Results 11  20
of
55
A Definitional TwoLevel Approach to Reasoning with HigherOrder Abstract Syntax
 Journal of Automated Reasoning
, 2010
"... Abstract. Combining higherorder abstract syntax and (co)induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, syntax, and reasoned about using tactical theorem proving and principles of (co ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
Abstract. Combining higherorder abstract syntax and (co)induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use it in a multilevel reasoning fashion, similar in spirit to other metalogics such as Linc and Twelf. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of nonstratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuationmachine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly
Combining generic judgments with recursive definitions
 in "23th Symp. on Logic in Computer Science", F. PFENNING (editor), IEEE Computer Society Press, 2008, p. 33–44, http://www.lix.polytechnique.fr/Labo/Dale.Miller/papers/lics08a.pdf US
"... Many semantical aspects of programming languages are specified through calculi for constructing proofs: consider, for example, the specification of structured operational semantics, labeled transition systems, and typing systems. Recent proof theory research has identified two features that allow di ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
Many semantical aspects of programming languages are specified through calculi for constructing proofs: consider, for example, the specification of structured operational semantics, labeled transition systems, and typing systems. Recent proof theory research has identified two features that allow direct, logicbased reasoning about such descriptions: the treatment of atomic judgments as fixed points (recursive definitions) and an encoding of binding constructs via generic judgments. However, the logics encompassing these two features have thus far treated them orthogonally. In particular, they have not contained the ability to form definitions of objectlogic properties that themselves depend on an intrinsic treatment of binding. We propose a new and simple integration of these features within an intuitionistic logic enhanced with induction over natural numbers and we show that the resulting logic is consistent. The pivotal part of the integration allows recursive definitions to define generic judgments in general and not just the simpler atomic judgments that are traditionally allowed. The usefulness of this logic is illustrated by showing how it can provide elegant treatments of objectlogic contexts that appear in proofs involving typing calculi and arbitrarily cascading substitutions in reducibility arguments.
Case analysis of higherorder data
"... Abstract. We discuss coverage checking for data that is dependently typed and is defined using higherorder abstract syntax. Unlike previous work on coverage checking that required objects to be closed, we consider open data objects, i.e. objects that may depend on some context. Our work may therefo ..."
Abstract

Cited by 13 (11 self)
 Add to MetaCart
Abstract. We discuss coverage checking for data that is dependently typed and is defined using higherorder abstract syntax. Unlike previous work on coverage checking that required objects to be closed, we consider open data objects, i.e. objects that may depend on some context. Our work may therefore provide insights into coverage checking in Twelf, and serve as a basis for coverage checking in functional languages such as Delphin and Beluga. More generally, our work is a foundation for proofs by case analysis in systems that reason about higherorder abstract syntax. 1
Cyclic proofs for firstorder logic with inductive definitions
 In TABLEAUX’05, volume 3702 of LNCS
, 2005
"... Abstract. We consider a cyclic approach to inductive reasoning in the setting of firstorder logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sect ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
Abstract. We consider a cyclic approach to inductive reasoning in the setting of firstorder logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sections. Soundness is guaranteed by a wellfoundedness condition formulated globally in terms of traces over the proof tree, following an idea due to Sprenger and Dam. However, in contrast to their work, our proof system does not require an extension of logical syntax by ordinal variables. A fundamental question in our setting is the strength of the cyclic proof system compared to the more familiar use of a noncyclic proof system using explicit induction rules. We show that the cyclic proof system subsumes the use of explicit induction rules. In addition, we provide machinery for manipulating and analysing the structure of cyclic proofs, based primarily on viewing them as generating regular infinite trees, and also formulate a finitary trace condition sufficient (but not necessary) for soundness, that is computationally and combinatorially simpler than the general trace condition. 1
Combining de Bruijn indices and higherorder abstract syntax in Coq
 Proceedings of TYPES 2006, volume 4502 of Lecture Notes in Computer Science
, 2006
"... Abstract. The use of higherorder abstract syntax is an important approach for the representation of binding constructs in encodings of languages and logics in a logical framework. Formal metareasoning about such object languages is a particular challenge. We present a mechanism for such reasoning, ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
Abstract. The use of higherorder abstract syntax is an important approach for the representation of binding constructs in encodings of languages and logics in a logical framework. Formal metareasoning about such object languages is a particular challenge. We present a mechanism for such reasoning, formalized in Coq, inspired by the Hybrid tool in Isabelle. At the base level, we define a de Bruijn representation of terms with basic operations and a reasoning framework. At a higher level, we can represent languages and reason about them using higherorder syntax. We take advantage of Coq’s constructive logic by formulating many definitions as Coq programs. We illustrate the method on two examples: the untyped lambda calculus and quantified propositional logic. For each language, we can define recursion and induction principles that work directly on the higherorder syntax. 1
MultiLevel MetaReasoning with Higher Order Abstract Syntax
 Foundations of Software Science and Computation Structures, volume 2620 of Lecture Notes in Computer Science
, 2003
"... Abstract. Combining Higher Order Abstract Syntax (HOAS) and (co)induction is well known to be problematic. In previous work [1] we have described the implementation of a tool called Hybrid, within Isabelle HOL, which allows object logics to be represented using HOAS, and reasoned about using tactica ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Abstract. Combining Higher Order Abstract Syntax (HOAS) and (co)induction is well known to be problematic. In previous work [1] we have described the implementation of a tool called Hybrid, within Isabelle HOL, which allows object logics to be represented using HOAS, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. In this paper we describe how to use it in a multilevel reasoning fashion, similar in spirit to other metalogics such F Oλ ∆IN and Twelf. By explicitly referencing provability, we solve the problem of reasoning by (co)induction in presence of nonstratifiable hypothetical judgments, which allow very elegant and succinct specifications. We demonstrate the method by formally verifying the correctness of a compiler for (a fragment) of MiniML, following [10]. To further exhibit the flexibility of our system, we modify the target language with a notion of nonwellfounded closure, inspired by Milner & Tofte [19] and formally verify via coinduction a subject reduction theorem for this modified language. 1
Weak Normalization for the SimplyTyped LambdaCalculus in Twelf (Extended Abstract)
 In Logical Frameworks and Metalanguages (LFM 04), IJCAR
, 2004
"... Andreas Abel Department of Computer Science, Chalmers University of Technology Rannvagen 6, SWE41296 Goteborg, Sweden Abstract. Weak normalization for the simplytyped calculus is proven in Twelf, an implementation of the Edinburgh Logical Framework. Since due to prooftheoretical restrict ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
Andreas Abel Department of Computer Science, Chalmers University of Technology Rannvagen 6, SWE41296 Goteborg, Sweden Abstract. Weak normalization for the simplytyped calculus is proven in Twelf, an implementation of the Edinburgh Logical Framework. Since due to prooftheoretical restrictions Twelf Tait's computability method does not seem to be directly usable, a combinatorical proof is adapted and formalized instead.
Elimination of Negation in a Logical Framework
, 2000
"... Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgements, we adapt the idea of elimination of negation introduced in [21] for Horn logic to a fragment of higherorder HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.
Staged Computation with Names and Necessity
, 2005
"... Staging is a programming technique for dividing the computation in order to exploit the early availability of some arguments. In the early stages the program uses the available arguments to generate, at run time, the code for the late stages. The late stages may then be explicitly evaluated when app ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Staging is a programming technique for dividing the computation in order to exploit the early availability of some arguments. In the early stages the program uses the available arguments to generate, at run time, the code for the late stages. The late stages may then be explicitly evaluated when appropriate. A type system for staging should ensure that only welltyped expressions are generated, and that only expressions with no free variables are permitted for evaluation.
System Description: Delphin – A Functional Programming Language for Deductive Systems
"... Abstract. Delphin is a functional programming language [PS08] utilizing dependent higherorder datatypes. Delphin is a two level system, which cleanly separates data from computation. The data level is LF [HHP93], which allows for the specification of deductive systems following the judgmentsastyp ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Abstract. Delphin is a functional programming language [PS08] utilizing dependent higherorder datatypes. Delphin is a two level system, which cleanly separates data from computation. The data level is LF [HHP93], which allows for the specification of deductive systems following the judgmentsastypes methodology utilizing higherorder abstract syntax (HOAS). The computation level facilitates the manipulation of such encodings by providing a newness constructor to create parameters (fresh constants) and the ability to write functions over parameters, which we also call parameter functions. A wealth of documentation and examples are available online at