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Logic Programming in the LF Logical Framework
, 1991
"... this paper we describe Elf, a metalanguage intended for environments dealing with deductive systems represented in LF. While this paper is intended to include a full description of the Elf core language, we only state, but do not prove here the most important theorems regarding the basic building b ..."
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Cited by 175 (50 self)
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this paper we describe Elf, a metalanguage intended for environments dealing with deductive systems represented in LF. While this paper is intended to include a full description of the Elf core language, we only state, but do not prove here the most important theorems regarding the basic building blocks of Elf. These proofs are left to a future paper. A preliminary account of Elf can be found in [26]. The range of applications of Elf includes theorem proving and proof transformation in various logics, definition and execution of structured operational and natural semantics for programming languages, type checking and type inference, etc. The basic idea behind Elf is to unify logic definition (in the style of LF) with logic programming (in the style of Prolog, see [22, 24]). It achieves this unification by giving types an operational interpretation, much the same way that Prolog gives certain formulas (Hornclauses) an operational interpretation. An alternative approach to logic programming in LF has been developed independently by Pym [28]. Here are some of the salient characteristics of our unified approach to logic definition and metaprogramming. First of all, the Elf search process automatically constructs terms that can represent objectlogic proofs, and thus a program need not construct them explicitly. This is in contrast to logic programming languages where executing a logic program corresponds to theorem proving in a metalogic, but a metaproof is never constructed or used and it is solely the programmer's responsibility to construct objectlogic proofs where they are needed. Secondly, the partial correctness of many metaprograms with respect to a given logic can be expressed and proved by Elf itself (see the example in Section 5). This creates the possibilit...
Elf: A Language for Logic Definition and Verified Metaprogramming
 In Fourth Annual Symposium on Logic in Computer Science
, 1989
"... We describe Elf, a metalanguage for proof manipulation environments that are independent of any particular logical system. Elf is intended for metaprograms such as theorem provers, proof transformers, or type inference programs for programming languages with complex type systems. Elf unifies logic ..."
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Cited by 77 (8 self)
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We describe Elf, a metalanguage for proof manipulation environments that are independent of any particular logical system. Elf is intended for metaprograms such as theorem provers, proof transformers, or type inference programs for programming languages with complex type systems. Elf unifies logic definition (in the style of LF, the Edinburgh Logical Framework) with logic programming (in the style of Prolog). It achieves this unification by giving types an operational interpretation, much the same way that Prolog gives certain formulas (Hornclauses) an operational interpretation. Novel features of Elf include: (1) the Elf search process automatically constructs terms that can represent objectlogic proofs, and thus a program need not construct them explicitly, (2) the partial correctness of metaprograms with respect to a given logic can be expressed and proved in Elf itself, and (3) Elf exploits Elliott's unification algorithm for a calculus with dependent types. This research was...
Metalogical Frameworks
, 1992
"... In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the me ..."
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Cited by 57 (15 self)
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In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...
Proofassistants using Dependent Type Systems
, 2001
"... this article we will not attempt to describe all the dierent possible choices of type theories. Instead we want to discuss the main underlying ideas, with a special focus on the use of type theory as the formalism for the description of theories including proofs ..."
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Cited by 47 (4 self)
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this article we will not attempt to describe all the dierent possible choices of type theories. Instead we want to discuss the main underlying ideas, with a special focus on the use of type theory as the formalism for the description of theories including proofs
Integrating computer algebra into proof planning
 Journal of Automated Reasoning
, 1998
"... Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not e ..."
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Cited by 41 (26 self)
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Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not easily separable. In this contribution we advocate an integration of computer algebra into mechanised reasoning systems at the proof plan level. This approach allows to view the computer algebra algorithms as methods, that is, declarative representations of the problem solving knowledge speci c to a certain mathematical domain. Automation can be achieved in many cases bysearching for a hierarchic proof plan at the methodlevel using suitable domainspeci c control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows to solve a large class of problems that are not automatically solvable by separate systems. Our approach also gives an answer to the correctness problems inherent insuch an integration. We advocate an approach where the computer algebra system produces highlevel protocol information that can be processed by aninterface to derive proof plans. Such a proof plan in turn can be expanded to proofs at di erent levels of abstraction, so the approach iswellsuited for producing a highlevel verbalised explication as well as for a lowlevel machine checkable calculuslevel proof. We present an implementation of our ideas and exemplify them using an automatically solved example. Changes in the criterion of `rigour of the proof ' engender major revolutions in mathematics.
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.
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...
Program Tactics and Logic Tactics
 IN PROCEEDINGS 5TH INTNL. CONFERENCE ON LOGIC PROGRAMMING AND AUTOMATED REASONING (LPAR'94
, 1994
"... In this paper we present a first order classical metatheory, called MT, with the following properties: (1) tactics are terms of the language of MT (we call these tactics, Logic Tactics); (2) there exists a mapping between Logic Tactics and the tactics developed as programs within the GETFOL theor ..."
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Cited by 19 (10 self)
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In this paper we present a first order classical metatheory, called MT, with the following properties: (1) tactics are terms of the language of MT (we call these tactics, Logic Tactics); (2) there exists a mapping between Logic Tactics and the tactics developed as programs within the GETFOL theorem prover (we call these tactics, Program Tactics). MT is expressive enough to represent the most interesting tacticals, i.e. then, orelse, try, progress and repeat. repeat allows us to express Logic Tactics which correspond to Program Tactics which may not terminate. This work is part of a larger project which aims at the development and mechanization of a metatheory which can be used to reason about, extend and, possibly, modify the code implementing Program Tactics and the GETFOL basic inference rules.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 19 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re