Models for Inexact Reasoning


Welcome to the home page of Models for Inexact Reasoning, a basic course offered by the European Masters Program in Computational Logic at the School of Computer Science (Universidad Politécnica de Madrid).

Course Syllabus


Course Overview:

The automated management of imprecision and its associated uncertainty are of huge interest in present and future applications of Computational Intelligence. This course deals with advanced reasoning methods to handle uncertainty and imprecision. This includes certainty factors-based approaches, probabilistic reasoning methods, the Dempster-Shafer Theory, and Fuzzy Logic. Fuzzy Logic deals with the management of imprecision emphasizing in the meaning issue, and provides a sound theoretical framework and a wide range of practical applications. The main topics in Fuzzy Logic are the fuzzy sets standard theories, the study of linguistic modifiers and quantifiers, the approximate inference, and the granularity of the related imprecise concepts. Together with Neural Networks and Evolutive Algorithms, Fuzzy Logic is at the very heart of the Soft Computing discipline, one of the most fruitful areas of Computational Intelligence.


Locations and times:


All lectures will be held at Meeting Room #2 of the Department of Artificial Intelligence. Lectures are Thursdays 12 pm - 2 pm.


Student personal record forms:


All students must download and fill in the electronic student record form (including a recent photograph) and send it by email to Prof. Miguel García by October 21st, 2009.



Details of the schedule, slides and reading lists will be updated as the course progresses. The schedule and the readings are subject to change.

Date Topics Slides Who Recommended Readings and Additional Material
Oct 8

Course Presentation.

(0) Introduction to Uncertainty, Imprecision and Approximate Reasoning.

(1) Overview of Rule-based Systems.

(0) [PDF]

(1) [PDF]


Basic Knowledge Representation

First-Order Logic

Rule-Based Systems



(0, 1) References [1, 2, 4]

Oct 15
(2) Reasoning with Certainty Factors. The MYCIN Approach
(2) [PDF] 
(2) References [2, 4, 6] 
Oct 22
(2) Reasoning with Certainty Factors. The MYCIN Approach (cont.)

(3) Reasoning with Pseudo-Probabilities: The PROSPECTOR Approach 

(3) [PDF]

(3) References [2, 4, 7]  
Oct 29
(3) Reasoning with Pseudo-Probabilities: The PROSPECTOR Approach (cont.)

(4) The Dempster-Shafer Theory of Evidence 

(4) [PDF]

(4) References [2, 4, 6]  
Nov 5
(4) The Dempster-Shafer Theory of Evidence (cont.)

(5) The Dempster-Shafer Theory of Evidence - A Sample Scenario 

(5) [PDF]
(4) References [2, 4, 6]  
Nov 12
(6) Applications of uncertain reasoning: Information Retrieval
(4) References [20]  
Nov 19

(7) Description of assignment #1

IMPORTANT! Attendance to this lecture is COMPULSORY.

Assignment #1 must be submitted to Prof. Miguel García by Jan 29th 2009. Submissions not respecting the deadline WILL NOT BE ACCEPTED.


(7) References [20]   
Nov 26

(8) Introduction to Fuzzy Prolog




(8) References [8, 9, 10]   
Dec 3
(9) Fuzzy Logic - Lesson 1: Crisp and Fuzzy Sets

(9) [PDF]
(9) References [8, 9 ,10, 11, 12]   
Dec 10
(10) Fuzzy Logic - Lesson 2: Fuzzy Propositions

 (11) Fuzzy Logic - Lesson 5: Fuzzy Relations
(10) [PDF]

(11) [PDF]
(10, 11) References [8, 9 ,10, 11, 12]   
Dec 17
(12) R-Fuzzy Lecture
(12) [PDF] 
Additional material [HERE] and [HERE] 
Jan 14
(13)  Fuzzy Logic - Lesson 6: Inference from Conditional Fuzzy Propositions

 (14) Fuzzy Logic - Lesson 7: Fuzzy Expert Systems

 (15) Fuzzy Logic - Lesson 9: Selection of Fuzzy Implications

(13) [PDF]

(14) [PDF]

(15) [PDF]
(13-15) References [8, 9 ,10, 11, 12]   
Jan 21
(16)  Fuzzy Logic - Lesson 3: Fuzzy Quantifiers

 (17) Fuzzy Logic - Lesson 4: Fuzzy Hedges

 (18) Fuzzy Logic - Lesson 8: Fuzzy Controllers

(16) [PDF]

(17) [PDF]

(18) [PDF]
(16-18) References [8, 9 ,10, 11, 12]   




  1. S. Russell, P. Norvig: "Artificial Intelligence, A Modern Approach, 2nd Edition". Prentice-Hall (2003)
  2. A. Gómez, N. Juristo, C. Montes, J. Pazos: "Ingeniería del Conocimiento". Ceura (1997)
  3. J. Cuena: "Sistemas Inteligentes. Conceptos, Técnicas y Métodos de Construcción". Servicio de Publicaciones FI- UPM(1997)
  4. M. Stefik: “Introduction to Knowledge Systems”. Morgan Kaufmann (1995)
  5. F. Puppe: “Systematic Introduction to Expert Systems: Knowledge Representations and Problem-Solving Methods”. Springer-Verlag (1993)
  6. B.G. Buchanan, E.H. Shortliffe: "Rule-based Expert Systems: The MYCIN Experiments of the Stanford Heuristic Programming Project". Addison-Wesley (1984)
  7. R.O. Duda, P.E. Hart, N.J. Nilsson: "Subjective Bayesian Methods for Rule-Based Inference Systems". Technical Note 134. SRI International (1976)
  8. L.A. Zadeh: "Fuzzy Sets" Information and Control 8: 338-353 (1965)
  9. L.A. Zadeh: "Fuzzy Sets as a Basis for a Theory of Possibility". Information Sciences 8: 199-249; 301-357; 9: 43-80 (1975)
  10. G.J. Klir, U. St. Clair, B. Yuan: "Fuzzy Set Theory: Foundations and Applications". Prentice-Hall (1997)
  11. E. Trillas, S. Cubillo: "Primeras Lecciones de Lógica Borrosa". Servicio de Publicaciones de la Facultad de Informática de la UPM (1999)
  12. E. Trillas, C. Moraga, S. Guadarrama, S. Cubillo, E. Castiñeira: "Computing with Antonyms". Technical Report. Universidad Politécnica de Madrid.
  13. L. Sterling, E. Shapiro: "The Art of Prolog, 2nd Edition". MIT Press (1994)
  14. K. Apt: "From Logic Programming to Prolog". Prentice-Hall (1997)
  15. I. Bratko: "Prolog Programming for Artificial Intelligence". Addison-Wesley (1990)
  16. J. Lloyd: "Foundations on Logic Programming". Springer-Verlag (1991)
  17. W.F. Clocksin, C.S. Mellish: "Programming in Prolog". Springer-Verlag (1981)
  18. M. Ghallab, D. Nau, P. Traverso: "Automated Planning: Theory and Practice". Morgan-Kauffman (2004)
  19. S.M. Ross: "Introduction to Probability Models". Academic Press, 9th Edition (2006)
  20. R. Baeza-Yates, B. Ribeiro-Neto: "Modern Information Retrival". Addison-Wesley, Wokingham, UK (1999)


<mgremesal (at) fi.upm.es>