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Resolution Proof Technique in Linear Temporal Logic. Krzysztof Jan Kochut Louisiana State University and Agricultural & Mechanical College

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Resolution Proof Technique in Linear Temporal Logic. Krzysztof Jan Kochut Louisiana State University and Agricultural & Mechanical College Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1987 Resolution Proof Technique in Linear Temporal Logic. Krzysztof Jan Kochut Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Kochut, Krzysztof Jan, "Resolution Proof Technique in Linear Temporal Logic." (1987). LSU Historical Dissertations and Theses. 4456. https://digitalcommons.lsu.edu/gradschool_disstheses/4456 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS The most advanced technoiogy has been used to photo­ graph and reproduce this manuscript from the microfilm master. UMI films the original text directly from the copy submitted. Thus, some dissertation copies are in typewriter face, while others may be from a computer printer. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyrighted material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are re­ produced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each oversize page is available as one exposure on a standard 35 mm slide or as a 17" x 23" black and white photographic print for an additional charge. Photographs included in the original manuscript have been reproduced xerographically in this copy. 35 mm slides or 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. S&SUMIAccessing the World's Information since 1938 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA Order Number 8811416 Resolution proof technique in linear temporal logic Kochut, Krzysztof Jan, Ph.D. The Louisiana State University and Agricultural and Mechanical Col., 1987 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark V . 1. Glossy p h o to g rap h s or p a g e______ s 2. Colored illustrations, paper or print_______ 3. Photographs with dark background _____ 4. Illustrations are poor copy_______ 5. 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O ther ________________________________________________________________________________ UMI RESOLUTION PROOF TECHNIQUE IN LINEAR TEMPORAL LOGIC A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Computer Science by Krzysztof J. Kochut M.S., University of Warsaw, 1982 December 1987 ACKNOWLEDGEMENTS I would like to thank Dr. Sukhamay Kundu for hours of fruitful conversations. Without his constructive criticism my work would not be possible. I also want to extend my appreciation to Dr. Donald H. Kraft for his participation. Finally, I would like to thank my wife, Beata, for her constant support and encourage­ ment throughout my studies. Table of Contents 1. Introduction ..................................................................................................................... 1 2. Introduction to Temporal L o g ic ..................................................................................6 2.1. What is Temporal Logic? .........................................................................................9 2.2. A Quick Review of Some Temporal Logics ........................................................18 2.2.1. The Minimal System of Temporal Logic .........................................................18 2.2.2. Temporal Logic of Branching T im e................................................................. 20 2.2.3. Temporal Logic of Linear Time ...................................................................... 21 2.2.4. Interval Temporal Logic..................................................................................... 23 2.3. Prepositional Discrete Linear Temporal Logic (LTL) with the "Until" Operator................................................................................24 2.3.1. Language of the Prepositional Linear Temporal Logic.................................25 2.3.2. Semantics of the Prepositional Linear Temporal Logic.................................29 2.3.3. Axiomatic System of the Prepositional Linear Temporal Logic.................................................................................................32 3. Resolution in Prepositional Linear Temporal Logic ............................................36 3.1. What is the Resolution Method? ........................................................................... 37 3.2. Outline of the Resolution Proof Technique for Prepositional Linear Temporal Logic...........................................................40 3.3. Resolution Rules for Prepositional Linear Temporal Logic ........................... 49 3.4. Completeness and Soundness of the Resolution for Prepositional L T L..................................................................................... 54 3.5. Examples ....................................................................................................................93 3.6. Concluding Remarks ............................................................................................. 99 4. Ordered Linear Resolution Strategy for Prepositional LTL ....................................................................................106 4.1. OL-Resolution Rules for Propositional Linear Temporal Logic...............................................................................................109 4.2. Examples..................................................................................................................119 5. Resolution in Predicate Linear Temporal L o g ic .................................................. 125 5.1. Predicate Linear Temporal Logic with the "Until" Operator........................ 126 5.1.1. Syntax of Predicate Linear Temporal Logic .................................................. 126 5.1.2. Semantics of Predicate Linear Temporal Logic ........................................... 131 5.1.3. Conjunctive Normal Form for Predicate L T L ...............................................136 5.2. Resolution Rules for Predicate L T L ...................................................................143 5.3. Examples ..................................................................................................................154 6. Future Research and Conclusion ............................................................................158 7. Bibliography ...............................................................................................................162 Appendix A.......................................................................................................................172 Appendix B....................................................................................................................... 178 iv Abstract This dissertation presents a resolution proof technique for Propositional Linear Temporal Logic of discrete time with the "Until" operator. The presented proof tech­ nique stems from the resolution method developed by L. Farinas del Cerro and A. Cavalli. However, their method is incomplete, and their completeness proof, as origi­ nally reported, is incorrect. Unlike Farinas’s method, our proof technique incor­ porates the "Until" operator, which is a very powerful and useful in describing com­ plex temporal relationships which are common in many areas of computer science. Our technique is also proved complete. The presented resolution method for linear temporal logic is similar to classical resolutions: the main goal is to show unsatisfiability of a set of temporal clauses by locating, either directly or indirectly, a state which contains unsatisfiability. Subse­ quent resolvents of a refutation are obtained by resolving out complementary literals referring to the same instant of time. In order to increase the efficiency of the resolution proof technique, we have developed a refinement of the presented basic method. This refinement is similar to the well-known Ordered Linear (OL) strategy for classical resolution. We also present the lifting of the basic resolution method to predicate linear tem­ poral logic. Unlike First Order Logic,
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