An indexed model of recursive types for foundational proof-carrying code

Abstract

The proofs of "traditional" proof carrying code (PCC) are type-specialized in the sense that they require axioms about a specific type system. In contrast, the proofs of foundational PCC explicitly define all required types and explicitly prove all the required properties of those types assuming only a fixed foundation of mathematics such as higher-order logic. Foundational PCC is both more flexible and more secure than type-specialized PCC.For foundational PCC we need semantic models of type systems on von Neumann machines. Previous models have been either too weak (lacking general recursive types and first-class function-pointers), too complex (requiring machine-checkable proofs of large bodies of computability theory), or not obviously applicable to von Neumann machines. Our new model is strong, simple, and works either in λ-calculus or on Pentiums.

Functional Big-Step Semantics

“…We could omit these rules if we do not need or want to distinguish divergence from getting stuck, and this is often done with big-step semantics. 2 However, for the purposes of this paper, we are primarily interested in the (many) situations where the distinction is important -that is where the functional big-step approach has the largest benefit.…”

“…The technique of step-indexed logical relations [2] supports reasoning about programs that have recursive types, higher-order state, or other features that introduce aspects of circularity into a language's semantics [1,12]. The soundness of these relations is usually proved with respect to a small-step semantics, because the length of a small-step trace can be used to make the relation well-founded when following the structure of the language's cyclic constructs (e.g., when following a pointer cycle in the heap or unfolding a recursive type).…”

Parametricity, type equality, and higher-order polymorphism

“…If we wished to use these results to reason about Haskell implementations of gcast, we must extend our model to include more-in particular, general recursion and recursive types (Melliès & Vouillon, 2005;Johann & Voigtländer, 2004;Appel & McAllester, 2001;Ahmed, 2006b;Crary & Harper, 2007a). We believe that the techniques developed here are independent of those for advanced language features.…”

Construction of a Semantic Model for a Typed Assembly Language

“…Appel and Felty [8] applied this idea to PCC and gave a semantic model to types and machine instructions in higher-order logic. In the following years, semantic models of types have been extended to include recursive types [9] and mutable references [10]. With these models, it is possible to reason locally about types and operations on values, but unfortunately no model has been provided to typing judgments such as Ψ; ∆; Γ jmp r and no method is provided to construct the safety proof for an entire program.…”

“…The material in these three sections has been described by other papers [11,6,9] as part of the foundational PCC project; we briefly sketch them to set up a framework within which our proof method can be formally presented in section 5.…”