System Description: Twelf — A Meta-Logical Framework for Deductive Systems

Efficiency of a good but not linear nominal unification algorithm

“…Two theories provide this convenience: Miller's higher-order pattern unification [Miller, 1989] and Urban et al's nominal unification . Higher-order pattern unification, the foundation of Isabelle [Paulson, 1986], λProlog [Nadathur et al, 1988], and Twelf [Pfenning and Schürmann, 1999], handles a fragment of the βη-rules. Nominal unification, the unification modulo the α-rule, has inspired extensions of logic programming languages such as αProlog [Cheney and Urban, 2004] and αKanren [Byrd and Friedman, 2007], as well as theorem provers such as Nominal Isabelle [Urban and Tasson, 2005] and αLeanTAP [Near et al, 2008].…”

Eliminating Redundancy in Higher-Order Unification: A Lightweight Approach

“…Our approach is light-weight in the sense that we do not translate or change our internal representation of terms and types. This has the advantage that it can be seamlessly incorporated into the current implementation of the Twelf system [14] and it can be easily compared to other existing optimizations. Adapting this lightweight approach is not just a matter of practical engineering convenience, but a change to a different internal representation of terms impacts the foundation of LF itself and it remains unclear whether other algorithms such as mode, termination, and coverage checking would still remain correct.…”

“…We justify this optimization theoretically using a contextual modal type theory. 2) We have implemented this optimization as part of the Twelf system [14], and discuss its impact in practice. Our experimental results show that although the size of redundant arguments is large and there is a substantial number of them, their impact on run-time performance is surprisingly limited (roughly 20% improvement).…”

Small Proof Witnesses for LF

“…The Twelf system [22], an implementation of the logical framework LF [12], provides a general safety infrastructure to represent and execute safety policies via a higher-order logic program interpretation and has been employed in several proof-carrying code projects [4,8,3,9]. Higher-order logic programming extends first order logic programming along two orthogonal dimensions: First, dynamic assumptions may be generated and used during proof search.…”

“…This work continues where Necula and Rahul [16] left off saying "more experimental results are needed especially in the higher-order setting". Our work has been implemented and evaluated within the Twelf system [22] making it unnecessary to build separate proof checking engines. To obtain a practical scalable implementation, we use higher-order substitution tree indexing [24].…”