Nominal system T

Lecture Notes in Computer Science, 2004

This work describes the proof and uses of a theorem allowing definition of recursive functions over the type of λ-calculus terms, where terms with bound variables are identified up to α-equivalence. The theorem embodies what is effectively a principle of primitive recursion, and the analogues of this theorem for other types with binders are clear. The theorem's side-conditions require that the putative definition be well-behaved with respect to fresh name generation and name permutation. A number of examples over the type of λ-calculus terms illustrate the use of the new principle.

2002

We introduce a language based upon lambda calculus with products, coproducts and strictly positive inductive types that allows the definition of recursive terms. We present the implementation (foetus) of a syntactical check that ensures that all such terms are structurally recursive, i.e., recursive calls appear only with arguments structurally smaller than the input parameters of terms considered. To ensure the correctness of the termination checker, we show that all structurally recursive terms are normalizing with respect to a given operational semantics. To this end, we define a semantics on all types and a structural ordering on the values in this semantics and prove that all values are accessible with regard to this ordering. Finally, we point out how to do this proof predicatively using set based operators.

\emph{Induction-recursion} is a definitional principle in Martin-L{\"o}f Type Theory defining families $(\U , \T: \U\to D) : \Fam{D}$ where $D: \Setone$ is an arbitrary fixed (large) set (which in motivating examples is chosen to be $D=\Set$), and $\U : \Set$ is defined by induction while $\T$ is simultaneously defined by recursion on $\U$; the qualifier ``simultaneously'' means here that $\U$ may depend on values of the function $\T : \U \to D$. Two equivalent\footnote{These axiomatizations are equivalent if the underlying logical framework is chosen appropriately.} axiomatizations of this situation were proposed by Dybjer-Setzer in \cite{dybjer99}\cite{dybjer03}. In both cases a (large) set of \emph{codes} $\DS\; D\; D$ (respectively $\DS'$) for inductive-recursive definitions is defined such that each code $c : \DS\; D\; D$ decodes to an endofunctor $\sem{c} : \Fam{D}\to \Fam{D}$ between categories of families whose initial algebra is the family defined by this code $c$. The authors proved the consistency of their axiomatizations be giving a set-theoretic model. These axiomatizations $\DS$ and $\DS'$ are however not the only reasonable axiomatizations of induction-recursion. There are at least two ways to come to this conclusion: the more practical one is motivated by the observation that while in the reference theory of inductive definitions it is always possible to compose (in a semantically sound way) two inductive definitions to a single new one, this seems hardly to be the case for the preexisting axiomatizations of Induction-Recursion. The second-, more conceptual observation about the existing axiomatizations of induction-recursion is that it does not contain constructors for dependent products (or powers\footnote{There is a close relationship between dependent products-, and powers of codes.}) of codes but only for dependent sums of codes. Indeed, we show that these two observations are related by characterizing compositionality of Dybjer-Setzer induction-recursion in terms of the existence of powers of codes by sets. Departing from this characterization, we define- and explore two new axiomatizations of induction-recursion satisfying the mentioned characterization and for which we prove compositionality. In the first one, this is achieved by restricting to a subsystem\footnote{By ``subsystem'' we mean here that there is a semantics-preserving translation from $\UF$ to $\DS$.} $\UF$ of $\DS$ of \emph{uniform codes} for which powers of codes exist. Consistency of this system is established by a semantics-preserving embedding into the system $\DS$. The second axiomatization $\IR$ of \emph{polynomial codes} (so called since they are based on the idea of iterating polynomials\footnote{Polynomials are also called \emph{containers} and are a formalization for inductive definitions.}) defines a system into which $\DS$ can be embedded and which contains a constructor for dependent products- (and in particular powers) of codes. While for $\UF$ the existence of a model can be obtained by embedding it into $\DS$ for which Dybjer-Setzer themselves devised a (set-theoretic) model, we cannot argue in this way for a model of $\IR$ and instead we provide a new model for the latter having almost the same set-theoretical assumptions concerning large cardinals: while Dybjer-Setzer induction-recursion can be modeled in ZFC supplemented by a Mahlo cardinal and a $0$-inaccessible, we need ZFC plus a Mahlo cardinal and a $1$-inaccessible. Since the system $\IR$ does not simply arise by adding a constructor for dependent products of codes to $\DS$ caeteris paribus, but additionally requires redefining all other constructors, the question about constructors generating the image of the inclusion $\DS\hookrightarrow \IR$ imposes itself; we approach this question by defining an intermediary system lying between $\DS$ and $\IR$ and give a translation of this intermediary system into $\DS$. This intermediary system does not only arise by removal of the constructor for dependent products of codes since this did not yet enable us to define a desired translation to $\DS$, but by introduction of an additional uniformity constraint realized by an annotation of codes by binary trees. A common feature of both new systems $\UF$ and $\IR$ is that they admit a more flexible relation between codes and their subcodes than this is the case for $\DS$: sub-$\DS$-codes of a given $\DS$-code do all have the same type while sub-$\UF$-, and sub-$\IR$-codes are not constrained in this way. This latter feature reveals a more abstract way of arriving at the conclusion that Dybjer-Setzer induction-recursion is not the most general formulation of induction-recursion: like inductive definitions can be characterized by a set of operations on sets\footnote{Theses operations are called ``stictly positive operations'' (see \cite{dybjer1996Wtypes}).}, inductive-recursive definitions are also determined by a set of operations on families (namely those represented by the set of functors defined by codes) and Dybjer-Setzer's systems did not take into account operations that change the (large) set $D$ indexing families while the new systems $\UF$ and $\IR$ do so. In line with the idea of considering induction-recursion as contributing to the pursue of the project of formalizing the meta theory of type theory in type theory itself \cite[p.1]{Dybjer96internaltype}, we return to Dybjer-Setzer's original formalization $\DS$ and provide a \emph{relationally-parametric} model for it that we articulate as a categories-with-families model in the category of reflexive graphs; categories-with-families were proposed in loc.cit. as a formalization of type theory inside type theory that is additionally well adapted to category theoretic reasoning. Relational parametricity is an established and important proof technique to establish meta-theoretic properties of type theories.

Formal Aspects of Computing, 2002

The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of 'name-abstraction' and 'fresh name' that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science.

Theoretical Computer Science, 2004

We present a generalisation of first-order unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms α-equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a "nominal" approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijn-style representations) and yet get a version of this form of substitution that respects α-equivalence and possesses good algorithmic properties. We achieve this by adapting two existing ideas. The first one is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The second one is that the unification algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of classical first-order unification problems to this setting which retains the latter's pleasant properties: unification problems involving α-equivalence and freshness are decidable; and solvable problems possess most general solutions.

2008 23rd Annual IEEE Symposium on Logic in Computer Science, 2008

Many semantical aspects of programming languages, such as their operational semantics and their type assignment calculi, are specified by describing appropriate proof systems. Recent research has identified two proof-theoretic features that allow direct, logic-based 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: that is, they do not provide the ability to define object-logic 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 benefit of the integration is that it allows recursive definitions to not just encode simple, traditional forms of atomic judgments but also to capture generic properties pertaining to such judgments. 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 of arbitrarily cascading substitutions that play a role in reducibility arguments.

ACM SIGPLAN Notices, 2011

We characterize the data type of terms with bindings, freshness and substitution, as an initial model in a suitable Horn theory. This characterization yields a convenient recursive definition principle, which we have formalized in Isabelle/HOL and employed in a series of ...

Interactive Theorem Proving, 2010

We mechanise termination and correctness for two unification algorithms, written in a recursive descent style. One computes unifiers for first order terms, the other for nominal terms (terms including α-equivalent binding structure). Both algorithms work with triangular substitutions in accumulator-passing style: taking a substitution as input, and returning an extension of that substitution on success. This style of algorithm has performance benefits and has not been mechanised previously. The algorithms use nested recursion so the termination proofs are non-trivial; the termination relation is also slightly different from usual.

Logical Methods in Computer Science, 2012

Nominal Isabelle is a definitional extension of the Isabelle/HOL theorem prover. It provides a proving infrastructure for reasoning about programming language calculi involving named bound variables (as opposed to de-Bruijn indices). In this paper we present an extension of Nominal Isabelle for dealing with general bindings, that means term constructors where multiple variables are bound at once. Such general bindings are ubiquitous in programming language research and only very poorly supported with single binders, such as lambda-abstractions. Our extension includes new definitions of alpha-equivalence and establishes automatically the reasoning infrastructure for alphaequated terms. We also prove strong induction principles that have the usual variable convention already built in.

2006

We present a sound and complete proof technique, based on syntactic logical relations, for showing contextual equivalence of expressions in a λ-calculus with recursive types and impredicative universal and existential types. Our development builds on the step-indexed PER model of recursive types presented by Appel and McAllester. We have discovered that a direct proof of transitivity of that model does not go through, leaving the “PER” status of the model in question. We show how to extend the Appel-McAllester model to obtain a logical relation that we can prove is transitive, as well as sound and complete with respect to contextual equivalence. We then augment this model to support relational reasoning in the presence of quantified types. Step-indexed relations are indexed not just by types, but also by the number of steps available for future evaluation. This stratification is essential for handling various circularities, from recursive functions, to recursive types, to impredicat...