Extending Nunchaku to Dependent Type Theory

Nunchaku is a new higher-order counterexample generator based on a sequence of transformations from polymorphic higher-order logic to first-order logic. Unlike its predecessor Nitpick for Isabelle, it is designed as a stand-alone tool, with frontends for various proof assistants. In this short paper, we present some ideas to extend Nunchaku with partial support for dependent types and type classes, to make frontends for Coq and other systems based on dependent type theory more useful.Comment: In Proceedings HaTT 2016, arXiv:1606.0542

Categories with families and first-order logic with dependent sorts

First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-L\"of type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved for DFOL. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle.Comment: 83 page

Kripke Semantics for Martin-L\"of's Extensional Type Theory

It is well-known that simple type theory is complete with respect to non-standard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of interpretations of dependent type theory, i.e., to show that the interpretations of equal expressions are indeed equal. Several classes of models have been used to remedy this problem. We contribute to this investigation by giving a semantics that is standard, coherent, and sufficiently general for completeness while remaining relatively easy to compute with. Our models interpret types of Martin-L\"of's extensional dependent type theory as sets indexed over posets or, equivalently, as fibrations over posets. This semantics can be seen as a generalization to dependent type theory of the interpretation of intuitionistic first-order logic in Kripke models. This yields a simple coherent model theory, with respect to which simple and dependent type theory are sound and complete

A Declarative Foundation of λProlog with Equality

We build general model-theoretic semantics for higher-order logic programming languages. Usual semantics for first-order logic is two-level: i.e., at a lower level we define a domain of individuals, and then, we define satisfaction of formulas with respect to this domain. In a higher-order logic which includes the propositional type in its primitive set of types, the definition of satisfaction of formulas is mutually recursive with the process of evaluation of terms. As result of this in higher-order logic it is extremely difficult to define an effective semantics. For example to define T p operator for logic program P, we need a fixed domain without regard to interpretations. In usual semantics for higher-order logic, domain is dependent on interpretations. We overcome this problem and argue that our semantics provides a more suitable declarative basis for higher-order logic programming than the usual general model semantics. We develop a fix point semantics based on our model. We also show that a quotient of the domain of our model can be the domain of a model for higher-order logic programs with equality

A Semantics of ? into Dedukti

? is a semantical framework for formally describing the semantics of programming languages thanks to a BNF grammar and rewriting rules on configurations. It is also an environment that offers various tools to help programming with the languages specified in the formalism. For example, it is possible to execute programs thanks to the generated interpreter, or to check their properties thanks to the provided automatic theorem prover called the KProver. ? is based on la Matching Logic, a first-order logic with an application and fixed-point operators, extended with symbols to encode equality, typing and rewriting. This specific la Matching Logic theory is called Kore. Dedukti is a logical framework having for main goal the interoperability of proofs between different formal proof tools. Several translators to Dedukti exist or are under development, in order to automatically translate formalizations written, for instance, in Coq or PVS. Dedukti is based on the ??-calculus modulo theory, a ?-calculus with dependent types and extended with a primitive notion of computation defined by rewriting rules. The flexibility of this logical framework allows to encode many theories ranging from first-order logic to the Calculus of Constructions. In this article, we present a paper formalization of the translation from ? into Kore, and a paper formalization and an automatic translation tool, called KaMeLo, from Kore to Dedukti in order to execute programs in Dedukti

Using Histories to Implement Atomic Objects

In this paper we describe an approach of implementing atomicity. Atomicity requires that computations appear to be all-or-nothing and executed in a serialization order. The approach we describe has three characteristics. First, it utilizes the semantics of an application to improve concurrency. Second, it reduces the complexity of application-dependent synchronization code by analyzing the process of writing it. In fact, the process can be automated with logic programming. Third, our approach hides the protocol used to arrive at a serialization order from the applications. As a result, different protocols can be used without affecting the applications. Our approach uses a history tree abstraction. The history tree captures the ordering relationship among concurrent computations. By determining what types of computations exist in the history tree and their parameters, a computation can determine whether it can proceed

Reasoning about functional programs by combining interactive and automatic proofs

We propose a new approach to computer-assisted verification of lazy functional programs where functions can be defined by general recursion. We work in first-order theories of functional programs which are obtained by translating Dybjer's programming logic (Dybjer, P. [1985]. Program Verification in a Logical Theory of Constructions. In: Functional Programming Languages and Computer Architecture. Ed. by Jouannaud, J. P. Vol. 201. Lecture Notes in Computer Science. Springer, pp. 334–349) into a first-order theory, and by extending this programming logic with new (co-)inductive predicates. Rather than building a special purpose system, we formalise our theories in Agda, a proof assistant for dependent type theory which can be used as a generic theorem prover. Agda provides support for interactive reasoning by representing first-order theories using the propositions-as-types principle. Further support is provided by off-the-shelf automatic theorem provers for first-order-logic called by a Haskell program that translates our Agda representations of first-order formulae into the TPTP language understood by the provers. We show some examples where we combine interactive and automatic reasoning, covering both proofs by induction and co-induction. The examples include functions defined by structural recursion, simple general recursion, nested recursion, higher-order recursion, guarded and unguarded co-recursion.Proponemos un nuevo enfoque a la verificación asistida por computador de programas funcionales perezosos, en los cuales las funciones pueden ser definidas por recursión general. Empleamos teorías de primer orden para programas funcionales las cuales fueron obtenidas de traducir la lógica para la programación de Dybjer (Dybjer, P. [1985]. Program Verification in a Logical Theory of Constructions. En: Functional Programming Languages and Computer Architecture. Ed. by Jouannaud, J.-P. Vol. 201. Lecture Notes in Computer Science. Springer, págs. 334–349) a una teoría de primer orden, y de extender esta lógica para la programación con nuevos predicados (co-)inductivos. En lugar de construir un sistema para formalizar nuestras teorías, formalizamos éstas en Agda, un asistente de pruebas para teoría de tipos dependientes que puede ser usado como un demostrador de teoremas genérico. Agda proporciona soporte para el razonamiento interactivo representando las teorías de primer orden mediante el principio de propositions-as-types. Se obtiene soporte adicional mediante demostradores automáticos de teoremas genéricos para lógica de primer orden, los cuales son llamados por un programa desarrollado en Haskell, que traslada nuestra representación en Agda de las fórmulas de primer orden al lenguaje TPTP entendido por los demostradores automáticos. Mostramos ejemplos de combinación de razonamiento interactivo y automático en pruebas por inducción y por co-inducción. Nuestros ejemplos incluyen funciones definidas por recursión estructural, recursión general simple, recursión anidada, recursión de orden superior y co-recursión

Margrave: An Improved Analyzer for Access-Control and Configuration Policies

As our society grows more dependent on digital systems, policies that regulate access to electronic resources are becoming more common. However, such policies are notoriously difficult to configure properly, even for trained professionals. An incorrectly written access-control policy can result in inconvenience, financial damage, or even physical danger. The difficulty is more pronounced when multiple types of policy interact with each other, such as in routers on a network. This thesis presents a policy-analysis tool called Margrave. Given a query about a set of policies, Margrave returns a complete collection of scenarios that satisfy the query. Since the query language allows multiple policies to be compared, Margrave can be used to obtain an exhaustive list of the consequences of a seemingly innocent policy change. This feature gives policy authors the benefits of formal analysis without requiring that they state any formal properties about their policies. Our query language is equivalent to order-sorted first-order logic (OSL). Therefore our scenario-finding approach is, in general, only complete up to a user-provided bound on scenario size. To mitigate this limitation, we identify a class of OSL that we call Order-Sorted Effectively Propositional Logic (OS-EPL). We give a linear-time algorithm for testing membership in OS-EPL. Sentences in this class have the Finite Model Property, and thus Margrave\u27s results on such queries are complete without user intervention

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S