18 research outputs found
Strategic Issues, Problems and Challenges in Inductive Theorem Proving
Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP
On Irrelevance and Algorithmic Equality in Predicative Type Theory
Dependently typed programs contain an excessive amount of static terms which
are necessary to please the type checker but irrelevant for computation. To
separate static and dynamic code, several static analyses and type systems have
been put forward. We consider Pfenning's type theory with irrelevant
quantification which is compatible with a type-based notion of equality that
respects eta-laws. We extend Pfenning's theory to universes and large
eliminations and develop its meta-theory. Subject reduction, normalization and
consistency are obtained by a Kripke model over the typed equality judgement.
Finally, a type-directed equality algorithm is described whose completeness is
proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author,
titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement
A Saturation-Based Unification Algorithm for Higher-Order Rational Patterns
Higher-order unification has been shown to be undecidable. Miller discovered
the pattern fragment and subsequently showed that higher-order pattern
unification is decidable and has most general unifiers. We extend the algorithm
to higher-order rational terms (a.k.a. regular B\"{o}hm trees, a form of cyclic
-terms) and show that pattern unification on higher-order rational
terms is decidable and has most general unifiers. We prove the soundness and
completeness of the algorithm
Covering All the Bases: Type-Based Verification of Test Input Generators
Test input generators are an important part of property-based testing (PBT)
frameworks. Because PBT is intended to test deep semantic and structural
properties of a program, the outputs produced by these generators can be
complex data structures, constrained to satisfy properties the developer
believes is most relevant to testing the function of interest. An important
feature expected of these generators is that they be capable of producing all
acceptable elements that satisfy the function's input type and
generator-provided constraints. However, it is not readily apparent how we
might validate whether a particular generator's output satisfies this coverage
requirement. Typically, developers must rely on manual inspection and
post-mortem analysis of test runs to determine if the generator is providing
sufficient coverage; these approaches are error-prone and difficult to scale as
generators become more complex. To address this important concern, we present a
new refinement type-based verification procedure for validating the coverage
provided by input test generators, based on a novel interpretation of types
that embeds ``must-style'' underapproximate reasoning principles as a
fundamental part of the type system. The types associated with expressions now
capture the set of values guaranteed to be produced by the expression, rather
than the typical formulation that uses types to represent the set of values an
expression may produce. Beyond formalizing the notion of coverage types in the
context of a rich core language with higher-order procedures and inductive
datatypes, we also present a detailed evaluation study to justify the utility
of our ideas
Automated Deduction – CADE 28
This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions
Certificates for decision problems in temporal logic using context-based tableaux and sequent calculi.
115 p.Esta tesis trata de resolver problemas de Satisfactibilidad y Model Checking, aportando certificados del resultado. En ella, se trabaja con tres lógicas temporales: Propositional Linear Temporal Logic (PLTL), Computation Tree Logic (CTL) y Extended Computation Tree Logic (ECTL). Primero se presenta el trabajo realizado sobre Certified Satisfiability. Ahí se muestra una adaptación del ya existente método dual de tableaux y secuentes basados en contexto para satisfactibilidad de fórmulas PLTL en Negation Normal Form. Se ha trabajado la generación de certificados en el caso en el que las fórmulas son insactisfactibles. Por último, se aporta una prueba de soundness del método. Segundo, se ha optimizado con Sat Solvers el método de Certified Satisfiability para el contexto de Certified Model Checking. Se aportan varios ejemplos de sistemas y propiedades. Tercero, se ha creado un nuevo método dual de tableaux y secuentes basados en contexto para realizar Certified Satisfiability para fórmulas CTL yECTL. Se presenta el método y un algoritmo que genera tanto el modelo en el caso de que las fórmulas son satisfactibles como la prueba en el caso en que no lo sean. Por último, se presenta una implementación del método para CTL y una experimentación comparando el método propuesto con otro método de similares características
Foundations of Security Analysis and Design III, FOSAD 2004/2005- Tutorial Lectures
he increasing relevance of security to real-life applications, such as electronic commerce and Internet banking, is attested by the fast-growing number of research groups, events, conferences, and summer schools that address the study of foundations for the analysis and the design of security aspects. This book presents thoroughly revised versions of eight tutorial lectures given by leading researchers during two International Schools on Foundations of Security Analysis and Design, FOSAD 2004/2005, held in Bertinoro, Italy, in September 2004 and September 2005. The lectures are devoted to: Justifying a Dolev-Yao Model under Active Attacks, Model-based Security Engineering with UML, Physical Security and Side-Channel Attacks, Static Analysis of Authentication, Formal Methods for Smartcard Security, Privacy-Preserving Database Systems, Intrusion Detection, Security and Trust Requirements Engineering