formali ..."
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Cited by 16 (11 self)
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The goal of this paper is to present a new family of formal systems, so called multilanguage systems (MLsystems), which allow the use of multiple distinct first order languages and inference rules whose premises and consequences need not belong to the same language. MLsystems are argued to formalize naturally and elegantly notions like belief, knowledge and, more in general, various forms of propositional attitudes. Some instances of MLsystems are defined and proved equivalent to the modal logic K and some of Konolige's logics for belief.