Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation

We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step semantics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and equivalent nondeterministic denotational semantics (with alternative powerdomains to the Egli-Milner and Smyth constructions), D. Scott’s deterministic denotational semantics, the generalized and Dijkstra’s conservative/liberal predicate transformer semantics, the generalized/total and Hoare’s partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these semantics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarsk

Interaction Trees: Representing Recursive and Impure Programs in Coq

"Interaction trees" (ITrees) are a general-purpose data structure for representing the behaviors of recursive programs that interact with their environments. A coinductive variant of "free monads," ITrees are built out of uninterpreted events and their continuations. They support compositional construction of interpreters from "event handlers", which give meaning to events by defining their semantics as monadic actions. ITrees are expressive enough to represent impure and potentially nonterminating, mutually recursive computations, while admitting a rich equational theory of equivalence up to weak bisimulation. In contrast to other approaches such as relationally specified operational semantics, ITrees are executable via code extraction, making them suitable for debugging, testing, and implementing software artifacts that are amenable to formal verification. We have implemented ITrees and their associated theory as a Coq library, mechanizing classic domain- and category-theoretic results about program semantics, iteration, monadic structures, and equational reasoning. Although the internals of the library rely heavily on coinductive proofs, the interface hides these details so that clients can use and reason about ITrees without explicit use of Coq's coinduction tactics. To showcase the utility of our theory, we prove the termination-sensitive correctness of a compiler from a simple imperative source language to an assembly-like target whose meanings are given in an ITree-based denotational semantics. Unlike previous results using operational techniques, our bisimulation proof follows straightforwardly by structural induction and elementary rewriting via an equational theory of combinators for control-flow graphs.Comment: 28 pages, 4 pages references, published at POPL 202

Extending the Calculus of Constructions with Tarski's fix-point theorem

We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to potentially non-terminating functions. This is only possible if we extend the logical framework by adding the axioms that correspond to classical logic. We claim that the extended framework makes it possible to reason about terminating and non-terminating computations and we show that common facilities of the calculus of inductive construction, like program extraction can be extended to also handle the new functions

Formal Analysis of Concurrent Programs

In this thesis, extensions of Kleene algebras are used to develop algebras for rely-guarantee style reasoning about concurrent programs. In addition to these algebras, detailed denotational models are implemented in the interactive theorem prover Isabelle/HOL. Formal soundness proofs link the algebras to their models. This follows a general algebraic approach for developing correct by construction verification tools within Isabelle. In this approach, algebras provide inference rules and abstract principles for reasoning about the control flow of programs, while the concrete models provide laws for reasoning about data flow. This yields a rapid, lightweight approach for the construction of verification and refinement tools. These tools are used to construct a popular example from the literature, via refinement, within the context of a general-purpose interactive theorem proving environment

Emerging trends proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics: TPHOLs 2004

technical reportThis volume constitutes the proceedings of the Emerging Trends track of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004) held September 14-17, 2004 in Park City, Utah, USA. The TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research cate- gory, each of which was refereed by at least 3 reviewers selected by the program committee. Of these submissions, 21 were accepted for presentation at the con- ference and publication in volume 3223 of Springer?s Lecture Notes in Computer Science series. In keeping with longstanding tradition, TPHOLs 2004 also offered a venue for the presentation of work in progress, where researchers invite discussion by means of a brief introductory talk and then discuss their work at a poster session. The work-in-progress papers are held in this volume, which is published as a 2004 technical report of the School of Computing at the University of Utah

Mechanized Reasoning About how Using Functional Programs And Embeddings

Embedding describes the process of encoding a program\u27s syntax and/or semantics in another language---typically a theorem prover in the context of mechanized reasoning. Among different embedding styles, deep embeddings are generally preferred as they enable the most faithful modeling of the original language. However, deep embeddings are also the most complex, and working with them requires additional effort. In light of that, this dissertation aims to draw more attention to alternative styles, namely shallow and mixed embeddings, by studying their use in mechanized reasoning about programs\u27 properties that are related to how . More specifically, I present a simple shallow embedding for reasoning about computation costs of lazy programs, and a class of mixed embeddings that are useful for reasoning about properties of general computation patterns in effectful programs. I show the usefulness of these embedding styles with examples based on real-world applications