scholarly journals The category of EQ-algebras

2021 ◽  
Author(s):  
Rajab Ali Borzooei ◽  
Narges Akhlaghinia ◽  
Mona Aaly Kologani ◽  
Xiao Long Xin
Keyword(s):  
Algebraic Structure ◽  
Type Theory ◽  
Truth Values ◽  
Special Case

EQ-algebras were introduced by Nova ́k in [15] as an algebraic structure of truth values for fuzzy type theory (FFT). In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculate coprodut and pushout in a special case. Also, we construct a free EQ-algebra on a singleton.

scholarly journals Constructing some logical algebras from EQ-algebras

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2747-2760
Author(s):  
Rajab Borzooei ◽  
Narges Akhlaghinia ◽  
Xiao Xin ◽  
Mona Kologani
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Algebraic Structure ◽  
Type Theory ◽  
Stone Algebra ◽  
Algebraic Structures ◽  
De Morgan Algebra ◽  
Truth Values ◽  
Mv Algebras

EQ-algebras were introduced by Nov?ak in [16] as an algebraic structure of truth values for fuzzy type theory (FTT). Nov?k and De Baets in [18] introduced various kinds of EQ-algebras such as good, residuated, and lattice ordered EQ-algebras. In any logical algebraic structures, by using various kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, by means of fantastic filters of EQ-algebras we construct MV-algebras. Also, we study prelinear EQ-algebras and introduce a new kind of filter and named it prelinear filter. Then, we show that the quotient structure which is introduced by a prelinear filter is a distributive lattice-ordered EQ-algebras and under suitable conditions, is a De Morgan algebra, Stone algebra and Boolean algebra.

scholarly journals Modal descent

2020 ◽  
pp. 1-29
Author(s):  
Felix Cherubini ◽  
Egbert Rijke
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Orthogonal Factorization ◽  
Simple Description ◽  
Factorization System ◽  
Homotopy Type Theory ◽  
Orthogonal Factorization System ◽  
The Right ◽  
Special Case

Abstract Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated with any modality, of which the left class is the class of ○-equivalences and the right class is the class of ○-étale maps. This factorization system is called the modal reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the modal reflective factorization system of a modality. In the special case of the n-truncation, the modal reflective factorization system has a simple description: we show that the n-étale maps are the maps that are right orthogonal to the map $${\rm{1}} \to {\rm{ }}{{\rm{S}}^{n + 1}}$$ . We use the ○-étale maps to prove a modal descent theorem: a map with modal fibers into ○X is the same thing as a ○-étale map into a type X. We conclude with an application to real-cohesive homotopy type theory and remark how ○-étale maps relate to the formally etale maps from algebraic geometry.

scholarly journals Leibniz equality is isomorphic to Martin-Löf identity, parametrically

2020 ◽  
Vol 30 ◽  
Author(s):  
ANDREAS ABEL ◽  
JESPER COCKX ◽  
DOMINIQUE DEVRIESE ◽  
AMIN TIMANY ◽  
PHILIP WADLER
Keyword(s):  
Type Theory ◽  
General Relation ◽  
The Other ◽  
Proof Systems ◽  
Special Case ◽  
Leibniz Equality

Abstract Consider two widely used definitions of equality. That of Leibniz: one value equals another if any predicate that holds of the first holds of the second. And that of Martin-Löf: the type identifying one value with another is occupied if the two values are identical. The former dates back several centuries, while the latter is widely used in proof systems such as Agda and Coq. Here we show that the two definitions are isomorphic: we can convert any proof of Leibniz equality to one of Martin-Löf identity and vice versa, and each conversion followed by the other is the identity. One direction of the isomorphism depends crucially on values of the type corresponding to Leibniz equality satisfying functional extensionality and Reynolds’ notion of parametricity. The existence of the conversions is widely known (meaning that if one can prove one equality then one can prove the other), but that the two conversions form an isomorphism (internally) in the presence of parametricity and functional extensionality is, we believe, new. Our result is a special case of a more general relation that holds between inductive families and their Church encodings. Our proofs are given inside type theory, rather than meta-theoretically. Our paper is a literate Agda script.

scholarly journals FUZZY TYPE THEORY IN THE ANALYSIS OF ARGUMENTATION

2021 ◽  
pp. 37-47
Author(s):  
Oleg Domanov
Keyword(s):  
Fuzzy Logic ◽  
Type Theory ◽  
Proof Assistant ◽  
Truth Values

The article deals with a fazzy variant of P. Martin-Löf ’s intuitionistic type theory. It presents the overview of fuzzy type theory rules and an example of its application to the analysis of the persuasiveness of argumentation. In the latter, the truth values of fuzzy logic are interpreted as degrees of persuasiveness of statements and arguments. The formalization is implemented in the proof assistant Agda.

A RING-THEORETIC BASIS FOR LOGIC PROGRAMMING

1990 ◽  
Vol 01 (01) ◽  
pp. 23-48 ◽  
Author(s):  
V.S. SUBRAHMANIAN
Keyword(s):  
Logic Programming ◽  
Algebraic Structure ◽  
Complete Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Truth Values ◽  
Partially Ordered

Investigations into the semantics of logic programming have largely restricted themselves to the case when the set of truth values being considered is a complete lattice. While a few theorems have been obtained which make do with weaker structures, to our knowledge there is no semantical characterization of logic programming which does not require that the set of truth values be partially ordered. We derive here semantical results on logic programming over a space of truth values that forms a commutative pseudo-ring (an algebraic structure a bit weaker than a ring) with identity. This permits us to study the semantics of multi-valued logic programming having a (possibly) non-partially ordered set of truth values.

scholarly journals Sets in homotopy type theory

2015 ◽  
Vol 25 (5) ◽  
pp. 1172-1202 ◽  
Author(s):  
EGBERT RIJKE ◽  
BAS SPITTERS
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Connected Components ◽  
Homotopy Type Theory ◽  
Universe Of Sets ◽  
Univalent Foundations ◽  
Subobject Classifier ◽  
Theoretical Foundations ◽  
Special Case

Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.

scholarly journals New heavenly double copies

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Erick Chacón ◽  
Hugo García-Compeán ◽  
Andrés Luna ◽  
Ricardo Monteiro ◽  
Chris D. White
Keyword(s):  
Type Theory ◽  
Equation Of Motion ◽  
Field Theories ◽  
Classical Solutions ◽  
Related Field ◽  
Yang Mills ◽  
Hermitian Manifolds ◽  
Area Preserving ◽  
Double Copy ◽  
Special Case

Abstract The double copy relates scattering amplitudes and classical solutions in Yang-Mills theory, gravity, and related field theories. Previous work has shown that this has an explicit realisation in self-dual YM theory, where the equation of motion can be written in a form that maps directly to Plebański’s heavenly equation for self-dual gravity. The self-dual YM equation involves an area-preserving diffeomorphism algebra, two copies of which appear in the heavenly equation. In this paper, we show that this construction is a special case of a wider family of heavenly-type examples, by (i) performing Moyal deformations, and (ii) replacing the area-preserving diffeomorphisms with a less restricted algebra. As a result, we obtain a double-copy interpretation for hyper-Hermitian manifolds, extending the previously known hyper-Kähler case. We also introduce a double-Moyal deformation of the heavenly equation. The examples where the construction of Lax pairs is possible are manifestly consistent with Ward’s conjecture, and suggest that the classical integrability of the gravity-type theory may be guaranteed in general by the integrability of at least one of two gauge-theory-type single copies.

scholarly journals Implicative algebras: a new foundation for realizability and forcing

2020 ◽  
Vol 30 (5) ◽  
pp. 458-510
Author(s):  
Alexandre Miquel
Keyword(s):  
Algebraic Structure ◽  
Classical Logic ◽  
Salient Feature ◽  
Truth Values

AbstractWe introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model-theoretic constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this construction with the corresponding constructions in forcing and realizability, we conclude that the class of implicative triposes encompasses all forcing triposes (both intuitionistic and classical), all classical realizability triposes (in the sense of Krivine), and all intuitionistic realizability triposes built from partial combinatory algebras.

Logic Programming on a Topological Bilattice1

1988 ◽  
Vol 11 (2) ◽  
pp. 209-218 ◽  
Author(s):  
Melvin Fitting
Keyword(s):  
Logic Programming ◽  
Topological Spaces ◽  
Truth Values ◽  
Truth Value ◽  
Special Case

We investigate the semantics of logic programming using a generalized space of truth values. These truth values may be thought of as evidences for and against – possibly incomplete or contradictory. The truth value spaces we use essentially have the structure of M. Ginsberg’s bilattices, and arise from topological spaces. The simplest example is a four-valued logic, previously investigated by N. Belnap. The theory of this special case properly contains that developed in earlier research by the author, on logic programming using Kleene’s three-valued logic.

Extra Concepts and Domains of Truth Values in Interpreting Logical Languages in Artificial Intelligence Systems

Vestnik MEI ◽  
2020 ◽  
Vol 5 (5) ◽  
pp. 148-154
Author(s):  
Vadim N. Falk ◽  
Keyword(s):  
Bounded Lattice ◽  
Truth Values ◽  
Logical Operations ◽  
Countable Power ◽  
Number Representations ◽  
Definition Of ◽  
Symbol Sequences ◽  
Context Free ◽  
Special Case ◽  
Traditional Understanding

So-called extra concepts introduced to represent structurally defined objects and structures with unlimited complexity in their traditional understanding are suggested. The concepts of extra-word, extra-regular expression, context-free extra-grammar, and context-free extra-language are extensions of the well-known concepts used in the theory of formal languages. Extra words are a special case of symbol sequences; however, the set of all extra words in any alphabet is countable, whereas the set of all symbol sequences is not countable. The periodic codes of rational number representations in some positional numeration system are in fact extra words in this terminology. The concept of an extra-tuple is a generalization of the tuple concept, which implies the possibility of interpreting extra-tuples both as finite and as the indicated type of infinite sequences of elements of an arbitrary, not more than countable set, and it should be noted that the set of all possible sequences of such sort remains countable. By using the introduced concepts, a countable family of the domains of truth values has been specified for multivalued and countable-valued logics, each of which is a bounded lattice of finite or countable power with the traditional definition of basic logical operations of negation, conjunction, and disjunction. The hierarchical construction of the proposed truth domains makes it possible to introduce new logical operations in consideration that do not have analogues in the classical logic.

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