## Implementing the Meta-Theory of Deductive Systems (1992)

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Venue: | Proceedings of the 11th International Conference on Automated Deduction |

Citations: | 32 - 9 self |

### BibTeX

@INPROCEEDINGS{Pfenning92implementingthe,

author = {Frank Pfenning and Ekkehard Rohwedder},

title = {Implementing the Meta-Theory of Deductive Systems},

booktitle = {Proceedings of the 11th International Conference on Automated Deduction},

year = {1992},

pages = {537--551},

publisher = {Springer-Verlag LNAI}

}

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### Abstract

. We exhibit a methodology for formulating and verifying metatheorems about deductive systems in the Elf language, an implementation of the LF Logical Framework with an operational semantics in the spirit of logic programming. It is based on the mechanical verification of properties of transformations between deductions, which relies on type reconstruction and schema-checking. The latter is justified by induction principles for closed LF objects, which can be constructed over a given signature. We illustrate our technique through several examples, the most extensive of which is an interpretation of classical logic in minimal logic through a continuation-passing-style transformation on proofs. 1 Introduction Formal deductive systems have become an important tool in computer science. They are used to specify logics, type systems, operational semantics and other aspects of languages. The role of such specifications is three-fold. Firstly, inference rules serve as a high-level notation w...

### Citations

711 | A framework for defining logics
- Harper, Honsel, et al.
- 1987
(Show Context)
Citation Context ...ta-theory of a language. For example, the soundness of a type system with respect to an operational semantics is most easily expressed as a property of two inference systems. The LF Logical Framework =-=[16]-=- has been designed to provide an appropriate language for the high-level specification of deductive systems. In LF, judgments are represented as types and deductions as objects. The validity of a dedu... |

592 |
Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39:176–210 and 405–431
- Gentzen
- 1934
(Show Context)
Citation Context ...rence rules serve as a high-level notation which helps to explain the meaning of the language under consideration. This was one of Gentzen's original motivations for his calculus of natural deduction =-=[10]-=-. Secondly, they can form the basis for an implementation of a deductive system. For example, it is not difficult to translate an operational semantics presented in the style of natural deduction [18,... |

319 | Natural Semantics
- Kahn
- 1988
(Show Context)
Citation Context ...[10]. Secondly, they can form the basis for an implementation of a deductive system. For example, it is not difficult to translate an operational semantics presented in the style of natural deduction =-=[18, 13, 4]-=- into an implementation of an interpreter. Thirdly, deductive systems help in developing the meta-theory of a language. For example, the soundness of a type system with respect to an operational seman... |

220 |
Call-by-name, call-by-value, and the *-calculus
- Plotkin
- 1975
(Show Context)
Citation Context ... computational simulations. We have transcribed and verified 8 of these interpretations. In this section we will verify the type-soundness of the continuation-passing-style (CPS) transform of Plotkin =-=[25]-=-, which is one of Griffin's examples. Syntax. Once more, the first task will be to represent the logics under consideration. Here we deal with a propositional logic, which allows their direct interpre... |

209 |
Call-by-name, call-by-value and the λ-calculus
- Plotkin
- 1975
(Show Context)
Citation Context ... computational simulations. We have transcribed and verified 8 of these interpretations. In this section we will verify the type-soundness of the continuation-passing-style (CPS) transform of Plotkin =-=[25]-=-, which is one of Griffin’s examples. Syntax. Once more, the first task will be to represent the logics under consideration. Here we deal with a propositional logic, which allows their direct interpre... |

176 | Logic programming in the LF logical framework
- Pfenning
- 1991
(Show Context)
Citation Context ...proof-checking. Unification and proof search algorithms have been developed [7, 27, 28, 24] and it has been amenable to an operational interpretation which is realized in the Elf programming language =-=[21, 23]-=-. A wide range of deductive systems have been specified in LF and implemented in Elf [1, 15, 19, 20]. In this paper we investigate the use of Elf to implement the meta-theory of deductive systems, thu... |

114 |
Inductively defined types
- Coquand, Paulin-Mohring
(Show Context)
Citation Context ... z dbl (s n) = Qs n (dbl n) = dbl (s (s n)) Here, [x:A] B is Elf concrete syntax for x:A: B. Many types, such as the type of exp defined below, are not inductive in the usual sense (see, for example, =-=[6]-=-). However, we can still derive a form of an induction principle for those types, as we limit ourselves to closed LF terms constructed over a fixed given signature. 3 A Functional Language Fragment To... |

99 |
A simple applicative language: Mini-ml
- Clement, Despeyroux, et al.
- 1986
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Citation Context ... the transformations axiomatized in Stage 3. We have carried out this methodology for a number of examples, the most intricate of which is a verification of the subject reduction property for Mini-ML =-=[5, 20]-=-. Currently, Stage 4 is done mostly by hand, as we have not yet implemented a general schema-checker. As we will illustrate, in the current Elf implementation schema-checking can be directly achieved ... |

84 | Using typed lambda calculus to implement formal systems on a machine
- Avron, Honsell, et al.
- 1992
(Show Context)
Citation Context ...nd it has been amenable to an operational interpretation which is realized in the Elf programming language [21, 23]. A wide range of deductive systems have been specified in LF and implemented in Elf =-=[1, 15, 19, 20]-=-. In this paper we investigate the use of Elf to implement the meta-theory of deductive systems, thus addressing the third of the principal applications listed above. Our approach is based on three ob... |

78 | Elf: a language for logic definition and verified metaprogram- ming
- Pfenning
- 1989
(Show Context)
Citation Context ...proof-checking. Unification and proof search algorithms have been developed [7, 27, 28, 24] and it has been amenable to an operational interpretation which is realized in the Elf programming language =-=[21, 23]-=-. A wide range of deductive systems have been specified in LF and implemented in Elf [1, 15, 19, 20]. In this paper we investigate the use of Elf to implement the meta-theory of deductive systems, thu... |

75 |
Termination Proofs for Logic Programs
- Plümer
(Show Context)
Citation Context ...posed by the dependent types, we believe that in many cases such transformations could be constructed automatically. Other work in inductive theorem proving and logic programming such as, for example =-=[3, 12, 26]-=-, should be applicable in our setting to aid in the construction of such transformations. Closely related to the ideas presented here is work by Fribourg [9] in the simpler setting of Horn clauses. Ag... |

72 |
A proof-theoretic approach to logic programming. II. Programs as definitions
- Hallnäs, Schroeder-Heister
- 1991
(Show Context)
Citation Context ...posed by the dependent types, we believe that in many cases such transformations could be constructed automatically. Other work in inductive theorem proving and logic programming such as, for example =-=[3, 12, 26]-=-, should be applicable in our setting to aid in the construction of such transformations. Closely related to the ideas presented here is work by Fribourg [9] in the simpler setting of Horn clauses. Ag... |

65 | Unification and anti-unification in the calculus of constructions
- Pfenning
- 1991
(Show Context)
Citation Context ...ions can be checked automatically for validity. However, LF is a powerful basis for much more comprehensive tasks than mere proof-checking. Unification and proof search algorithms have been developed =-=[7, 27, 28, 24]-=- and it has been amenable to an operational interpretation which is realized in the Elf programming language [21, 23]. A wide range of deductive systems have been specified in LF and implemented in El... |

44 | Natural semantics and some of its metatheory in Elf
- Michaylov, Pfenning
- 1991
(Show Context)
Citation Context ...nd it has been amenable to an operational interpretation which is realized in the Elf programming language [21, 23]. A wide range of deductive systems have been specified in LF and implemented in Elf =-=[1, 15, 19, 20]-=-. In this paper we investigate the use of Elf to implement the meta-theory of deductive systems, thus addressing the third of the principal applications listed above. Our approach is based on three ob... |

41 | Compiler verification in LF
- Hannan, Pfenning
(Show Context)
Citation Context ...ML, ffl type-soundness of Mini-ML (sometimes called the subject reduction theorem), including polymorphism, ffl correctness of a compiler from Mini-ML to a variant of the Categorical Abstract Machine =-=[14]-=-, 2 ffl the deduction theorem for an axiomatic formulation of propositional logic in the style of Hilbert, ffl equivalence of natural deduction and Hilbert's calculus, ffl soundness of two theorem pro... |

37 |
Extensions to the rippling-out tactic for guiding inductive proofs
- Bundy, Harmelen, et al.
- 1990
(Show Context)
Citation Context ...posed by the dependent types, we believe that in many cases such transformations could be constructed automatically. Other work in inductive theorem proving and logic programming such as, for example =-=[3, 12, 26]-=-, should be applicable in our setting to aid in the construction of such transformations. Closely related to the ideas presented here is work by Fribourg [9] in the simpler setting of Horn clauses. Ag... |

36 |
The expressiveness of simple and second-order type structures
- Fortune, Leivant, et al.
- 1983
(Show Context)
Citation Context ...priori weaker than the other. For example, the functions provably total in second-order arithmetic are exactly the functions which can be defined using a schema of primitive recursion at higher types =-=[8]-=-. As we hope to illustrate in this paper, the latter choice is a very natural one, and many examples can be treated very elegantly. Moreover, it does not preclude the application of automated theorem ... |

25 | Extensions and Applications of Higher-Order Unification
- Elliott
- 1990
(Show Context)
Citation Context ...ions can be checked automatically for validity. However, LF is a powerful basis for much more comprehensive tasks than mere proof-checking. Unification and proof search algorithms have been developed =-=[7, 27, 28, 24]-=- and it has been amenable to an operational interpretation which is realized in the Elf programming language [21, 23]. A wide range of deductive systems have been specified in LF and implemented in El... |

24 |
Extracting logic programs from proofs that use extended Prolog execution and induction
- Fribourg
- 1990
(Show Context)
Citation Context ...c programming such as, for example [3, 12, 26], should be applicable in our setting to aid in the construction of such transformations. Closely related to the ideas presented here is work by Fribourg =-=[9]-=- in the simpler setting of Horn clauses. Again, his ideas could add to the degree of automation available within our methodology. The remainder of this paper is organized as follows: In Section 2 we p... |

19 | Hoare’s Logic in the LF
- Mason
- 1987
(Show Context)
Citation Context ...nd it has been amenable to an operational interpretation which is realized in the Elf programming language [21, 23]. A wide range of deductive systems have been specified in LF and implemented in Elf =-=[1, 15, 19, 20]-=-. In this paper we investigate the use of Elf to implement the meta-theory of deductive systems, thus addressing the third of the principal applications listed above. Our approach is based on three ob... |

18 |
and Computation in General Logic
- Proofs
- 1990
(Show Context)
Citation Context ...ions can be checked automatically for validity. However, LF is a powerful basis for much more comprehensive tasks than mere proof-checking. Unification and proof search algorithms have been developed =-=[7, 27, 28, 24]-=- and it has been amenable to an operational interpretation which is realized in the Elf programming language [21, 23]. A wide range of deductive systems have been specified in LF and implemented in El... |

15 |
From operational semantics to abstract machines: Preliminary results
- Hannan, Miller
- 1990
(Show Context)
Citation Context ...[10]. Secondly, they can form the basis for an implementation of a deductive system. For example, it is not difficult to translate an operational semantics presented in the style of natural deduction =-=[18, 13, 4]-=- into an implementation of an interpreter. Thirdly, deductive systems help in developing the meta-theory of a language. For example, the soundness of a type system with respect to an operational seman... |

14 |
Systems of polymorphic type assignment in LF
- HARPER
- 1990
(Show Context)
Citation Context |

8 |
Investigations into proof-search in a system of first-order dependent function types
- Pym, Wallen
- 1990
(Show Context)
Citation Context |

4 |
An implementation of the Elf core language in Standard ML. Available via ftp over the Internet
- Pfenning
- 1991
(Show Context)
Citation Context ... to synthesize an appropriate deduction transformation, such as tr or cps above. We currently have a small prototype implementation of schema-checking as an extension of the current Elf core language =-=[22]-=-. Several further verifications of standard meta-theorems in logic and computer science using the methods described here are subject of current work. 3 We omit from this discussion the unit refinement... |

2 |
Logical interpretations as computational simulations. Draft paper. Talk given at the North
- Griffin
- 1991
(Show Context)
Citation Context ...vr m (valpair (q1 Q) (q2 Q))) -? vr m q -? vr (evalpi1 m) (Qpi1 e v1 v2 m q q1 q2). This succeeds with substitution Qpi1 = [e][v1][v2][m][q][q1][q2] q1 q. 4 Logic Interpretations and CPS Transform In =-=[11]-=-, Griffin presents a number of interpretations between logics and shows how they can be viewed as computational simulations. We have transcribed and verified 8 of these interpretations. In this sectio... |

1 |
Burstall and Furio Honsell. Operational semantics in a natural deduction setting
- Rod
- 1991
(Show Context)
Citation Context ...[10]. Secondly, they can form the basis for an implementation of a deductive system. For example, it is not difficult to translate an operational semantics presented in the style of natural deduction =-=[18, 13, 4]-=- into an implementation of an interpreter. Thirdly, deductive systems help in developing the meta-theory of a language. For example, the soundness of a type system with respect to an operational seman... |

1 |
Modularity in the LF logical framework. Submitted. Available as
- Harper, Pfenning
- 1991
(Show Context)
Citation Context ... could be used to explain the meta-theorems and their proofs. Schema-checking as presented in this paper verifies properties of signatures. Therefore, our work draws upon a calculus for LF signatures =-=[17]-=-. In that paper, it is also shown how some simple meta-theoretic properties can be witnessed directly by realizations (functions between signatures). The limitations of this alternative approach are a... |

1 |
and Computation in General Logic.PhDthesis,University of Edinburgh
- Proofs
- 1990
(Show Context)
Citation Context ...ions can be checked automatically for validity. However, LF is a powerful basis for much more comprehensive tasks than mere proof-checking. Unification and proof search algorithms have been developed =-=[7, 27, 28, 24]-=- and it has been amenable to an operational interpretation which is realized in the Elf programming language [21, 23]. A wide range of deductive systems havesbeen specified in LF and implemented in El... |