1. Finite-Trace Linear Temporal Logic: Coinductive Completeness
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Grigore Rosu
University of Illinois at Urbana-Champaign

2. Overview ───────  ∘    On finite traces
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Overview
On finite traces
Translating finite-trace LTL to infinite-trace LTL
Borrow decision procedures and results (PSPACE completeness for validity, etc.)
Borrow complete deduction system
Comes at a price, we do not want to go that way
Direct results
Direct decision procedure
Direct complete proof system
Coinduction

3. Finite Traces ───────  Ubiquitous in runtime verification
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Finite Traces
Examples abound in the literature
Automata theory, languages (regular, context-free, Turing complete)
Program verification (Hoare logic – partial correctness)
Log analysis (all logs are finite)
Philosophy (are there really any infinite traces? what is infinity, anyway?)
Ubiquitous in runtime verification
Many variants of LTL with finite trace models proposed
Goal: Coinduction for finite-trace reasoning
Approach: simplest LTL variant; no new LTL variant proposed

4. ∘ holds one one-element traces
Finite-Trace LTL
Weak interpretation:
∘ holds one one-element traces

5. Finite-Trace LTL  &  Infinite-Trace LTL

6. Translating Finite-Trace to Infinite-Trace LTL
So we can borrow decision procedures and complete deduction from infinite-trace LTL. But original formula more than double in size, and we lose intuition for its meaning. We want direct results!
Translating Finite-Trace to Infinite-Trace LTL
Extend each finite trace with infinitely many $ events ($ = nothing)
Transform to , where defined as follows:
Example:
becomes

7. First Result for Finite-Trace LTL
Technical, see paper
Complete atom traces
Fischer-Ladner closure
Tableaux-based decision procedure for satisfiability
No need to check for ultimately periodic sequences
Direct PSPACE-complete decision procedure for satisfiability

8. Second Result: Complete Deduction
But let’s firs recall complete deduction for infinite-trace LTL, starting with modal logic

9. Modal logic Complete Proof System

10. Infinite-trace LTL = two modal logics, plus more
Lichtenstein & Pnueli 2000, Dam & Guelev 2004+, …
Unsound: “” does not hold for finite-trace LTL.
Everything else stays sound.
Incomplete:
cannot prove

11. Induction equivalently stated as below
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Stays sound for finite-trace LTL

12. Main Contribution: Coinduction proof rule for finite trace reasoning
If happy tomorrow implies happy today,
then happy forever.
With weak next “∘”
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13. Gödel-Löb rule in “provability logic”
“□” means
“ provable”
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14. Interesting Observation: Coinduction = Induction + Finiteness
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15. Second Result: Complete Deduction for Finite-Trace LTL
Very technical, see paper

16. Conclusion and Future Work
Finite-traces important in RV and not only
Straightforward finite-trace LTL variant
Translation to infinite-trace LTL undesirable
Direct decision procedure
Coinduction yields complete deduction
Not implemented yet: future work

17. Coinduction is Cool!
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