Larch Prover

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The Larch Prover, or LP for short, is an interactive theorem proving system for multi-sorted first-order logic. It was used at MIT and elsewhere during the 1990s to reason about designs for circuits, concurrent algorithms, hardware, and software. [1]

Contents

Unlike most theorem provers, which attempt to find proofs automatically for correctly stated conjectures, LP was intended to assist users in finding and correcting flaws in conjecturesthe predominant activity in the early stages of the design process. It worked efficiently on large problems, had many important user amenities, and could be used by relatively naïve users.

Development

LP was developed by Stephen Garland and John Guttag at the MIT Laboratory for Computer Science with assistance from James Horning and James Saxe at the DEC Systems Research Center, as part of the Larch project on formal specifications. [2] It extended the REVE 2 equational term rewriting system developed by Pierre Lescanne, [3] Randy Forgaard [4] with assistance from David Detlefs and Katherine Yelick. It supports proofs by equational term rewriting (for terms with associative-commutative operators), cases, contradiction, induction, generalization, and specialization.

LP was written in the CLU programming language.

Sample LP Axiomatization

declare sorts E, S declare variables e, e1, e2: E, x, y, z: S declare operators   {}:                   -> S   {__}:            E    -> S   insert:          E, S -> S   __ \union __:    S, S -> S   __ \in __:       E, S -> Bool   __ \subseteq __: S, S -> Bool   ..  set name setAxioms assert   sort S generated by {}, insert;   {e} = insert(e, {});   ~(e \in {});   e \in insert(e1, x) <=> e = e1 \/ e \in x;   {} \subseteq x;   insert(e, x) \subseteq y <=> e \in y /\ x \subseteq y;   e \in (x \union y) <=> e \in x \/ e \in y   .. set name extensionality assert \A e (e \in x <=> e \in y) => x = y 

Sample LP Proofs

set name setTheorems prove e \in {e} qed  prove \E x \A e (e \in x <=> e = e1 \/ e = e2)   resume by specializing x to insert(e2, {e1}) qed  % Three theorems about union (proved using extensionality)  prove x \union {} = x   instantiate y by x \union {} in extensionality qed  prove x \union insert(e, y) = insert(e, x \union y)   resume by contradiction     set name lemma     critical-pairs *Hyp with extensionality qed  prove ac \union     resume by contradiction       set name lemma       critical-pairs *Hyp with extensionality     resume by contradiction       set name lemma       critical-pairs *Hyp with extensionality qed  % Three theorems about subset  set proof-methods =>, normalization  prove e \in x /\ x \subseteq y => e \in y by induction on x       resume by case ec = e1c         set name lemma         complete qed  prove x \subseteq y /\ y \subseteq x => x = y     set name lemma     prove e \in xc <=> e \in yc by <=>         complete         complete     instantiate x by xc, y by yc in extensionality qed  prove (x \union y) \subseteq z <=> x \subseteq z /\ y \subseteq z by induction on x qed  % An alternate induction rule  prove sort S generated by {}, {__}, \union     set name lemma     resume by induction       critical-pairs *GenHyp with *GenHyp       critical-pairs *InductHyp with lemma qed 

Bibliography

Pascal André, Annya Romanczuk, Jean-Claude Royer, and Aline Vasconcelos, "Checking the consistency of UML class diagrams using Larch Prover", Proceedings of the 2000 International Conference on Rigorous Object-Oriented Methods, page 1, York, UK, BCS Learning & Development Ltd., Swindon, GBR, January 2000.

Boutheina Chetali, "Formal verification of concurrent programs using the Larch Prover", IEEE Transactions on Software Engineering 24:1, pages 4662, January 1998. doi: 10.1109/32.663997.

Manfred Broy, "Experiences with software specification and verification using LP, the Larch proof assistant", Formal Methods in System Design 8:3, pages 221272, 1996.

Urban Engberg, Peter Grønning, and Leslie Lamport, "Mechanical Verification of Concurrent Systems with TLA", Computer-Aided Verification, G. v. Bochmann and D. K. Probst editors, Proceedings of the Fourth International Conference CAV'92), Lecture Notes in Computer Science 663, Springer-Verlag, June 1992, pages 4455.

Urban Engberg, Reasoning in the Temporal Logic of Actions, BRICS Dissertation Series DS 961, Department of Computer Science, University of Aarhus, Denmark, August 1996. ISSN 1396-7002.

Stephen J. Garland and John V. Guttag, "Inductive methods for reasoning about abstract data types," Fifteenth Annual ACM Symposium on Principles of Programming Languages, pages 219228, San Diego, CA, January 1988.

Stephen J. Garland and John V. Guttag, "LP: The Larch Prover," Ninth International Conference on Automated DeductionLecture Notes in Computer Science 310, pages 748749, Argonne, Illinois, May 1988. Springer-Verlag.

Stephen J. Garland, John V. Guttag, and Jørgen Staunstrup, "Verification of VLSI circuits using LP," The Fusion of Hardware Design and Verification, pages 329–345, Glasgow, Scotland, July 46, 1988. IFIP WG 10.2, North Holland.

Stephen J. Garland and John V. Guttag, "An overview of LP, the Larch Prover," Third International Conference on Rewriting Techniques and ApplicationsLecture Notes in Computer Science 355, pages 137151, Chapel Hill, NC, April 1989. Springer-Verlag.

Stephen J. Garland and John V. Guttag, "Using LP to debug specifications," Programming Concepts and Methods, Sea of Galilee, Israel, April 25, 1990. IFIP WG 2.2/2.3, North-Holland.

Stephen J. Garland and John V. Guttag, A Guide to LP: the Larch Prover, MIT Laboratory for Computer Science, December 1991. Also published as Digital Equipment Corporation Systems Research Center Report 82, 1991.

Victor Luchangco, Ekrem Söylemez, Stephen Garland, and Nancy Lynch, "Verifying timing properties of concurrent algorithms," FORTE '94: Seventh International Conference on Formal Description Techniques, pages 259–273, Berne, Switzerland, October 47, 1994. Chapman & Hall.

Ursula Martin and Michael Lai, "Some experiments with a completion theorem prover", Journal of Symbolic Computation 13:1, 1992, pages 81100, ISSN 0747-7171.

Ursula Martin and Jeannette M. Wing, editors, First International Workshop on Larch, Proceedings of the First International Workshop non Larch, Dedham, Massachusetts, July 1315 1992, Workshops in Computing, Springer-Verlag, 1992.

Toh Ne Win, Michael D. Ernst, Stephen J. Garland, Dilsun Kirli, and Nancy Lynch, Using simulated execution in verifying distributed algorithms," Software Tools for Technology Transfer 6:1, Lenore D. Zuck, Paul C. Attie, Agostino Cortesi, and Supratik Mukhopadhyay (editors), pages 6776. Springer-Verlag, July 2004.

Tsvetomir P. Petrov, Anya Pogosyants, Stephen J. Garland, Victor Luchangco, and Nancy A. Lynch, "Computer-assisted verification of an algorithm for concurrent timestamps," Formal Description Techniques IX: Theory, Application, and Tools (FORTE/PSTV), Reinhard Gotzhein and Jan Bredereke (editors), pages 2944, Kaiserslautern, Germany, October 811, 1996. Chapman & Hall.

James B. Saxe, Stephen J. Garland, John V. Guttag, and James J. Horning, "Using transformations and verification in circuit design," Formal Methods in System Design 3:3 (December 1993), pages 181209.

Jørgen F. Søgaard-Anderson, Stephen J. Garland, John V. Guttag, Nancy A. Lynch, and Anya Pogosyants, "Computed-assisted simulation proofs," Fifth Conference on Computer-Aided Verification (CAV '03), Costas Courcoubetis (editor), Lecture Notes in Computer Science 697, pages 305319, Elounda, Greece, June 1993. Springer-Verlag.

Jørgen Staunstrup, Stephen J. Garland, and John V. Guttag, "Localized verification of circuit descriptions," Automatic Verification Methods for Finite State Systems, Lecture Notes in Computer Science 407, pages 349364, Grenoble, France, June 1989. Springer-Verlag.

Jørgen Staunstrup, Stephen J. Garland, and John V. Guttag, "Mechanized verification of circuit descriptions using the Larch Prover", Theorem Provers in Circuit Design, Victoria Stavridou, Thomas F. Melham, and Raymond T. Boute (editors), IFIP Transactions A-10, pages 277299, Nijmegen, The Netherlands, June 2224, 1992. North-Holland.

Mark T. Vandevoorde and Deepak Kapur, "Distributed Larch Prover (DLP): an experiment in parallelizing a rewrite-rule based prover", International Conference on Rewriting Techniques and Applications RTA 1996, Lecture Notes in Computer Science 1103, pages 420423. Springer-Verlag.

Frédéric Voisin, "A new proof manager and graphic interface for the Larch prover", International Conference on Rewriting Techniques and Applications RTA 1996, Lecture Notes in Computer Science 1103, pages 408411. Springer-Verlag.

Jeannette M. Wing and Chun Gong, Experience with the Larch Prover, ACM SIGSOFT Software Engineering Notes 15:44, September 1990, pages 140143 https://doi.org/10.1145/99571.99835

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References

  1. 1985 Larch survey, Çarnegie Mellon University
  2. John V. Guttag and James Horning with S. J. Garland, K. D. Jones, A. Modet, and J. M. Wing, Larch: Languages and Tools for Formal Specification, Springer-Verlag Texts and Monographs in Computer Science, 1993. ISBN 978-1-4612-2704-5.
  3. Pierre Lescanne and "Computer experiments with the REVE term rewriting system generator," Proceedings of the 10th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, POPL '83, Austin, Texas, Association for Computing Machinery, New York, NY, pages 99108.
  4. Randy Forgaard and John Guttag, "REVE: a term rewriting system generator with a failure-resistant Knuth-Bendix", Proceedings of a Workshop on Term Rewriting, edited by D. Kapur and D. Musser, April 1984, pages 531.