Paraconsistent Relational Model: A Quasi-Classic Logic Approach 1 - PowerPoint PPT Presentation

Paraconsistent Relational Model: A Quasi-Classic Logic Approach 1 Authors: Badrinath Jayakumar* and Rajshekhar Sunderraman Institute: Georgia State University, Georgia, USA Corresponding author: bjayakumar2@cs.gsu.edu

Overview 2  Quasi-classic logic and quasi-classic models for logic programs  Paraconsistent relational model  The advantages of paraconsistent relational model  Quasi-classic models using paraconsistent relational model  The future works and short comings

Quasi-Classic Logic 3  It is a paraconsistent logic.  Unlike Belnap’s four- valued logic [5], Hunter’s quasi -classic logic [2] supports disjunctive syllogism, disjunction introduction, etc.  It is moves one step towards classical logic.  Its power comes from the resolution inference rule.

Quasi-Classic Logic Programs (1) 4  Z.Zhang’s quasi-classic logic programs [1] inspired from Hunter’s quasi - classic logic notion and Sakma’s paraconsistent minimal models notion [3]. Z.Zhang’s quasi-classic logic program determines minimal quasi-classic  models based on the set inclusion. Logic rules of the form:  Literals are either positive or negative atoms.

Quasi-Classic Logic Programs (2) 5 always terminates in finite time. 

Paraconsistent Relational Model (1) 6  The normal relation stores only information that is believed to be true.  The paraconsistent relation [4] stores information that is believed to be true and believed to be false. we define two types of algebraic operators:   Set Theoretic: union ( ), complement ( (unary)), intersection ( ), and difference ( (binary)).  Relation Theoretic: Join ( ), selection ( ), and projection ( ).

Paraconsistent Relational Model (2) 7 Normal Relation (Closed World Assumption): Paraconsistent Relation (Open World Assumption):

Paraconsistent Relational Model (3) 8

Paraconsistent Relational Model (4) 9

Paraconsistent Relational Model 10 Example (5) Example 1. Let R = and S = . Then (union) (intersection)

Paraconsistent Relational Model 11 Example (6)

Paraconsistent Relational Model (7) 12

Paraconsistent Relational Model (8) 13

Paraconsistent Relational Model 14 Example(9) Example 2. Let R = and S= . Here attributes are ordered sequence and tuples are lists of values.

Paraconsistent Relational Model 15 Example(10)

Advantages of Using Paraconsistent 16 Relational Model  Three main advantages:  works with a set of tuples instead of a tuple at a time,  can apply various laws of equality,  suits good for query intensive applications.

Quasi-Classic Models Construction (1) 17 Here, we consider positive extended disjunctive deductive databases.  The model construction involves two steps:   associate every literal to a paraconsistent relation and construct an equation for every clause;  solve the equations.

Quasi-Classic Models Construction (2) 18  It is hard to represent disjunctive information in paraconsistent relation.  We introduce disjunctive paraconsistent relation. Paraconsistent Relation Disjunctive Paraconsistent Relation We allow sometimes the conjunctive tuple in the positive part.

Quasi-Classic Models Construction 19 Example (3) Converting the rules into equations: 1. 2. LHS in both equations are the same. So,

Quasi-Classic Models Construction 20 Example (4) First, facts are added to the paraconsistent relation.   Copies are created. Copies are the same, but have different relation name.

Quasi-Classic Models Construction 21 Example (5) Mapping both definite tuples and disjunctive tuples from LHS of the  equation to the disjunctive paraconsistent relation.  We renamed the attribute before we map.  Inconsistency is in the disjunctive relation.

Quasi-Classic Models Construction 22 Example (6) Applying focus, which removes complementary tuples from the disjunctive  relation with respect to SModel.

Quasi-Classic Models Construction 23 Example (7)  The disjunctive paraconsistent relation contains disjunctive information which leads to more relations called proper disjunctive paraconsistent relations. Therefore,

Quasi-Classic Models Construction 24 Example (8) Relationalizing: removing paraconsistent unions among paraconsistent relations. Then, create an exact relation in DModel for every relation in SModel.

Quasi-Classic Models Construction 25 Example (9)

Quasi-Classic Models Construction 26 Example (10) Minimize removes redundant sets.

Quasi-Classic Models Construction 27 Example (11) Minimal model by size implies minimal model by set inclusion (vice versa is not true).

Future Works/Short Comings 28 The algorithm does not work in the presence of disjunctive facts, and  constants and duplicate variables in disjunctive literals. The algorithm finds only strong models and no constrains/recursions are  allowed. The algorithm could be extended to allow default negation.  The algorithm lacks the proof of correctness and complexity. 

29 Thank You

Bibliography 30 [1]. Zhang, Z.; Lin, Z.; and Ren, S. 2009. Quasi-classical model semantics for logic programs – a paraconsistent approach. In Foundations of Intelligent Systems. Springer. 181 – 190. [2]. Hunter, A. 2000. Reasoning with contradictory information using quasi- classical logic. Journal of Logic and Computa- tion 10(5):677 – 703. [3]. Sakama, C., and Inoue, K. 1995. Paraconsistent stable se- mantics for extended disjunctive programs. Journal of Logic and Computation 5(3):265 – 285. [4]. Bagai, R., and Sunderraman, R. 1995. A paraconsistent relational data model. International Journal of Computer Mathematics 55(1-2):39 – 55. [5]. Belnap Jr, N. D. 1977. A useful four-valued logic. In Mod- ern uses of multiple-valued logic. Springer. 5 – 37.