Archive for the 'Nominal Isabelle' Category

Proof pearl: A new foundation for Nominal Isabelle

Ack!  Sorry for the severe lack of updates lately.  The blog hasn’t died (I’m in the process of moving countries).  Anyway…

By Christian Urban and Brian Huffman, from ITP 2010, available from Christian Urban’s website:

Pitts et al introduced a beautiful theory about names and binding based on the notions of permutation and support. The engineering challenge is to smoothly adapt this theory to a theorem prover environment, in our case Isabelle/HOL. We present a formalisation of this work that differs from our earlier approach in two important respects: First, instead of representing permutations as lists of pairs of atoms, we now use a more abstract representation based on functions. Second, whereas the earlier work modeled different sorts of atoms using different types, we now introduce a unified atom type that includes all sorts of atoms. Interestingly, we allow swappings, that is permutations build from two atoms, to be ill-sorted. As a result of these design changes, we can iron out inconveniences for the user, considerably simplify proofs and also drastically reduce the amount of custom ML-code. Furthermore we can extend the capabilities of Nominal Isabelle to deal with variables that carry additional information. We end up with a pleasing and formalised theory of permutations and support, on which we can build an improved and more powerful version of Nominal Isabelle.

Revisiting cut-elimination: one difficult proof is really a proof

By Christian Urban and Bozhi Zhu, from RTA 2008, available from Christian Urban’s website:

Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of term-rewriting systems. The first author used such a logical relation argument to establish strong normalising for a cut-elimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation, which implements proof transformation that permute cuts over other inference rules. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the informal proof given by the first author in his PhD-thesis. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.

Abstracting syntax

By Brian Aydemir, Stephan A. Zdancewic and Stephanie Weirich, University of Pennsylvania Technical Report, available from the UPenn Tech Report website:

Binding is a fundamental part of language specification, yet it is both difficult and tedious to get right. In previous work, we argued that an approach based on locally nameless representation and a particular style for defining inductive relations can provide a portable, transparent, lightweight methodology to define the semantics of binding. Although the binding infrastructure required by this approach is straightforward to develop, it leads to duplicated effort and code as the number of binding forms in a language increases.

In this paper, we critically compare a spectrum of approaches that attempt to ameliorate this tedium by unifying the treatment of variables and binding. In particular, we compare our original methodology with two alternative ideas: First, we define variable binding in the object language via variable binding in a reusable library. Second, we present a novel approach that collapses the syntactic categories of the object language together, permitting variables to be shared between them.

Our main contribution is a careful characterization of the benefits and drawbacks of each approach. In particular, we use multiple solutions to the POPLMARK challenge in the Coq proof assistant to point out specic consequences with respect to the size of the binding infrastructure, transparency of the definitions, impact to the metatheory of the object language, and adequacy of the object language encoding.

A head-to-head comparison of de Bruijn indices and names

Stefan Berghofer and Christian Urban, from LFMTP 2006, available from the Nominal Isabelle research group’s website:

Often debates about pros and cons of various techniques for formalising lambda-calculi rely on subjective arguments, such as de Bruijn indices are hard to read for humans or nominal approaches come close to the style of reasoning employed in informal proofs. In this paper we will compare four formalisations based on de Bruijn indices and on names from the nominal logic work, thus providing some hard facts about the pros and cons of these two formalisation techniques. We conclude that the relative merits of the different approaches, as usual, depend on what task one has at hand and which goals one pursues with a formalisation.