Formal verification in Coq of program properties involving the global state effect

The syntax of an imperative language does not mention explicitly the state, while its denotational semantics has to mention it. In this paper we present a framework for the verification in Coq of properties of programs manipulating the global state effect. These properties are expressed in a proof system which is close to the syntax, as in effect systems, in the sense that the state does not appear explicitly in the type of expressions which manipulate it. Rather, the state appears via decorations added to terms and to equations. In this system, proofs of programs thus present two aspects: properties can be verified {\em up to effects} or the effects can be taken into account. The design of our Coq library consequently reflects these two aspects: our framework is centered around the construction of two inductive and dependent types, one for terms up to effects and one for the manipulation of decorations

Second-Order Algebraic Theories

Fiore and Hur recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding

Initial Semantics for Reduction Rules

We give an algebraic characterization of the syntax and operational semantics of a class of simply-typed languages, such as the language PCF: we characterize simply-typed syntax with variable binding and equipped with reduction rules via a universal property, namely as the initial object of some category of models. For this purpose, we employ techniques developed in two previous works: in the first work we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In the second work we characterize untyped syntax with reduction rules as initial object in a category of models. In the present work, we combine the techniques used earlier in order to characterize simply-typed syntax with reduction rules as initial object in a category. The universal property yields an operator which allows to specify translations---that are semantically faithful by construction---between languages over possibly different sets of types. As an example, we upgrade a translation from PCF to the untyped lambda calculus, given in previous work, to account for reduction in the source and target. Specifically, we specify a reduction semantics in the source and target language through suitable rules. By equipping the untyped lambda calculus with the structure of a model of PCF, initiality yields a translation from PCF to the lambda calculus, that is faithful with respect to the reduction semantics specified by the rules. This paper is an extended version of an article published in the proceedings of WoLLIC 2012.Comment: Extended version of arXiv:1206.4547, proves a variant of a result of PhD thesis arXiv:1206.455

Heterogeneous substitution systems revisited

Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is made up from least or greatest fixed points. We extend this work in two directions: we continue the analysis by creating more categorical structure, in particular by organizing substitution systems into a category and studying its properties, and we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover.Comment: 24 page

Extended Initiality for Typed Abstract Syntax

Initial Semantics aims at interpreting the syntax associated to a signature as the initial object of some category of 'models', yielding induction and recursion principles for abstract syntax. Zsid\'o proves an initiality result for simply-typed syntax: given a signature S, the abstract syntax associated to S constitutes the initial object in a category of models of S in monads. However, the iteration principle her theorem provides only accounts for translations between two languages over a fixed set of object types. We generalize Zsid\'o's notion of model such that object types may vary, yielding a larger category, while preserving initiality of the syntax therein. Thus we obtain an extended initiality theorem for typed abstract syntax, in which translations between terms over different types can be specified via the associated category-theoretic iteration operator as an initial morphism. Our definitions ensure that translations specified via initiality are type-safe, i.e. compatible with the typing in the source and target language in the obvious sense. Our main example is given via the propositions-as-types paradigm: we specify propositions and inference rules of classical and intuitionistic propositional logics through their respective typed signatures. Afterwards we use the category--theoretic iteration operator to specify a double negation translation from the former to the latter. A second example is given by the signature of PCF. For this particular case, we formalize the theorem in the proof assistant Coq. Afterwards we specify, via the category-theoretic iteration operator, translations from PCF to the untyped lambda calculus