130,148 research outputs found
Second-Order Algebraic Theories
Fiore and Hur recently introduced a conservative extension of universal
algebra and equational logic from first to second order. Second-order universal
algebra and second-order equational logic respectively provide a model theory
and a formal deductive system for languages with variable binding and
parameterised metavariables. This work completes the foundations of the subject
from the viewpoint of categorical algebra. Specifically, the paper introduces
the notion of second-order algebraic theory and develops its basic theory. Two
categorical equivalences are established: at the syntactic level, that of
second-order equational presentations and second-order algebraic theories; at
the semantic level, that of second-order algebras and second-order functorial
models. Our development includes a mathematical definition of syntactic
translation between second-order equational presentations. This gives the first
formalisation of notions such as encodings and transforms in the context of
languages with variable binding
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Second-Order Algebraic Theories
Second-order universal algebra and second-order equational logic respectively provide a model theory
and a formal deductive system for languages with variable binding and parameterised metavariables.
This dissertation completes the algebraic foundations of second-order languages from the viewpoint of
categorical algebra.
In particular, the dissertation introduces the notion of second-order algebraic theory. A main role in
the definition is played by the second-order theory of equality M, representing the most elementary
operators and equations present in every second-order language. We show that M can be described
abstractly via the universal property of being the free cartesian category on an exponentiable object.
Thereby, in the tradition of categorical algebra, a second-order algebraic theory consists of a cartesian
category M and a strict cartesian identity-on-objects functor M: M →M that preserves the universal
exponentiable object of M.
At the syntactic level, we establish the correctness of our definition by showing a categorical equivalence
between second-order equational presentations and second-order algebraic theories. This equivalence,
referred to as the Second-Order Syntactic Categorical Type Theory Correspondence, involves distilling
a notion of syntactic translation between second-order equational presentations that corresponds to the canonical notion of morphism between second-order algebraic theories. Syntactic translations provide a mathematical formalisation of notions such as encodings and transforms for second-order languages.
On top of the aforementioned syntactic correspondence, we furthermore establish the Second-Order
Semantic Categorical Type Theory Correspondence. This involves generalising Lawvere’s notion of
functorial model of algebraic theories to the second-order setting. By this semantic correspondence,
second-order functorial semantics is shown to correspond to the model theory of second-order universal algebra.
We finally show that the core of the theory surrounding Lawvere theories generalises to the second order as well. Instances of this development are the existence of algebraic functors and monad morphisms in the second-order universe. Moreover, we define a notion of translation homomorphism that allows us to establish a 2-categorical type theory correspondence
Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional programming are
tricky to handle because of their cyclicity. This paper presents an
investigation of categorical, algebraic, and computational foundations of
cyclic datatypes. Our framework of cyclic datatypes is based on second-order
algebraic theories of Fiore et al., which give a uniform setting for syntax,
types, and computation rules for describing and reasoning about cyclic
datatypes. We extract the "fold" computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby, the rules
are correct by construction. We prove strong normalisation using the General
Schema criterion for second-order computation rules. Rather than the fixed
point law, we particularly choose Bekic law for computation, which is a key to
obtaining strong normalisation. We also prove the property of "Church-Rosser
modulo bisimulation" for the computation rules. Combining these results, we
have a remarkable decidability result of the equational theory of cyclic data
and fold.Comment: 38 page
Towards the most general scalar-tensor theories of gravity: a unified approach in the language of differential forms
We use a description based on differential forms to systematically explore
the space of scalar-tensor theories of gravity. Within this formalism, we
propose a basis for the scalar sector at the lowest order in derivatives of the
field and in any number of dimensions. This minimal basis is used to construct
a finite and closed set of Lagrangians describing general scalar-tensor
theories invariant under Local Lorentz Transformations in a pseudo-Riemannian
manifold, which contains ten physically distinct elements in four spacetime
dimensions. Subsequently, we compute their corresponding equations of motion
and find which combinations are at most second order in derivatives in four as
well as arbitrary number of dimensions. By studying the possible exact forms
(total derivatives) and algebraic relations between the basis components, we
discover that there are only four Lagrangian combinations producing second
order equations, which can be associated with Horndeski's theory. In this
process, we identify a new second order Lagrangian, named kinetic Gauss-Bonnet,
that was not previously considered in the literature. However, we show that its
dynamics is already contained in Horndeski's theory. Finally, we provide a full
classification of the relations between different second order theories. This
allows us to clarify, for instance, the connection between different
covariantizations of Galileons theory. In conclusion, our formulation affords
great computational simplicity with a systematic structure. As a first step we
focus on theories with second order equations of motion. However, this new
formalism aims to facilitate advances towards unveiling the most general
scalar-tensor theories.Comment: 28 pages, 1 figure, version published in PRD (minor changes
A New Derivation of the Picard-Fuchs Equations for Effective Super Yang-Mills Theories
A new method to obtain the Picard-Fuchs equations of effective
supersymmetric gauge theories in 4 dimensions is developed. It includes both
pure super Yang-Mills and supersymmetric gauge theories with massless matter
hypermultiplets. It applies to all classical gauge groups, and directly
produces a decoupled set of second-order, partial differential equations
satisfied by the period integrals of the Seiberg-Witten differential along the
1-cycles of the algebraic curves describing the vacuum structure of the
corresponding theory.Comment: Latex version, 43 pages, a few cosmetic changes and some references
adde
Strongly Normalising Cyclic Data Computation by Iteration Categories of Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional
programming are tricky to handle because of their cyclicity. This
paper presents an investigation of categorical, algebraic, and
computational foundations of cyclic datatypes. Our framework of
cyclic datatypes is based on second-order algebraic theories of Fiore
et al., which give a uniform setting for syntax, types, and
computation rules for describing and reasoning about cyclic datatypes.
We extract the ``fold\u27\u27 computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby,
the rules are correct by construction. Finally, we prove strong
normalisation using the General Schema criterion for second-order
computation rules. Rather than the fixed point law, we particularly
choose Bekic law for computation, which is a key to obtaining strong
normalisation
MacNeille Completion and Buchholz\u27 Omega Rule for Parameter-Free Second Order Logics
Buchholz\u27 Omega-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut elimination for higher order logics, Maehara and Okada\u27s algebraic proofs are of particular interest, since the essence of their arguments can be algebraically described as the (Dedekind-)MacNeille completion together with Girard\u27s reducibility candidates. Interestingly, it turns out that the Omega-rule, formulated as a rule of logical inference, finds its algebraic foundation in the MacNeille completion.
In this paper, we consider a family of sequent calculi LIP = cup_{n >= -1} LIP_n for the parameter-free fragments of second order intuitionistic logic, that corresponds to the family ID_{<omega} = cup_{n <omega} ID_n of arithmetical theories of inductive definitions up to omega. In this setting, we observe a formal connection between the Omega-rule and the MacNeille completion, that leads to a way of interpreting second order quantifiers in a first order way in Heyting-valued semantics, called the Omega-interpretation. Based on this, we give a (partly) algebraic proof of cut elimination for LIP_n, in which quantification over reducibility candidates, that are genuinely second order, is replaced by the Omega-interpretation, that is essentially first order. As a consequence, our proof is locally formalizable in ID-theories
Boolean algebras, Morita invariance, and the algebraic K-theory of Lawvere theories
The algebraic K-theory of Lawvere theories is a conceptual device to
elucidate the stable homology of the symmetry groups of algebraic structures
such as the permutation groups and the automorphism groups of free groups. In
this paper, we fully address the question of how Morita equivalence classes of
Lawvere theories interact with algebraic K-theory. On the one hand, we show
that the higher algebraic K-theory is invariant under passage to matrix
theories. On the other hand, we show that the higher algebraic K-theory is not
fully Morita invariant because of the behavior of idempotents in non-additive
contexts: We compute the K-theory of all Lawvere theories Morita equivalent to
the theory of Boolean algebras.Comment: 20 pages. This updated paper discusses the work on Morita equivalence
of Lawvere theories that appeared in version one. In order to better
highlight the two separate directions of the results in that first version,
the material on assembly maps has been incorporated into a second paper,
arXiv:2112.0700
BRST, Generalized Maurer-Cartan Equations and CFT
The paper is devoted to the study of BRST charge in perturbed two dimensional
conformal field theory. The main goal is to write the operator equation
expressing the conservation law of BRST charge in perturbed theory in terms of
purely algebraic operations on the corresponding operator algebra, which are
defined via the OPE. The corresponding equations are constructed and their
symmetries are studied up to the second order in formal coupling constant. It
appears that the obtained equations can be interpreted as generalized
Maurer-Cartan ones. We study two concrete examples in detail: the bosonic
nonlinear sigma model and perturbed first order theory. In particular, we show
that the Einstein equations, which are the conformal invariance conditions for
both these perturbed theories, expanded up to the second order, can be
rewritten in such generalized Maurer-Cartan form.Comment: LaTeX2e, elsart.cls, 36 pages, typos corrected, references and
acknowledgements adde
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