30 research outputs found
Initial Semantics for Strengthened Signatures
We give a new general definition of arity, yielding the companion notions of
signature and associated syntax. This setting is modular in the sense requested
by Ghani and Uustalu: merging two extensions of syntax corresponds to building
an amalgamated sum. These signatures are too general in the sense that we are
not able to prove the existence of an associated syntax in this general
context. So we have to select arities and signatures for which there exists the
desired initial monad. For this, we follow a track opened by Matthes and
Uustalu: we introduce a notion of strengthened arity and prove that the
corresponding signatures have initial semantics (i.e. associated syntax). Our
strengthened arities admit colimits, which allows the treatment of the
\lambda-calculus with explicit substitution.Comment: In Proceedings FICS 2012, arXiv:1202.317
Variable binding, symmetric monoidal closed theories, and bigraphs
This paper investigates the use of symmetric monoidal closed (SMC) structure
for representing syntax with variable binding, in particular for languages with
linear aspects. In our setting, one first specifies an SMC theory T, which may
express binding operations, in a way reminiscent from higher-order abstract
syntax. This theory generates an SMC category S(T) whose morphisms are, in a
sense, terms in the desired syntax. We apply our approach to Jensen and
Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This
leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at
Concur '0
Cartesian closed 2-categories and permutation equivalence in higher-order rewriting
We propose a semantics for permutation equivalence in higher-order rewriting.
This semantics takes place in cartesian closed 2-categories, and is proved
sound and complete
C-system of a module over a monad on sets
This is the second paper in a series that aims to provide mathematical
descriptions of objects and constructions related to the first few steps of the
semantical theory of dependent type systems.
We construct for any pair , where is a monad on sets and is
a left module over , a C-system (contextual category) and
describe a class of sub-quotients of in terms of objects directly
constructed from and . In the special case of the monads of expressions
associated with nominal signatures this construction gives the C-systems of
general dependent type theories when they are specified by collections of
judgements of the four standard kinds
Innocent strategies as presheaves and interactive equivalences for CCS
Seeking a general framework for reasoning about and comparing programming
languages, we derive a new view of Milner's CCS. We construct a category E of
plays, and a subcategory V of views. We argue that presheaves on V adequately
represent innocent strategies, in the sense of game semantics. We then equip
innocent strategies with a simple notion of interaction. This results in an
interpretation of CCS.
Based on this, we propose a notion of interactive equivalence for innocent
strategies, which is close in spirit to Beffara's interpretation of testing
equivalences in concurrency theory. In this framework we prove that the
analogues of fair and must testing equivalences coincide, while they differ in
the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014
Heterogeneous substitution systems revisited
Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical
description of substitution systems capable of capturing syntax involving
binding which is independent of whether the syntax is made up from least or
greatest fixed points. We extend this work in two directions: we continue the
analysis by creating more categorical structure, in particular by organizing
substitution systems into a category and studying its properties, and we
develop the proofs of the results of the cited paper and our new ones in
UniMath, a recent library of univalent mathematics formalized in the Coq
theorem prover.Comment: 24 page
A C-system defined by a universe category
This is a major update of the previous version. The methods of the paper are
now fully constructive and the style is "formalization ready" with the emphasis
on the possibility of formalization both in type theory and in constructive set
theory without the axiom of choice.
This is the third paper in a series started in 1406.7413. In it we construct
a C-system starting from a category together with a
morphism , a choice of pull-back squares based on
for all morphisms to and a choice of a final object of . Such a
quadruple is called a universe category. We then define universe category
functors and construct homomorphisms of C-systems defined by
universe category functors. As a corollary of this construction and its
properties we show that the C-systems corresponding to different choices of
pull-backs and final objects are constructively isomorphic.
In the second part of the paper we provide for any C-system CC three
constructions of pairs where is a universe
category and is an isomorphism.
In the third part we define, using the constructions of the previous parts,
for any category with a final object and fiber products a C-system
and an equivalence
High-level signatures and initial semantics
We present a device for specifying and reasoning about syntax for datatypes,
programming languages, and logic calculi. More precisely, we study a notion of
signature for specifying syntactic constructions.
In the spirit of Initial Semantics, we define the syntax generated by a
signature to be the initial object---if it exists---in a suitable category of
models. In our framework, the existence of an associated syntax to a signature
is not automatically guaranteed. We identify, via the notion of presentation of
a signature, a large class of signatures that do generate a syntax.
Our (presentable) signatures subsume classical algebraic signatures (i.e.,
signatures for languages with variable binding, such as the pure lambda
calculus) and extend them to include several other significant examples of
syntactic constructions.
One key feature of our notions of signature, syntax, and presentation is that
they are highly compositional, in the sense that complex examples can be
obtained by assembling simpler ones. Moreover, through the Initial Semantics
approach, our framework provides, beyond the desired algebra of terms, a
well-behaved substitution and the induction and recursion principles associated
to the syntax.
This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi,
which, in turn, was directly inspired by some earlier work of
Ghani-Uustalu-Hamana and Matthes-Uustalu.
The main results presented in the paper are computer-checked within the
UniMath system.Comment: v2: extended version of the article as published in CSL 2018
(http://dx.doi.org/10.4230/LIPIcs.CSL.2018.4); list of changes given in
Section 1.5 of the paper; v3: small corrections throughout the paper, no
major change