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

Code	Subject	
MII0006	Automated Theorem Proving Systems	

Section	Semester	Hours: C+S+L	Category	Type	
Computer Science - in English	4	2+0+2	speciality	optional	
Computer Science - in English	6	2+0+2	speciality	optional	

Teaching Staff in Charge	Lect. LUPEA Mihaiela, Ph.D., lupea@cs.ubbcluj.ro	
Aims	- To study proof methods, which solve specific decisional problems in propositional logic and first-order logic: check if a statement is a theorem or if a statement (the conjecture) is a logical consequence of a set of statements (the axioms and hypotheses)
- To implement ATP (automated theorem prover) systems based on the studied methods.
	
Content	1. Automated theorem proving systems: architecture, examples, applications.
2. Basic notions of classical logics (propositional logic and first-order logic).
3. Data structures used to represent and manipulate logical formulas.
4. Binary decision diagrams in propositional logic.
5. Semantic tableaux method &#8211; a refutation proof method
   - A new approach of semantic tableaux method;
   - Considerations for implementation of an ATP based on this method.
6. Sequent and anti-sequent  calculi&#8211; two complementary direct proof systems
   - Sequent calculus &#8211; used to check the derivability in propositional/first-order logic.
   - Anti-sequent calculus- used to check the non-derivability and to build anti-models.
   - Considerations for implementation of an ATP based on these methods.
7. Semantic trees 
   - Heuristics and tree-searching methods used to implement an efficient proof procedure based on the construction of semantic trees.
8. Model elimination calculus &#8211; a refutation  proof method
   - Connection tableaux used to find a refutation in clausal-form model elimination  procedure
9. Resolution method &#8211; a refutation proof method
   - Refinements of resolution
      &middot;	lock resolution
      &middot;	linear resolution (input, unit, ordered); 
      &middot;	semantic resolution (hyper-resolution, the set-of support-strategy, ordered)
   - Considerations for implementation of ATP systems based on different refinements of 
       resolution.

Laboratory: 
1. Study and work with some dedicated ATP systems.
2. Projects for teams of 3 students: implementation of ATP systems based on the studied methods.
	
Bibliography	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: 50%
- laboratory activity: 50%