Cut-elimination for a logic with definitions and induction

McDowell, Raymond; Miller, Dale

doi:10.1016/s0304-3975(99)00171-1

Cited by 82 publications

(136 citation statements)

References 24 publications

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“…are allowed in the body of definitions, inconsistencies can be easily constructed. In order to avoid such inconsistencies, we introduce the notion of level of a formula to define a proper stratification on definitions, as done in [MM00,Tiu04]. To each predicate p we associate a natural number lvl(p), the level of p. The notion of level is then extended to formulas.…”

Section: Flat Linear Logicmentioning

confidence: 99%

“…It is well known that proving cut-elimination for a logic with definitions and induction is not easy [MM00]. The method developed for cut-elimination of Llinda (see [Pim05]) is based on some of the ideas present in [Tiu04] and uses a particular notion of rank of cut formulas that depends on the level of the formula and on the shape of the derivation itself.…”

Section: Given a Definition Clause ∀X[px △mentioning

confidence: 99%

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On the Specification of Sequent Systems

Pimentel

Miller

2005

Logic for Programming, Artificial Intelligence, and Reasoning

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Abstract. Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the meta-theory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programming-style search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semi-automated proof of cut-elimination for Girard's Logic of Unicity (LU).

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Section: Flat Linear Logicmentioning

confidence: 99%

Section: Given a Definition Clause ∀X[px △mentioning

confidence: 99%

On the Specification of Sequent Systems

Pimentel

Miller

2005

Logic for Programming, Artificial Intelligence, and Reasoning

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“…Moreover, even we restrict ourselves to an intuitionistic setting, we still need to be very careful with (and, to some extent, make compromises on) the foundations of the logic in order for axioms like (*) to be sound. This is because, while the behavior of the intuitionistic connectives accommodates such axioms adequately, other mechanisms pertaining to recursive definitions are not a priori guaranteed to preserve adequacy -see [33], [39]. So what can one make of a clause such as (*) in a framework with meta-reasoning capabilities?…”

Section: Induction Principle For Type Inferencementioning

confidence: 99%

Strong Normalization for System F by HOAS on Top of FOAS

Popescu

Gunter

Osborn

2010

2010 25th Annual IEEE Symposium on Logic in Computer Science

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Abstract-We present a point of view concerning HOAS (Higher-Order Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (First-Order Abstract Syntax). As such, HOAS is not only an encoding technique, but also a higher-order view of a first-order reality. A rich collection of concepts and proof principles is developed inside the standard mathematical universe to give technical life to this point of view. The exercise consists of a new proof of Strong Normalization for System F. HOAS makes our proof considerably more direct than previous proofs. The concepts and results presented here have been formalized in the theorem prover Isabelle/HOL. [49] and has ever since been extensively developed in frameworks with a wide variety of features and flavors. We distinguish two main (overlapping) directions in these developments. Keywords--(I) First, the employment of a chosen meta logic as a pure logical framework, used for defining object systems for the purpose of reasoning inside those systems. A standard example is higher-order logic (HOL) as the meta logic and firstorder logic (FOL) as the object system. Thanks to affinities between the mechanisms of these two logics, one obtains an adequate encoding of FOL in HOL by merely declaring in HOL types and constants and stating the FOL axioms and rules as HOL axioms -then the mechanisms for building FOL deductions (including substitution, instantiation, etc.) are already present in the meta logic, HOL.-(II) Second, the employment of the meta-logic to reason about the represented object systems, i.e., to represent not only the object systems, but also (some of) their metatheory. (E.g., cut elimination is a property about Gentzenstyle FOL, not expressible in a standard HOAS-encoding of FOL into HOL.) While direction (I) has been quasi-saturated by the achievement of quasi-maximally convenient logical frameworks (such Edinburgh LF [31] and generic Isabelle [49]), this second direction undergoes these days a period of active research. We distinguish two main approaches here:

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“…The logic G allows for the direct specification of recursive predicate definitions and to interpret them either as a least and or greatest fixed point in the sense of [2,5,22,31]. The rules for induction and co-induction use higher-order predicate schema variables in their premises in order to range over possible pre-and post-fixed points.…”

Section: The Reasoning Logicmentioning

confidence: 99%

Reasoning about Computations Using Two-Levels of Logic

Miller

2010

Programming Languages and Systems

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Abstract. We describe an approach to using one logic to reason about specifications written in a second logic. One level of logic, called the "reasoning logic", is used to state theorems about computational specifications. This logic is classical or intuitionistic and should contain strong proof principles such as induction and co-induction. The second level of logic, called the "specification logic", is used to specify computation. While computation can be specified using a number of formal techniques-e.g., Petri nets, process calculus, and state machines-we shall illustrate the merits and challenges of using logic programming-like specifications of computation.

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Cut-elimination for a logic with definitions and induction

Cited by 82 publications

References 24 publications

On the Specification of Sequent Systems

On the Specification of Sequent Systems

Strong Normalization for System F by HOAS on Top of FOAS

Reasoning about Computations Using Two-Levels of Logic

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