44 research outputs found
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
A formal proof of modal completeness for provability logic
This work presents a formalized proof of modal completeness for G\"odel-L\"ob
provability logic (GL) in the HOL Light theorem prover. We describe the code we
developed, and discuss some details of our implementation, focusing on our
choices in structuring proofs which make essential use of the tools of HOL
Light and which differ in part from the standard strategies found in main
textbooks covering the topic in an informal setting. Moreover, we propose a
reflection on our own experience in using this specific theorem prover for this
formalization task, with an analysis of pros and cons of reasoning within and
about the formal system for GL we implemented in our code
Extension and Tangential CRF Conditions in Quaternionic Analysis
We prove some extension theorems for quaternionic holomorphic functions in
the sense of Fueter. Starting from the existence theorem for the nonhomogeneous
Cauchy-Riemann-Fueter Problem, we prove that an -valued function
on a smooth hypersurface, satisfying suitable tangential conditions, is
locally a jump of two -holomorphic functions. From this, we obtain,
in particular, the existence of the solution for the Dirichlet Problem with
smooth data. We extend these results to the continous case. In the final part,
we discuss the octonian case.Comment: 22 page