839 research outputs found
ASMs and Operational Algorithmic Completeness of Lambda Calculus
We show that lambda calculus is a computation model which can step by step
simulate any sequential deterministic algorithm for any computable function
over integers or words or any datatype. More formally, given an algorithm above
a family of computable functions (taken as primitive tools, i.e., kind of
oracle functions for the algorithm), for every constant K big enough, each
computation step of the algorithm can be simulated by exactly K successive
reductions in a natural extension of lambda calculus with constants for
functions in the above considered family. The proof is based on a fixed point
technique in lambda calculus and on Gurevich sequential Thesis which allows to
identify sequential deterministic algorithms with Abstract State Machines. This
extends to algorithms for partial computable functions in such a way that
finite computations ending with exceptions are associated to finite reductions
leading to terms with a particular very simple feature.Comment: 37 page
A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems
Basic proof-search tactics in logic and type theory can be seen as the
root-first applications of rules in an appropriate sequent calculus, preferably
without the redundancies generated by permutation of rules. This paper
addresses the issues of defining such sequent calculi for Pure Type Systems
(PTS, which were originally presented in natural deduction style) and then
organizing their rules for effective proof-search. We introduce the idea of
Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the
syntax of a permutation-free sequent calculus for propositional logic due to
Herbelin, which is strongly related to natural deduction and already well
adapted to proof-search. The operational semantics is adapted from Herbelin's
and is defined by a system of local rewrite rules as in cut-elimination, using
explicit substitutions. We prove confluence for this system. Restricting our
attention to PTSC, a type system for the ground terms of this system, we obtain
the Subject Reduction property and show that each PTSC is logically equivalent
to its corresponding PTS, and the former is strongly normalising iff the latter
is. We show how to make the logical rules of PTSC into a syntax-directed system
PS for proof-search, by incorporating the conversion rules as in
syntax-directed presentations of the PTS rules for type-checking. Finally, we
consider how to use the explicitly scoped meta-variables of PTSCalpha to
represent partial proof-terms, and use them to analyse interactive proof
construction. This sets up a framework PE in which we are able to study
proof-search strategies, type inhabitant enumeration and (higher-order)
unification
Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker
Since the proof of the four color theorem in 1976, computer-generated proofs
have become a reality in mathematics and computer science. During the last
decade, we have seen formal proofs using verified proof assistants being used
to verify the validity of such proofs.
In this paper, we describe a formalized theory of size-optimal sorting
networks. From this formalization we extract a certified checker that
successfully verifies computer-generated proofs of optimality on up to 8
inputs. The checker relies on an untrusted oracle to shortcut the search for
witnesses on more than 1.6 million NP-complete subproblems.Comment: IMADA-preprint-c
Ontology: A Linked Data Hub for Mathematics
In this paper, we present an ontology of mathematical knowledge concepts that
covers a wide range of the fields of mathematics and introduces a balanced
representation between comprehensive and sensible models. We demonstrate the
applications of this representation in information extraction, semantic search,
and education. We argue that the ontology can be a core of future integration
of math-aware data sets in the Web of Data and, therefore, provide mappings
onto relevant datasets, such as DBpedia and ScienceWISE.Comment: 15 pages, 6 images, 1 table, Knowledge Engineering and the Semantic
Web - 5th International Conferenc
SCC: A Service Centered Calculus
We seek for a small set of primitives that might serve as a basis for formalising and programming service oriented applications over global computers. As an outcome of this study we introduce here SCC, a process calculus that features explicit notions of service definition, service invocation and session handling. Our proposal has been influenced by Orc, a programming model for structured orchestration of services, but the SCC’s session handling mechanism allows for the definition of structured interaction protocols, more complex than the basic request-response provided by Orc. We present syntax and operational semantics of SCC and a number of simple but nontrivial programming examples that demonstrate flexibility of the chosen set of primitives. A few encodings are also provided to relate our proposal with existing ones
Hierarchical models for service-oriented systems
We present our approach to the denotation and representation of hierarchical graphs: a suitable algebra of hierarchical graphs and two domains of interpretations. Each domain of interpretation focuses on a particular perspective of the graph hierarchy: the top view (nested boxes) is based on a notion of embedded graphs while the side view (tree hierarchy) is based on gs-graphs. Our algebra can be understood as a high-level language for describing such graphical models, which are well suited for defining graphical representations of service-oriented systems where nesting (e.g. sessions, transactions, locations) and linking (e.g. shared channels, resources, names) are key aspects
Call-by-value non-determinism in a linear logic type discipline
We consider the call-by-value lambda-calculus extended with a may-convergent
non-deterministic choice and a must-convergent parallel composition. Inspired
by recent works on the relational semantics of linear logic and non-idempotent
intersection types, we endow this calculus with a type system based on the
so-called Girard's second translation of intuitionistic logic into linear
logic. We prove that a term is typable if and only if it is converging, and
that its typing tree carries enough information to give a bound on the length
of its lazy call-by-value reduction. Moreover, when the typing tree is minimal,
such a bound becomes the exact length of the reduction
A Vernacular for Coherent Logic
We propose a simple, yet expressive proof representation from which proofs
for different proof assistants can easily be generated. The representation uses
only a few inference rules and is based on a frag- ment of first-order logic
called coherent logic. Coherent logic has been recognized by a number of
researchers as a suitable logic for many ev- eryday mathematical developments.
The proposed proof representation is accompanied by a corresponding XML format
and by a suite of XSL transformations for generating formal proofs for
Isabelle/Isar and Coq, as well as proofs expressed in a natural language form
(formatted in LATEX or in HTML). Also, our automated theorem prover for
coherent logic exports proofs in the proposed XML format. All tools are
publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014
- …