Unification Type Research Articles

Free Constructions in Hoops via $$\ell $$-Groups

AbstractLattice-ordered abelian groups, or abelian$$\ell $$ ℓ -groups in what follows, are categorically equivalent to two classes of 0-bounded hoops that are relevant in the realm of the equivalent algebraic semantics of many-valued logics: liftings of cancellative hoops and perfect MV-algebras. The former generate the variety of product algebras, and the latter the subvariety of MV-algebras generated by perfect MV-algebras, that we shall call $$\textsf{DLMV}$$ DLMV . In this work we focus on these two varieties and their relation to the structures obtained by forgetting the falsum constant 0, i.e., product hoops and DLW-hoops. As main results, we first show a characterization of the free algebras in these two varieties as particular weak Boolean products; then, we show a construction that freely generates a product algebra from a product hoop and a DLMV-algebra from a DLW-hoop. In other words, we exhibit the free functor from the two algebraic categories of hoops to the corresponding categories of 0-bounded algebras. Finally, we use the results obtained to study projective algebras and unification problems in the two varieties (and the corresponding logics); both varieties are shown to have (strong) unitary unification type, and as a consequence they are structurally and universally complete.

The polyhedral geometry of Wajsberg hoops

Abstract We show that the algebraic category of finitely presented Wajsberg hoops is equivalent to a non-full subcategory of finitely presented MV-algebras. We use this connection to show how methods and techniques developed to study MV-algebras can be adapted to study Wajsberg hoops, as well. In particular, we show that finitely presented Wajsberg hoops are dually equivalent to a subcategory of rational polyhedra with $\mathbb {Z}$-maps. We use the duality to provide a geometrical characterization of finitely generated projective and exact Wajsberg hoops. As applications, we study logical properties of the $0$-free fragment of Łukasiewicz logic, seen as a substructural logic. We show that, while deducibility in the fragment is equivalent to deducibility among 0-free formulas in Łukasiewicz logic, the same is not true for the admissibility of rules: there are rules written in the $0$-free language that are admissible in Wajsberg hoops but not in MV-algebras. Moreover, we show that the unification type of Wajsberg hoops is nullary, while the exact unification type is unitary, therefore showing decidability of admissible rules in the fragment.

Patch Locale of a Spectral Locale in Univalent Type Theory

Stone locales together with continuous maps form a coreflective subcategory of spectral locales and perfect maps. A proof in the internal language of an elementary topos was previously given by the second-named author. This proof can be easily translated to univalent type theory using resizing axioms. In this work, we show how to achieve such a translation without resizing axioms, by working with large, locally small, and small complete frames with small bases. This turns out to be nontrivial and involves predicative reformulations of several fundamental concepts of locale theory.

Palatal rugae as an unique and stable marker in personal identification-An interracial pilot study.

In post-mortem scenarios, often it is a very difficult process to establish a person's identity. Rugae are unique in that they are protected from trauma as they are insulated from heat by tongue and buccal pad of fat unlike fingerprint or lip print that is prone to destruction. This study was aimed to compare the palatal rugae among people of different races. The sole objectives of the study were to assess the predominant pattern in the selected groups, reliability of rugae pattern in personal identification, to evaluate reliability of sex determination and to compare the total number of rugae on right and left sides of the palate among the males and females. A total of 90 subjects were enrolled into the study and divided into three groups that are African, Dravidian and Mongoloid population. Shapes of rugae present were analyzed according to the classification given by Kapali et al. (1997) and Thomas & Kotze (1983). The predominant rugae shape in African and Dravidian population was wavy pattern, whereas Mongoloid race was predominant in curve pattern. African and Dravidian males were predominant in wavy pattern when compared to Mongoloid males where unification type was more predominant. A statistically significant association between the rugae shape in three populations exists, although subtle yet definite.

Projectivity and unification in substructural logics of generalized rotations

We develop a unifying approach to study projectivity and unification in substructural logics corresponding to varieties of residuated lattices generated by generalized rotation constructions. These include many interesting varieties especially in the realm of mathematical fuzzy logics. Our main results pertain what we shall call radical-determined varieties of rotations, which include all of the most relevant varieties in this framework. We characterize free algebras in a radical-determined variety V of rotations in terms of weak Boolean products of rotations of free algebras in the variety of radicals, the latter being the intersections of maximal filters of the algebras in V. Then we use such description to study projectivity in these varieties of rotations, characterizing finitely generated projective algebras. Moreover, we show that the strong unitary unification type of a variety of radicals implies the strong unitary type for the generated variety of rotations, which can be used to deduce the decidability of the admissibility of rules. As relevant applications of our general results, we obtain that product logic and nilpotent minimum logic have (strong) unitary unification type.

Unification types in Euclidean modal logics

AbstractWe prove that $\textbf {K}5$ and some of its extensions that do not contain $\textbf {K}4$ are of unification type $1$.

Remarks about the unification types of some locally tabular normal modal logics

AbstractIt is already known that unifiable formulas in normal modal logic $\textbf {K}+\square ^{2}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{2}\bot $ are unitary. In this paper, we prove that for all $d{\geq }3$, unifiable formulas in normal modal logic $\textbf {K}+\square ^{d}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{d}\bot $ are unitary.

Finitary unification in locally tabular modal logics characterized

We provide necessary and sufficient conditions for finitary unification in locally tabular modal logics, solely in terms of Kripke models. We apply the conditions to establish the unification types of logics determined by simple finite frames. In particular, we show that unification is finitary (or unitary) in the logic determined by the fork (frame F4, see Fig. 6), the rhombus (frame F5), in GL.3m, GrzBd2, S4Bd2 and other logics; whereas it is nullary in the logic of F6, and of the pentagon FN5. In Appendix analogous results are given for superintuitionistic logics.

On Church’s thesis in cubical assemblies

AbstractWe show that Church’s thesis, the axiom stating that all functions on the naturals are computable, does not hold in the cubical assemblies model of cubical type theory. We show that nevertheless Church’s thesis is consistent with univalent type theory by constructing a lex modality in cubical assemblies such that Church’s thesis holds in the corresponding reflective subuniverse.

Constructive sheaf models of type theory

AbstractWe provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.

The Scott model of PCF in univalent type theory

AbstractWe develop the Scott model of the programming language PCF in univalent type theory. Moreover, we work constructively and predicatively. To account for the non-termination in PCF, we use the lifting monad (also known as the partial map classifier monad) from topos theory, which has been extended to univalent type theory by Escardó and Knapp. Our results show that lifting is a viable approach to partiality in univalent type theory. Moreover, we show that the Scott model can be constructed in a predicative and constructive setting. Other approaches to partiality either require some form of choice or quotient inductive-inductive types. We show that one can do without these extensions.

The simplicial model of Univalent Foundations (after Voevodsky)

We present Voevodsky’s construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Martin-Löf type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

Univalent polymorphism

We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category EFF. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as such provide a context in which one can interpret many notions from homotopy theory and Homotopy Type Theory. Within the path category EFF one can identify a class of discrete fibrations which is closed under push forward along arbitrary fibrations (in other words, this class is polymorphic or closed under impredicative quantification) and satisfies propositional resizing. This class does not have a univalent representation, but if one restricts to those discrete fibrations whose fibres are propositions in the sense of Homotopy Type Theory, then it does. This means that, modulo the usual coherence problems, it can be seen as a model of the Calculus of Constructions with a univalent type of propositions. We will also build a more complicated path category in which the class of discrete fibrations whose fibres are sets in the sense of Homotopy Type Theory has a univalent representation, which means that this will be a model of the Calculus of Constructions with a univalent type of sets.

Remarks about the unification type of several non-symmetric non-transitive modal logics

AbstractThe problem of unification in a normal modal logic $L$ can be defined as follows: given a formula $\varphi$, determine whether there exists a substitution $\sigma$ such that $\sigma (\varphi )$ is in $L$. In this paper, we prove that for several non-symmetric non-transitive modal logics, there exists unifiable formulas that possess no minimal complete set of unifiers.

The Application of Palatal Rugae for Sex Discrimination in Forensic Medicine in a Selected Iranian Population.

Palatal rugae have been shown to be associated with racial and geographical variations and are known to be useful in sex identification. The present study aimed to assess palatal rugae patterns in a sample Iranian population and to compare the results with those from other studied populations. During summer 2017, 130 pre-orthodontic plaster casts from patients aged 17-25 years (65 males and 65 females) were obtained from the Department of Orthodontics, School of Dentistry, Shiraz University of Medical Sciences, Shiraz, Iran. The castes were evaluated using the classification given by Thomas and Kotze followed by recording the rugae lengths and shapes. The data were analyzed with the SPSS statistical software package (version 15.0) using the unpaired t test and Chi-square test. P<0.05 were considered statistically significant. Additionally, discriminant function analysis was applied to determine the applicability of palatal rugae patterns as a tool to aid sex identification. The rugae count showed an insignificant difference between the Iranian males and females. The primary rugae were most common in both sexes followed by the secondary and fragmentary. Significant differences were observed between the sexes and the number of primary rugae in males (P=0.03) and fragmentary rugae in females (P=0.04) on the left side of the palate. A significant difference in the straight and diverging unification types was observed, which was highest in males (P=0.01) and females (P=0.03), respectively. Discriminant function analysis enabled sex identification with an accuracy of 70%. Palatal rugae is shown to be an appropriate tool for sex identification in an Iranian population. Further studies with a larger sample size are required for a comprehensive outcome.

Two decision problems in Contact Logics

AbstractContact Logics provide a natural framework for representing and reasoning about regions in several areas of computer science. In this paper, we focus our attention on reasoning methods for Contact Logics and address the satisfiability problem and the unifiability problem. Firstly, we give sound and complete tableaux-based decision procedures in Contact Logics and we obtain new results about the decidability/complexity of the satisfiability problem in these logics. Secondly, we address the computability of the unifiability problem in Contact Logics and we obtain new results about the unification type of the unifiability problem in these logics.

Unification in epistemic logics

Epistemic logics are essential to the design of logical systems that capture elements of reasoning about knowledge. In this paper, we study the computability of unifiability and the unification types in several epistemic logics.

Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.

Palatal Rugae Patterns Among a Sudanese Population Sample

Palatal Rugae Patterns in a Sudanese Population Sample Palatal rugae are ridges resulted from symmetrical and irregular mucosal folds of tissue and located in the alveolar third of the palate. Palatal rugae identification can have a significant role in forensic identification where it is difficult to use the normal identification methods. The aim of this study is to identify and compare the palatal rugal morphological patterns in a sample of adult males and females of Sudanese population. Maxillary impression models obtained from 100 subjects (50 males and 50 females) of age group 21 – 23 years, of the dental students of University of Science and Technology, Sudan. The data were assessed based on the Thomas and Kotze classification (1983) for the length, shape and prevalence. The Independent - t - Test was used for the statistical analysis. Ethical approvals declared from the University of Science and Technology, Sudan and University of the Western Cape South, South Africa to conduct the study. Statistical analysis showed that females had significantly more number of total rugae on the right side of the palate than in males. The prevalence of the palatine rugae numbers was significantly more in females than in males. Primary rugae were distributed more significantly in females more than males. On the right side of the palate, the straight type of rugae was significantly predominant in females than in males whilst the wavy shape was more in males than in females. On the left side of the palate, the curved type of rugae was significantly more presented in males than in females whilst the diverging unification type was more in females than in males. The distribution of number of rugae shapes in both males and females were seen as straight followed by wavy and least was the circular. The total number of straight rugae was significantly more in females than in males, as well as circular rugae and diverging rugae. In contrast, curved rugae were significantly more in males than in females. There were 11 individuals who did not comply with the Thomas and Kotze classification thus was recorded as unspecified. Page iv This Sudanese study showed significant differences between males and females in the number, length and shapes of rugae. Hence, the individuality and uniqueness of palatine rugae can be used as a tool for forensic dental identification, and possibly distinction between males and females. There is also a possibility that there are ethnic differences in rugal patterns.

A Syntactic Approach to Unification in Transitive Reflexive Modal Logics

This paper contains a proof-theoretic account of unification in transitive reflexive modal logics, which means that the reasoning is syntactic and uses as little semantics as possible. New proofs of theorems on unification types are presented and these results are extended to negationless fragments. In particular, a syntactic proof of Ghilardi’s result that S4 has finitary unification is provided. In this approach the relation between classical valuations, projective unifiers, and admissible rules is clarified.