Math 670 Fall 1998 MWF 10:30-12 DRL 4C4

Fall 1998

Math 670, MWF 10:30-12 DRL 4C4

Professor Andre Scedrov

Office: Room 4E6 in David Rittenhouse Laboratory
Telephone: eight five nine eight three ( Math. Dept. Office: eight eight one seven eight )
Fax: three four zero six three
E-mail: lastname at math
Office Hours: By appointment

Textbooks

R. Goldblatt. "Logics of Time and Computation". CSLI Lecture Notes, 1992. ISBN 0-937-07394-6.
Z. Manna and A. Pnueli. "The Temporal Logic of Reactive and Concurrent Systems: Specification". Springer-Verlag, 1992, ISBN 0-387-97664-7.
Z. Manna and A. Pnueli. "Temporal Verification of Reactive Systems: Safety". Springer-Verlag, 1995, ISBN 0-387-94459-1.

Other Sources

E.A. Emerson and C.S. Jutla. On simultaneously determinizing and complementing omega-automata. In: Fourth Annual IEEE Symposium on Logic in Computer Science, 1989, pp. 333-342.
G. Mints. "A Short Introduction to Modal Logic". CSLI Lecture Notes, 1992. ISBN 0-937-07375-X.
C. Stirling. Modal and temporal logics. In: Vol 2 of "Handbook of Logic in Computer Science, Vol. 2", ed. S. Abramsky, D. Gabbay, T. Maibaum, Oxford University Press, pages 477-563, 1992.
C. Stirling. Games and modal mu-calculus. Springer LNCS 1055, pages 298-312, 1996.
C. Stirling. Bisimulation, model checking, and other games. Notes for Mathfit Instructional Meeting on Games and Computation, Edinburgh, June 1997.
M. Vardi. An automata-theoretic approach to linear temporal logic. Banff Conference, 1994.
M. Vardi. Alternating automata and program verification. Springer LNCS Vol. 1000.

Topics Covered So Far

Goldblatt Sections 1, 2, 3, 4, 8 (up to 8.15), and 9: Propositional Modal Logic: Frames and Models, Theories, Canonical Models and Completeness, Filtrations, Finite Frame Property, Decidability, Linear Frames, Cluster Analysis. Propositional Temporal Logic: Linear Temporal Logic, State Sequences and Models, Axiomatization, Soundness, Completeness, Finite Frame Property, Decidability. Branching-Time Temporal Logic: Computational Trees and Models, Axiomatization, Soundness, Completeness, Finite Model Property, Decidability.
Büchi Automata, Rabin Automata, Alternating Automata, Complexity of Nonemptiness and Nonuniversality Problems. Temporal Logic and Automata on Infinite Words. Complexity of the Temporal Logic Satisfiability Problem. Complexity of the Verification Problem for Finite-State Programs. Rabin Tree Automata. Program Synthesis and Realizability. Sources: papers by Vardi listed above.
Complementation construction for Büchi Automata, Soundness and Completeness. Source: Paper by Emerson and Jutla listed above.

Take-Home Midterm Due in Class Monday, October 26

Goldblatt p. 9, Exercises 1.4: 1, 3, 4, 5, and 6.
Goldblatt p. 12, Exercise 1.11.
Goldblatt p. 13, Exercises 1.14: 1 and 2.
Goldblatt p. 15, Exercise 1.15.
Goldblatt p. 18, Exercises 2.2: 13, 14, 15, 16, and 17.
Goldblatt p. 19, Exercises 2.3: 1 - 9 all.
Goldblatt p. 20, Exercises 2.4: 1 - 3 all.
Goldblatt p. 21, Exercises 2.7: 1, 3 only.
Goldblatt p. 22, Exercises 2.8: 2, 3, 4, 5 only.
Goldblatt p. 23, Exercises 2.10: 1, 2, 3, 4 all.
Goldblatt p. 27, Exercises 3.7: only in the two cases of Schemas L and 10.
Goldblatt p. 27, Exercises 3.9: 3, 4 only.
Goldblatt p. 30, Exercises 3.11: 1 - 3 all.
Goldblatt pp. 70-71, Exercises 8.8: 1 - 6 all.
Goldblatt p. 72, Exercise 8.9.
Goldblatt p. 75, Exercise 8.15: 2, 3 only.

This is the complete list of problems due in class on October 26.

Take-Home Final Due in DRL 4E6 Tuesday, December 15 at 10 a.m.

Goldblatt p. 89, Exercises 9.3: 1 - 6 all.
Goldblatt p. 90, Exercises 9.4: 1 - 3 all.
Goldblatt p. 90, Exercise 9.5.
Goldblatt p. 101, Exercise 9.15.
Goldblatt p. 101, Exercises 9.16: 2 - 3 only.
Goldblatt p. 103, Exercise 9.18.
Goldblatt p. 104, Exercise 9.19.
Goldblatt p. 105, Exercise 9.22.
Goldblatt p. 107, Exercise 9.24.
Goldblatt p. 108, Exercise 9.26.
What can you say about the decidability and complexity of the nonuniversality problem for alternating automata on finite words? Prove your claims.
Prove in detail that for any alternating Büchi automaton there exists a nondeterministic Büchi automaton with the same associated infinitary language.
What can you say about the decidability and complexity of the nonuniversality problem for alternating Büchi automata? Prove your claims.
Finish the proof that the construction given in class of the alternating Büchi automaton associated to a given temporal formula has the desired property that the automaton accepts an infinite word iff this word when considered as a temporal model satisfies the formula.
Work out the construction of the associated alternating Büchi automaton for each of the following five temporal formulas, where p and q are atomic propositional symbols:
necessarily p,
possibly p,
p -> possibly q,
necessarily possibly q,
necessarily(p -> possibly q).
In each case prove in detail that the infinitary language of the alternating Büchi automaton is precisely the set of linear temporal logic models satisfying the formula.
Prove in detail PSPACE-hardness of the satisfiablity problem for linear temporal logic by describing a logarithmic-space algorithm that given a PSPACE-bounded Turing machine M and a finite word w outputs a linear temporal logic formula such that the formula is satisfiable iff M accepts w. Prove the desired properties of the reduction.

This is the complete list of problems due in in DRL 4E6 Tuesday, December 15 at 10 a.m.