1. Behavioral Rewrite Systems and Behavioral Productivity
Grigore Rosu (University of Illinois at Urbana-Champaign, USA)
Dorel Lucanu (University Alexandru Ioan Cuza, Iasi, Romania)

2. Thanks Kokichi!
For my PhD with Joseph Goguen at the University of California
I was supported by the CafeOBJ project as a Research Assistant
For advice and guiding during my PhD and … many times afterwards
Today’s presentation is a result of my work under the CafeOBJ project

3. Behavioral Abstraction
How a system behaves under a given set of obsevations/experiments
Abstracts away from internal implementation details
Important examples of behavioral abstraction (proposed in early 80’s)
Behavioral equivalence (Reichel, Tarlecki, Goguen, Wirsing, Hennicker,…)
Indistinguishability under experiments
CafeOBJ, BOBJ, Bmaude, CIRC, (co)CASL, etc
Productivity (Dijsktra, and many others afterwards)
Unlimited progress (infinite behavior = limit of continuous evaluation/rewriting)
Originally for streams
We want a unified, common ground for the above

4. Behavioral Equivalence (~1980)
Behavioral signature is a pair (∑,∆) where
∑ is an algebraic signature
∆ is a set of ∑-contexts, called derivatives (observers)
If δ[*:h] ϵ ∆ then sort h is hidden
H is the hidden sorts; V is the rest of sorts, called visible
∆-experiment is a ∆ -context of visible sort
Behavioral specification B = (∑,∆,E), with E a set of ∑-equations
Behavioral equivalence B |≡ t=t’ iff B |= C[t]=C[t’] for all ∆-experiments C
ϵ ∆ :h * =
for all t t’ *

5. Behavioral Specification – Example: Streams
Sorts Data (say Bit) and Stream ∑ ∆
or , etc. E

6. Behavioral Equivalence – Example: Streams
We can priove streams s and s’ are behavioral equivalent iff for all n
Examples of behavioral equivalence:

7. Productivity (Dijkstra ~1980)
Originally defined for streams; many definitions since then
Stream TRS
Sorts Data and Stream
Operation _:_ : Data x Stream -> Stream, plus others
Rewrite rules:
Stream TRS R is productive iff for all streams s

8. The Two Notions are Related
So both behavioral equivalence and productivity aim at defining infinite behaviors. Yet, despite common intuitions and ultimate goal, the two notions have lived separate lives.

9. Our Main Contribution
We attempt to bring behavioral equivalence and productivity together
Introduce the notion of behavioral rewrite system
Define behavioral productivity for such systems
Reduce productivity to standard termination
Investigate the complexity of the behavioral productivty
We counted more than 10 different variations of productivity
Instead of unifying them, we attempt to capture the essence of the notion
Maybe the relationship to behavioral equivalence gives insights on what the right notion or productivity should be?

10. Behavioral Rewrite Systems
Same like behavioral specification, but replace = with →; e.g.: ∑ ∆
or , etc. R

11. Behavioral Termination, Coinductivity, Productivity
Behavioral TRS (∑,∆,R) terminates iff (∑,R) terminates as normal TRS
(∑,∆,R) is coinductive iff δ[f(x)] is not a normal form for any f : s -> h in ∑ - ∆ and for any derivative/observer δ ϵ ∆
Dual to sufficient completeness: behavior of f defined for all observers
(∑,∆,R) productive for t iff for any ∆-experiment C, we have C[t] =>* d for some ∑|V U ∆ term d; (∑,∆,R) productive iff productive for all t
All these notions work with any behavioral rewrite systems, not only with streams; for example, with infinite trees, processes, etc.

12. Examples Behaviorally terminating, coinductive and productive:
So is the following:
but not with some conventional (adhoc) definitions of productivity
Behaviorally coinductive, productive, but non-terminating:
Tricky non-coinductive and non-productive examples; see paper

13. Behavioral Productivity vs Stream Productivity
Theorem: For the particular behavioral TRS of streams, behavioral productivity is stream productivity.
Stream productivity  Behavioral productivity
Add
Behavioral productivity  Stream productivity
Add lazy constructor and rule
But behavioral productivity works with any behavioral rewrite system, so it is more general than stream productivity

14. Termination and Coinductivity  Productivity
Theorem: If B=(∑,∆,R) is behaviorally terminating and coinductive, then B is productive.
Coinductivity is trivial to check, and termination of (∑,R) can be checked using conventional termination tools (APPROVE, etc.)
Related result by Zantema stating that termination implies model- theoretical well-definedness for certain restricted stream TRSes.

15. Behavioral Productivity is ∏20-complete
Theorem: Behavioral productivity is ∏20-complete
productive(t) iff (for any experiment C)
(there exists terminating rewriting sequence for C[t]) …
infinite …
∑1 =
r.e.
∏1 =
r.e. r. ∑2 ∏2 x

16. Conclusion Behavioral Rewrite Systems Behavioral Productivity Results:
Rewriting used to evaluate observations/experiments
Behavioral Productivity
Each experiment can be fully evaluated
Results:
Behavioral productivity generalizes stream productivity
Reduction of behavioral productivity to ordinary termination
So off-the-shelf termination tools can be used to check behavioral productivity
Behavioral productivity = behavioral well-definedness (see paper)
Behavioral productivity is ∏20