Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master

SUBJECT

Code
Subject
MI369 Automated Theorem Proving
Section
Semester
Hours: C+S+L
Category
Type
Formals Methods in Programming - in English
2
2+2+1
compulsory
Teaching Staff in Charge
Lect. LUPEA Mihaiela, Ph.D.,  lupeacs.ubbcluj.ro
Aims
- To study proof methods, specific to propositional and first-order logics, which solve the decision problems: check if a statement is a theorem or if a statement is a logical consequence of a set of axioms and hypotheses;
- To implement ATP (automated theorem prover) systems based on the studied methods.
- To apply the ATP systems for programs specification and verification.
Content
1. Basic notions of classical logics (propositional logic and first-order logic)
2. Formalization of common-sense reasoning and mathematical reasoning using these
classical logics.
3. Semantic tableaux method – a refutation proof method
- A new approach of semantic tableaux method;
- Considerations for implementation of an ATP based on this method, which solves the
following decision problems:
- build all the models of a consistent formula
- check if a formula is a tautology
- check if a formula is a logical consequence of a set of axioms and
hypotheses;
4.Sequent and antisequent calculi– two complementary direct proof systems
- Sequent calculus – used to check the derivability in propositional/first-order
logic
- Antisequent calculus- used to check the non-derivability and to build anti-models
- Considerations for implementation of an ATP based on these methods;
5. Theorem prover based on semantic trees and Herbrand Theorem
- Heuristics and tree-searching methods used to implement an efficient proof
procedure based on the construction of semantic trees
6. Model elimination calculus – a refutation proof method
- Connection tableaux used to find a refutation in clausal-form model elimination
procedure
7. Resolution method – a refutation proof method
- Refinements of resolution
· lock resolution
· linear resolution (input, unit, ordered); tree-searching methods and heuristics
used to find linear refutations;
· semantic resolution (hyperresolution, the set-of support-strategy,ordered)
- Considerations for implementation of ATP systems based on different refinements
of resolution.
8. Applications of theorem proving in programs specification and verification.
References
1. M. Ben-Ari: Mathematical Logic for Computer Science, Ed. Springer, 2001.
2. C.L.Chang, R.C.T.Lee: Symbolic Logic and Mechanical Theorem Proving, Academic Press, 1973.
3. M.Fitting: First-order Logic and Automated Theorem Proving, Texts and Monographs in Computer Science, Springer Verlag, 1990, Second Edition 1996.
4. M.R. Genesereth, N.J. Nilsson: Logical Foundations of Artificial Intelligence, Morgan Kaufman, 1992.
5. D.A. Duffy: Principles of automated theorem proving, John Willey & Sons, 1991.
6. L.C. Paulson: Logic and Proof, Univ. Cambridge, 2000, course on-line.
7. M. Possega: Deduction Systems, Institute of Informatics, 2002, course on-line.
8. S.Reeves, M.Clarke: Logic for computer science, Addison Wesley Publisher Ltd, 1990.
9. D.Tatar: Inteligenta artificiala: demonstrarea automata si NLP, Editura Microinformatica, Cluj-Napoca, 2001.
10. R.M.Smullyan: First-order logic, Revised Edition, Dover Press, New York, 1996.
Assessment
The final grade is obtained based on:
- written exam: 40%
- seminar activity: 10%
- presentation of a theoretical paper: 25%
- individual project- implementation of an ATP system: 25%
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject