Deductive
systems for BigData integration

Radu BUCEA-MANEA-ȚONIȘ^{1}

^{1} Hyperion University, Titu Maiorescu University

**Abstract.** *The globalization is associated with an increased data to be
processed from E-commerce transactions. The specialists are looking for
different solutions, such as BigData, Hadoop, Datawarehoues, but it seems that
the future is the predicative logic implemented through deductive database
technology. It has to be done the swift from imperative languages, to not
declaratively languages used for the application development. The deductive
databases are very useful in the student teaching programs, too. Thus, the
article makes a consistent literature review in the field and shows practical
examples of using predicative logic in deductive systems, in order to integrate
different kind of data types**. *

Keywords: deductive systems, predicative
logic, Datalog, parse tree, grammars

JEL Codes: *M15*

1.
Introduction

The volume of data collected and the type of data that has to be integrated and analyzed is in constant increasing trend. The knowledge is presented in lots of formats: relational datasets, XML databases, rule bases, ontologies, and this make the classical approach not reliable. In order to find a common denominator, the predicative logic seems to be a solution for correct and appropriate data interpretation in modern companies From different version of databases, the predicative logic can extract common features or differences, based on predicate dependency and rule predicate graphs, for example. The ontologies, frequently used in intelligent systems, used in semantic web application help the examination of the use of abstractions of rule bases by predicate dependency and rule predicate graphs [Seipel, 2017].

The article shows an appropriate method to extract information from different types of data, using predicative logic through deductive databases.

2.
Literature review

Deductive
systems were originally designed for educational purposes, helping students to
translate the relational data structure model in a deductive data model, where
atomic data are arranged in predicates which can be understood as relations,
i.e., relational tables [Senz-Prez, 2018]. Students are thought to integrate
different kind of data type in a predicative logic. In this article the author
presents interesting examples for students, helping them to understand the
syntax.

It seems that
modern e-learning platforms, such as Moodle, Blackboard, Academika, provides the
information in a fixed-sequence, without customizing it in accordance with
students differences in background knowledge. The Zone of Proximal Development
(ZPD) theory states that the frustration, boredom and incomprehensiveness of
the information can be prevented. In this regard was developed a new framework
that facilitate the users online distance interaction. The adaptive e-learning
systems include real-time dynamic adaptation and context modelling in addition
to the learner model, the domain model and the pedagogical strategy [Maravanyika,
2017].

New technologies (IoT), such as
Virtual Reality, can offer a complete experience in the e-learning system, e.g.
a better (3D) view that supports demonstration and promotion, the development
of creativity, increased understanding, less study time and fun during learning
[Manea, 2018].

Deductive systems are used in
business field, too. Facebook is a good example of application that integrates
different types of data. Facebook developed a new query protocol that provides
a unified interface between the client and the server for data fetching and
manipulation application layer, in order to work with different types of data
handling of various sources and to combine, e.g., relational with NoSQL
databases. This protocol can be used as interface
for deductive databases, too [Nogatz, 2017].

Another
modern solution to manipulate different format of date is Open Rule Bench. Is
dedicated to rule engines, including deductive databases. An example of
translation of Datalog to C++ based on a method that "pushes" derived
tuples immediately to places where they are used was provided by [Brass,2017].

Other
studies demonstrate that the semantics of deductive databases can be
implemented in the spiking neural P systems model, allowing the integration of
symbolic reasoning systems based on logic and connectionist systems based on
the functioning of living neurons [Diaz-Pernil, 2018].

Further
bellow we demonstrate how natural language can be translated in first order
logic statements and stored into a knowledge database.

3.
Formal systems

In the essay on Universal Language, GW Leibnitz proposes
replacing words with numbers so that the language form corresponds to the
logical sense. For example, if the animal word is associated with figure 2,
and the rational word is associated with figure 3, it results that man will
be the product of 2 and 3, so 6. When constructing sentences, the author
proposes as a grammatical rule the exact division of the subject number to the
predicate number in in the case of true affirmations. Basically, the new
rational language will make "any reasoning a kind of arithmetic calculation"
so that the correspondence that exists between things and ideas is taken into
account. The author's approach follows the emphasis on the relation between
words and concepts, because we can have a simple, finite alphabet that would
render the multitude of infinite concepts [Leibnitz, 2015].

A formal system
is (D, R) where ** d** is a set of data structures, and

**is a set of rules that determines what transitions between objects in**

*R***are allowed. The characteristics of a formal system are:**

*D*Completeness: (D, R) does everything that the set of rules requires.

Correctness: (D, R) does nothing to ban the set of rules.

Simplicity: the formal system must meet its set of rules or come closer to meeting the set of rules with minimal complexity.

Naturalness: the formal system should be in line with the intuition.

Formal systems based on text mimics how natural language is generated and semantically regulated by grammar.

3.1.
General formal systems based on text

General formal systems based on text are formal systems whose data structures are strings/symbols, and whose transformations are rewriting the rules.

Let Σ be a finite symbol alphabet. Let Σ* a set with all strings of length finite and which can be formed using symbols from Σ. The elements Σ* will be called words on the Σ set.

A rewrite rule S is an expression of α ⇝ β, where α and β are in Σ*. Note that S* contains the word Null. A rule α ⇝ β means that it is allowed to replace α with β in any context.

*Exemple:*

Suppose we have the alphabet Σ = {p, g,-}, where xp-gx- x={-} , and the following rule P, after [Hofstadter, 2015]:

xpygz ⇝ xpy-gz-

Prove that the W be the word --p-g--- Σ*

To apply the set of rules P to the word W, we replace each part of W with the corresponding P rule. We keep -- as --, replace - with --, and --- with ---- . Therefore we have the property W ⇒ --p--g---- and the W is a theorem because evaluates TRUE the axiom xp-gx-.

Artificial languages are generated using formal grammars.

*Grammars*

A grammar is a rewriting system P along with an initial word I. Such grammar (P, I) generates language.

Given the following grammar, lets find a parse tree for the string 1 + 2 * 3[Nelson, 2017]:

<E> --> <D>

<E> --> <F>

<E> --> <G>

<E> --> <H>

<E> --> <I>

<E> --> ( <E> )

<F> --> <E> + <E>

<H> --> <E> - <E>

<G> --> <E> * <E>

<I> --> <E> / <E>

<D> --> 0 | 1 | 2 | ... 9

The parse tree is:

Fig. 1: The
parse tree for the string 1 + 2 * 3

E(F(E(N
1),G(E(N 2),E(N 3)))--> E(E(N 1),E(E(N 2),E(N 3)))

A
more intuitive way of expressing more complex statements like those found in
natural languages is predicative
logic.

*Predicative logic*

A predicate is a function whose code is the truth values {T, F}.

For example, the uncle (X, Y), who asserts that X is uncle for Y, defines a predicate. For any value pair X and Y, this statement will be either TRUE or FALSE.

The number of variables that appear in a predicate and that can be instantiated in this way is called the predicate arithmetic.

For example, the red predicate (X) means that X is red and has the arithmetic 1, and the predicate between (X, Y, Z) means that X is between Y and Z, having the arithmetic 3.

Monadic predicates, such as hair (X), are called properties.

The **Logical operators** used in the predicative logic are presented
below:**
**1.

**, symbolically written as ∧**

*and*2.

**, symbolically written as ∨**

*or*3.

**, symbolically written as**

*non*4.

**, symbolically written as →**

*implies*5.

**, symbolically written as ↔**

*if and only if*6.

**, symbolically written as ∀**

*for all*7.

**, symbolically written as ∃**

*exists*Below
we present an example from [Bird & all, 2009], explaining how the sentence Everybody
admires someone is transformed in predicative logic

It seems there are two predicates involved here admire (x, y) that
says ** x**
admires

**, and person (x) that says x is a person. We mean that if x is a person, then there must be at least one-person**

*y***that**

*y***will admire.**

*x*There are (at least) two ways of expressing this in first-order logic:

a.
all x.(person(x) -> exists y.(person(y) & admire(x,y)))

b.
exists y.(person(y) & all x.(person(x) -> admire(x,y)))

The next step is to demonstrate how the predicative logic helps querying databases using Datalog language.

3.2.
Deductive databases

Deductive databases are a set of basic relationships containing
explicit tuples [Date, 2005].

A query is -> the evaluation of a Boolean expression over
explicit relations and tuples or the demonstration as the specified formula
represents the logical consequence of the basic axioms, so it is a theorem.

A database is form of basic axioms set and deductive axioms. There
are two types of databases:

extensive
database which is a set of basic axioms,

intensive
database which is a form of deductive axioms and integrity restrictions.

The characteristics of deductive
databases are listed below:

Uniformity of representation

Operational uniformity

Semantic modeling

Extended application

The article
has the aim to demonstrate the benefits of using deductive databases. These
benefits are listed below:

Representation of disjointed information

Reflection of negative information

Performing recursive queries

*Datalog
language*

Further bellow theres an example of how Datalog is used to query a knowledge database:

Fz_concitadini(fx,fy)<=F(fx,nfx,sfx,of)
AND F(fy,nfy,sfy,of) AND NOT(fx=fy)

Fz_bun(f,sf,of)
<= F(f,nf,sf,of) AND sf>50

4.
Conclusions

Deductive systems prove to be the right solution for many issues
in modern society, from teaching students natural language processing, to
business fields, such as Facebook that integrated different types of data based
on predicative logic, through a friendly interface of expert systems,
artificial intelligence applications and E-Commerce applications.

In the near future, arithmetical operations by using
aggregate functions, useful in On-Line
Analytical Pro-cessing applications involving data mining and data warehouses
well be done exclusively by evaluating first order logic expressions and
lambda calculus.

5.
References

[1] C. Nelson
Randal (2017) University of Rochester, NY 14627-0226, Online course available
at: https://www.cs.rochester.edu/~nelson/courses/csc_173/grammars/parsetrees.html

[2] C. Sung-Pil,
M. Sung-Hyon, Terminological paraphrase extraction from scientific literature
based on predicate argument tuples, *Journal
of Information Science*, **38**(6),
2012, http://journals.sagepub.com/doi/pdf/10.1177/0165551512459920

[3] C.J. Date, *Baze de date*, Plus, 2005.

[4] D,
Diaz-Pernil, MA, Gutierrez-Naranjo, Semantics of deductive databases with
spiking neural P systems, *Neurocomputing*,
2018, **272**: 365-373, DOI:
10.1016/j.neucom.2017.07.007.

[5] D.
Richardson, (2006) *Formal systems, logic
and semantics*, On line course of Department of Computer Science, University
of Bath, http://www.cs.bath.ac.uk/pb/EMCL/DS/DS-Ref-2011/c19.pdf

[6] D. Seipel,
Knowledge Engineering for Hybrid Deductive Databases, *Electronic Proceedings in Theoretical Computer Science*, 2017, **234**: 1-12, DOI: 10.4204/EPTCS.234.1.

[7] F. Nogatz, D.
Seipel, Implementing GraphQL as a Query Language for Deductive Databases in
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[8] F.
Senz-Prez, Relational calculi in a deductive system, *Expert Systems With Applications, *2018, **97**: 106116, DOI: https://doi.org/10.1016/j.eswa.2017.12.007.

[9] G. W.
Leibnitz, Elementele caracteristicii universale, 1679, Limba universală,
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[10] M.
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for Personalised Teaching and Learning on E-Learning Platforms, Book Group
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[11] Rocsana
Bucea-Manea-Țoniș, M. Andronie, M. Iatagan, E-LEARNING IN THE ERA OF
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[12] S. Bird,
E.Klein, E. Loper, E. *Natural Language
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[13] S. Brass, H.
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