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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 ..."
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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.
A Proof Search Specification of the πCalculus
 IN 3RD WORKSHOP ON THE FOUNDATIONS OF GLOBAL UBIQUITOUS COMPUTING
, 2004
"... We present a metalogic that contains a new quantifier (for encoding "generic judgment") and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite πcalculus within this metalogic ..."
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Cited by 21 (11 self)
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We present a metalogic that contains a new quantifier (for encoding "generic judgment") and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite πcalculus within this metalogic. Since we
A Framework for Specifying, Prototyping, and Reasoning about Computational Systems
, 2009
"... In this thesis we are interested in a framework for specifying, prototyping, and reasoning about systems that describe computations over formal objects such as formulas, proofs, and programs. The computations of interest include those like evaluation and typing in a programming language, provability ..."
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Cited by 17 (4 self)
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In this thesis we are interested in a framework for specifying, prototyping, and reasoning about systems that describe computations over formal objects such as formulas, proofs, and programs. The computations of interest include those like evaluation and typing in a programming language, provability in a logic, and behavior in a concurrency system. The development of these computational systems is often an iterative
Proof search specifications of bisimulation and modal logics for the πcalculus
 ACM Trans. on Computational Logic
"... We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allo ..."
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Cited by 15 (9 self)
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We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within πcalculus expressions and their executions (proofs). We shall illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no sideconditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for onestep transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode the πcalculus with replications, in an extended logic with induction and coinduction.
Nominal renaming sets
"... Abstract. Nominal techniques are based on the idea of sets with a finitelysupported atomspermutation action. We consider the idea of nominal renaming sets, which are sets with a finitelysupported atomsrenaming action; renamings can identify atoms, permutations cannot. We show that nominal renaming ..."
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Cited by 13 (8 self)
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Abstract. Nominal techniques are based on the idea of sets with a finitelysupported atomspermutation action. We consider the idea of nominal renaming sets, which are sets with a finitelysupported atomsrenaming action; renamings can identify atoms, permutations cannot. We show that nominal renaming sets exhibit many of the useful qualities found in (permutative) nominal sets; an elementary setsbased presentation, inductive datatypes of syntax up to binding, cartesian closure, and being a topos. Unlike is the case for nominal sets, the notion of namesabstraction coincides with functional abstraction. Thus we obtain a concrete presentation of sheaves on
A Comparison of Formalizations of the MetaTheory of a Language with Variable Bindings in Isabelle
 Supplemental Proceedings of the 14th International Conference on Theorem Proving in Higher Order Logics
, 2001
"... Abstract. Theorem provers can be used to reason formally about programming languages and there are various general methods for the formalization of variable binding operators. Hence there are choices for the style of formalization of such languages, even within a single theorem prover. The choice of ..."
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Cited by 6 (2 self)
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Abstract. Theorem provers can be used to reason formally about programming languages and there are various general methods for the formalization of variable binding operators. Hence there are choices for the style of formalization of such languages, even within a single theorem prover. The choice of formalization can affect how easy or difficult it is to do automated reasoning. The aim of this paper is to compare and contrast three formalizations (termed de Bruijn, weak HOAS and full HOAS) of a typical functional programming language. Our contribution is a detailed report on our formalizations, a survey of related work, and a final comparative summary, in which we mention a novel approach to a hybrid de Bruijn/HOAS syntax. 1
The Australian National University
"... We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allo ..."
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We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within πcalculus expressions and their executions (proofs). We illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no sideconditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for onestep transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode
A Theory of Inductive Definitions With αequivalence: Semantics, Implementation, Programming Language.
"... This document was compiled from L ATEX source on 10 August 2000. Copies will be printed, bound, and submitted for the title of PhD in Mathematics from Cambridge University, England. Other copies will be passed to those interested. Those interested are invited to write to me at Trinity College, Camb ..."
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This document was compiled from L ATEX source on 10 August 2000. Copies will be printed, bound, and submitted for the title of PhD in Mathematics from Cambridge University, England. Other copies will be passed to those interested. Those interested are invited to write to me at Trinity College, Cambridge, or email m.j.gabbay@dpmms.cam.ac.uk. I remind the reader that my examiners may well suggest corrections to this document so it need not necessarily be the final version of my thesis. If the reader is wondering, DPMMS stands for the “Department of Pure Maths