499 research outputs found
Initial Algebra Semantics for Cyclic Sharing Tree Structures
Terms are a concise representation of tree structures. Since they can be
naturally defined by an inductive type, they offer data structures in
functional programming and mechanised reasoning with useful principles such as
structural induction and structural recursion. However, for graphs or
"tree-like" structures - trees involving cycles and sharing - it remains
unclear what kind of inductive structures exists and how we can faithfully
assign a term representation of them. In this paper we propose a simple term
syntax for cyclic sharing structures that admits structural induction and
recursion principles. We show that the obtained syntax is directly usable in
the functional language Haskell and the proof assistant Agda, as well as
ordinary data structures such as lists and trees. To achieve this goal, we use
a categorical approach to initial algebra semantics in a presheaf category.
That approach follows the line of Fiore, Plotkin and Turi's models of abstract
syntax with variable binding
From nominal to higher-order rewriting and back again
We present a translation function from nominal rewriting systems (NRSs) to
combinatory reduction systems (CRSs), transforming closed nominal rules and
ground nominal terms to CRSs rules and terms, respectively, while preserving
the rewriting relation. We also provide a reduction-preserving translation in
the other direction, from CRSs to NRSs, improving over a previously defined
translation. These tools, together with existing translations between CRSs and
other higher-order rewriting formalisms, open up the path for a transfer of
results between higher-order and nominal rewriting. In particular, techniques
and properties of the rewriting relation, such as termination, can be exported
from one formalism to the other.Comment: 41 pages, journa
An Algebra of Hierarchical Graphs and its Application to Structural Encoding
We define an algebraic theory of hierarchical graphs, whose axioms
characterise graph isomorphism: two terms are equated exactly when
they represent the same graph. Our algebra can be understood as
a high-level language for describing graphs with a node-sharing, embedding
structure, and it is then well suited for defining graphical
representations of software models where nesting and linking are key
aspects. In particular, we propose the use of our graph formalism as a
convenient way to describe configurations in process calculi equipped
with inherently hierarchical features such as sessions, locations, transactions,
membranes or ambients. The graph syntax can be seen as an
intermediate representation language, that facilitates the encodings of
algebraic specifications, since it provides primitives for nesting, name
restriction and parallel composition. In addition, proving soundness
and correctness of an encoding (i.e. proving that structurally equivalent
processes are mapped to isomorphic graphs) becomes easier as it can
be done by induction over the graph syntax
Mechanising syntax with binders in Coq
Mechanising binders in general-purpose proof assistants such as Coq is cumbersome and difficult. Yet binders, substitutions, and instantiation of terms with substitutions are a critical ingredient of many programming languages. Any practicable mechanisation of the meta-theory of the latter hence requires a lean formalisation of the former. We investigate the topic from three angles: First, we realise formal systems with binders based on both pure and scoped de Bruijn algebras together with basic syntactic rewriting lemmas and automation. We automate this process in a compiler called Autosubst; our final tool supports many-sorted, variadic, and modular syntax. Second, we justify our choice of realisation and mechanise a proof of convergence of the sigma calculus, a calculus of explicit substitutions that is complete for equality of the de Bruijn algebra corresponding to the lambda calculus. Third, to demonstrate the practical usefulness of our approach, we provide concise, transparent, and accessible mechanised proofs for a variety of case studies refined to de Bruijn substitutions.Die Mechanisierung von Bindern in universellen Beweisassistenten wie Coq ist arbeitsaufwändig und schwierig. Binder, Substitutionen und die Instantiierung von Substitutionen sind jedoch kritischer Bestandteil
vieler Programmiersprachen. Deshalb setzt eine praktikable Mechanisierung der Metatheorie von Programmiersprachen eine elegante Formalisierung von Bindern voraus. Wir nähern uns dem Thema aus drei Richtungen an:
Zuerst realisieren wir formale Systeme mit Bindern mit Hilfe von reinen und indizierten de Bruijn Algebren, zusammen mit grundlegenden syntaktischen Gleichungen und Automatisierung. Wir automatisieren diesen Prozess in
einem Kompilierer namens Autosubst. Unser finaler Kompilierer unterstützt Sortenlogik, variadische Syntax und modulare Syntax. Zweitens rechtfertigen wir unsere Repräsentation und mechanisieren einen Beweis der Konvergenz
des SP-Kalküls, einem Kalkül expliziter Substitutionen der bezüglich der Gleichheit der puren de Bruijn Algebra des -Kalküls vollständig ist. Drittens entwickeln wir kurze, transparente und leicht zugängliche mechanisierte
Beweise für diverse Fallstudien, die wir an de Bruijn Substitutionen angepasst haben. Wir weisen so die praktische Anwendbarkeit unseres Ansatzes nach
A Universal Machine for Biform Theory Graphs
Broadly speaking, there are two kinds of semantics-aware assistant systems
for mathematics: proof assistants express the semantic in logic and emphasize
deduction, and computer algebra systems express the semantics in programming
languages and emphasize computation. Combining the complementary strengths of
both approaches while mending their complementary weaknesses has been an
important goal of the mechanized mathematics community for some time. We pick
up on the idea of biform theories and interpret it in the MMTt/OMDoc framework
which introduced the foundations-as-theories approach, and can thus represent
both logics and programming languages as theories. This yields a formal,
modular framework of biform theory graphs which mixes specifications and
implementations sharing the module system and typing information. We present
automated knowledge management work flows that interface to existing
specification/programming tools and enable an OpenMath Machine, that
operationalizes biform theories, evaluating expressions by exhaustively
applying the implementations of the respective operators. We evaluate the new
biform framework by adding implementations to the OpenMath standard content
dictionaries.Comment: Conferences on Intelligent Computer Mathematics, CICM 2013 The final
publication is available at http://link.springer.com
On the mathematical synthesis of equational logics
We provide a mathematical theory and methodology for synthesising equational
logics from algebraic metatheories. We illustrate our methodology by means of
two applications: a rational reconstruction of Birkhoff's Equational Logic and
a new equational logic for reasoning about algebraic structure with
name-binding operators.Comment: Final version for publication in Logical Methods in Computer Scienc
Bisimilarity of Open Terms
The standard way of lifting a binary relation, R, from closed terms of an algebra to open terms is to define its closed-instance extension, R_{ci}, which holds for a given pair of open terms if and only if R holds for all their closed instantiations. In this paper, we study alternatives for the case of (strong) bisimulation: we define semantic models for open terms, so-called conditional transition systems, and define bisimulation directly on those models. It turns out that this can be done in at least two different ways, giving rise to formal hypothesis bisimulation \sim_{fh} (due to De Simone) and hypothesis-preserving bisimilarity \sim_{hp}. For open terms, we have (strict) inclusions \sim_{fh} \subset \sim_{hp} \subset \sim_{ci}; for closed terms, the three relations coincide. We show that each of these relations is a congruence in the usual sense, and we give an alternative characterisation of \sim_{hp} in terms of non-conditional transitions. Finally, we study the issue of recursive congruence: we give general theorems for the congruence of each of the above variants with respect to the recursion combinator, where, however, the results we achieve for \sim_{fh} and \sim_{hp} hold in a more general setting than the one for \sim_{ci}.\u
Superdevelopments for Weak Reduction
We study superdevelopments in the weak lambda calculus of Cagman and Hindley,
a confluent variant of the standard weak lambda calculus in which reduction
below lambdas is forbidden. In contrast to developments, a superdevelopment
from a term M allows not only residuals of redexes in M to be reduced but also
some newly created ones. In the lambda calculus there are three ways new
redexes may be created; in the weak lambda calculus a new form of redex
creation is possible. We present labeled and simultaneous reduction
formulations of superdevelopments for the weak lambda calculus and prove them
equivalent
Formal mechanization of device interactions with a process algebra
The principle emphasis is to develop a methodology to formally verify correct synchronization communication of devices in a composed hardware system. Previous system integration efforts have focused on vertical integration of one layer on top of another. This task examines 'horizontal' integration of peer devices. To formally reason about communication, we mechanize a process algebra in the Higher Order Logic (HOL) theorem proving system. Using this formalization we show how four types of device interactions can be represented and verified to behave as specified. The report also describes the specification of a system consisting of an AVM-1 microprocessor and a memory management unit which were verified in previous work. A proof of correct communication is presented, and the extensions to the system specification to add a direct memory device are discussed
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