An automata-based automatic verification environment

With the continuing growth of computer systems including safety-critical computer control systems, the need for reliable tools to help construct, analyze, and verify such systems also continues to grow. The basic motivation of this work is to build such a formal verification environment for computer-based systems. An example of such a tool is the Design Oriented Verification and Evaluation (DOVE) created by Australian Defense Science and Technology Organization. One of the advantages of DOVE is that it combines ease of use provided by a graphical user interface for describing specifications in the form of extended state machines with the rigor of proving linear temporal logic properties in a robust theorem prover, Isabelle which was developed at Cambridge University, UK, and TU Munich, Germany. A different class of examples is that of model checkers, such as SPIN and SMV. In this work, we describe our technique to increase the utility of DOVE by extending it with the capability to build systems by specifying components. This added utility is demonstrated with a concrete example from a real project to study aspects of the control unit for an infusion pump being built at the Walter Reid Army Institute of Research. Secondly, we provide a formulation of linear temporal logic (LTL) in the theorem prover Isabelle. Next, we present a formalization of a variation of the algorithm for translating LTL into Büchi automata. The original translation algorithm is presented in Gerth et al and is the basis of model checkers such as SPIN. We also provide a formal proof of the termination and correctness of this algorithm. All definitions and proofs have been done fully formally within the generic theorem prover Isabelle, which guarantees the rigor of our work and the reliability of the results obtained. Finally, we introduce the automata theoretic framework for automatic verification as our future works

Tool support for reasoning in display calculi

We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of meta-tools, that allows us to adapt the tool for a wide variety of user defined calculi

General Recursion via Coinductive Types

A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations, implementation of operational semantics, formalization of domain theory, and inductive definition of domain predicates. Here, a different solution is proposed: exploiting coinductive types to model infinite computations. To every type A we associate a type of partial elements Partial(A), coinductively generated by two constructors: the first, return(a) just returns an element a:A; the second, step(x), adds a computation step to a recursive element x:Partial(A). We show how this simple device is sufficient to formalize all recursive functions between two given types. It allows the definition of fixed points of finitary, that is, continuous, operators. We will compare this approach to different ones from the literature. Finally, we mention that the formalization, with appropriate structural maps, defines a strong monad.Comment: 28 page

Hybrid - a definitional two-level approach to reasoning with higher-order abstract syntax

Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic.We describe the theory and the practice of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many of these difficulties. It allows object logics to be represented using higher-order abstract syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of \u3bb-terms providing a definitional layer that allows the user to represent object languages using higher-order abstract syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use Hybrid in a multi-level reasoning fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly more complex object logic whose encoding is elegantly expressed using features of the new specification logic

Wave Equation Numerical Resolution: a Comprehensive Mechanized Proof of a C Program

We formally prove correct a C program that implements a numerical scheme for the resolution of the one-dimensional acoustic wave equation. Such an implementation introduces errors at several levels: the numerical scheme introduces method errors, and floating-point computations lead to round-off errors. We annotate this C program to specify both method error and round-off error. We use Frama-C to generate theorems that guarantee the soundness of the code. We discharge these theorems using SMT solvers, Gappa, and Coq. This involves a large Coq development to prove the adequacy of the C program to the numerical scheme and to bound errors. To our knowledge, this is the first time such a numerical analysis program is fully machine-checked.Comment: No. RR-7826 (2011

An institution theory of formal meta-modelling in graphically extended BNF

Meta-modelling plays an important role in model driven software development. In this paper, a graphic extension of BNF (GEBNF) is proposed to define the abstract syntax of graphic modelling languages. From a GEBNF syntax definition, a formal predicate logic language can be induced so that meta-modelling can be performed formally by specifying a predicate on the domain of syntactically valid models. In this paper, we investigate the theoretical foundation of this metamodelling approach. We formally define the semantics of GEBNF and its induced predicate logic languages, then apply Goguen and Burstall’s institution theory to prove that they form a sound and valid formal specification language for meta-modelling

Type Theories with Explicit Universe Polymorphism

The aim of this paper is to refine and extend proposals by Sozeau and Tabareau and by Voevodsky for universe polymorphism in type theory. In those systems judgments can depend on explicit constraints between universe levels. We here present a system where we also have products indexed by universe levels and by constraints. Our theory has judgments for internal universe levels, built up from level variables by a successor operation and a binary supremum operation, and also judgments for equality of universe levels.Comment: This paper was presented at Types'2022 and has been submitted to its postconference proceeding