"Babes-Bolyai" University of Cluj-Napoca
Faculty of Mathematics and Computer Science
| Automated Theorem Proving |
ter |
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| Teaching Staff in Charge |
Lect. LUPEA Mihaiela, Ph.D., lupea cs.ubbcluj.ro |
| Aims |
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- 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 |
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1. Basic notions of classical logics (propositional logic and first-order logics)
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 |
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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 |
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The final grade is obtained based on:
- written exam: 40% - seminar activity: 10% - 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 |