73 research outputs found
Cut Elimination for a Logic with Induction and Co-induction
Proof search has been used to specify a wide range of computation systems. In
order to build a framework for reasoning about such specifications, we make use
of a sequent calculus involving induction and co-induction. These proof
principles are based on a proof theoretic (rather than set-theoretic) notion of
definition. Definitions are akin to logic programs, where the left and right
rules for defined atoms allow one to view theories as "closed" or defining
fixed points. The use of definitions and free equality makes it possible to
reason intentionally about syntax. We add in a consistent way rules for pre and
post fixed points, thus allowing the user to reason inductively and
co-inductively about properties of computational system making full use of
higher-order abstract syntax. Consistency is guaranteed via cut-elimination,
where we give the first, to our knowledge, cut-elimination procedure in the
presence of general inductive and co-inductive definitions.Comment: 42 pages, submitted to the Journal of Applied Logi
An Open Challenge Problem Repository for Systems Supporting Binders
A variety of logical frameworks support the use of higher-order abstract
syntax in representing formal systems; however, each system has its own set of
benchmarks. Even worse, general proof assistants that provide special libraries
for dealing with binders offer a very limited evaluation of such libraries, and
the examples given often do not exercise and stress-test key aspects that arise
in the presence of binders. In this paper we design an open repository ORBI
(Open challenge problem Repository for systems supporting reasoning with
BInders). We believe the field of reasoning about languages with binders has
matured, and a common set of benchmarks provides an important basis for
evaluation and qualitative comparison of different systems and libraries that
support binders, and it will help to advance the field.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
Scheduling of an aircraft fleet
Scheduling is the task of assigning resources to operations. When the resources are mobile vehicles, they describe routes through the served stations. To emphasize such aspect, this problem is usually referred to as the routing problem. In particular, if vehicles are aircraft and stations are airports, the problem is known as aircraft routing. This paper describes the solution to such a problem developed in OMAR (Operative Management of Aircraft Routing), a system implemented by Bull HN for Alitalia. In our approach, aircraft routing is viewed as a Constraint Satisfaction Problem. The solving strategy combines network consistency and tree search techniques
Two Applications of Logic Programming to Coq
The logic programming paradigm provides a flexible setting for representing, manipulating, checking, and elaborating proof structures. This is particularly true when the logic programming language allows for bindings in terms and proofs. In this paper, we make use of two recent innovations at the intersection of logic programming and proof checking. One of these is the foundational proof certificate (FPC) framework which provides a flexible means of defining the semantics of a range of proof structures for classical and intuitionistic logic. A second innovation is the recently released Coq-Elpi plugin for Coq in which the Elpi implementation of ?Prolog can send and retrieve information to and from the Coq kernel. We illustrate the use of both this Coq plugin and FPCs with two example applications. First, we implement an FPC-driven sequent calculus for a fragment of the Calculus of Inductive Constructions and we package it into a tactic to perform property-based testing of inductive types corresponding to Horn clauses. Second, we implement in Elpi a proof checker for first-order intuitionistic logic and demonstrate how proof certificates can be supplied by external (to Coq) provers and then elaborated into the fully detailed proof terms that can be checked by the Coq kernel
An Improved Implementation and Abstract Interface for Hybrid
Hybrid is a formal theory implemented in Isabelle/HOL that provides an
interface for representing and reasoning about object languages using
higher-order abstract syntax (HOAS). This interface is built around an HOAS
variable-binding operator that is constructed definitionally from a de Bruijn
index representation. In this paper we make a variety of improvements to
Hybrid, culminating in an abstract interface that on one hand makes Hybrid a
more mathematically satisfactory theory, and on the other hand has important
practical benefits. We start with a modification of Hybrid's type of terms that
better hides its implementation in terms of de Bruijn indices, by excluding at
the type level terms with dangling indices. We present an improved set of
definitions, and a series of new lemmas that provide a complete
characterization of Hybrid's primitives in terms of properties stated at the
HOAS level. Benefits of this new package include a new proof of adequacy and
improvements to reasoning about object logics. Such proofs are carried out at
the higher level with no involvement of the lower level de Bruijn syntax.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
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