90 research outputs found
Free Monads, Intrinsic Scoping, and Higher-Order Preunification
Type checking algorithms and theorem provers rely on unification algorithms.
In presence of type families or higher-order logic, higher-order
(pre)unification (HOU) is required. Many HOU algorithms are expressed in terms
of -calculus and require encodings, such as higher-order abstract
syntax, which are sometimes not comfortable to work with for language
implementors. To facilitate implementations of languages, proof assistants, and
theorem provers, we propose a novel approach based on the second-order abstract
syntax of Fiore, data types \`a la carte of Swierstra, and intrinsic scoping of
Bird and Patterson. With our approach, an object language is generated freely
from a given bifunctor. Then, given an evaluation function and making a few
reasonable assumptions on it, we derive a higher-order preunification procedure
on terms in the object language. More precisely, we apply a variant of
-unification for second-order syntax. Finally, we briefly demonstrate an
application of this technique to implement type checking (with type inference)
for Martin-L\"of Type Theory, a dependent type theory
E-Unification for Second-Order Abstract Syntax
Higher-order unification (HOU) concerns unification of (extensions of) ?-calculus and can be seen as an instance of equational unification (E-unification) modulo ??-equivalence of ?-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to ?-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce E-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general E-unification. We prove that the procedure is sound and complete
Magnetic structure and phase diagram in a spin-chain system: CaCoO
The low-temperature structure of the frustrated spin-chain compound
CaCoO is determined by the ground state of the 2D Ising model on
the triangular lattice. At high-temperatures it transforms to the honeycomb
magnetic structure. It is shown that the crossover between the two magnetic
structures at 12 K arises from the entropy accumulated in the disordered
chains. This interpretation is in an agreement with the experimental data.
General rules for for the phase diagram of frustrated Ising chain compounds are
formulated.Comment: 4 pages, 2 figure
Formalizing the -Categorical Yoneda Lemma
Formalized -category theory forms a core component of various libraries of
mathematical proofs. However, more sophisticated results in fields from
algebraic topology to theoretical physics, where objects have "higher
structure," rely on infinite-dimensional categories in place of -dimensional
categories, and -category theory has thusfar proved unamenable to
computer formalization.
Using a new proof assistant called Rzk, which is designed to support
Riehl-Shulman's simplicial extension of homotopy type theory for synthetic
-category theory, we provide the first formalizations of results from
-category theory. This includes in particular a formalization of the
Yoneda lemma, often regarded as the fundamental theorem of category theory, a
theorem which roughly states that an object of a given category is determined
by its relationship to all of the other objects of the category. A key feature
of our framework is that, thanks to the synthetic theory, many constructions
are automatically natural or functorial. We plan to use Rzk to formalize
further results from -category theory, such as the theory of limits and
colimits and adjunctions.Comment: To appear in CPP 202
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