Discounted Duration Calculus Work in Progress H. Ody Joint work - - PowerPoint PPT Presentation

Discounted Duration Calculus

Work in Progress

• H. Ody

Joint work with M. Fränzle and M. R. Hansen

October 19, 2015

Ody Discounted DC October 19, 2015 1 / 12

Motivation

Discounting in Temporal Logics

‘Eventually properties’ are common Booting Eventually the system is ready

Ody Discounted DC October 19, 2015 1 / 12

Motivation

Discounting in Temporal Logics

‘Eventually properties’ are common Booting Eventually the system is ready Access to shared Resource After a request, Eventually we get a grant

Ody Discounted DC October 19, 2015 1 / 12

Motivation

Discounting in Temporal Logics

‘Eventually properties’ are common Booting Eventually the system is ready Access to shared Resource After a request, Eventually we get a grant Often Soon is better than Eventually Want quantitative statements about eventually properties Truth value is in interval [0, 1]

Ody Discounted DC October 19, 2015 1 / 12

Motivation

Discounting in Duration Caluclus

Example - Energy Consumption

Working consumes energy Idle conserves energy Property: The energy lasts long

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2 4 6 8 10 1 2 3 4 working idle time

work

Motivation

Discounting in Duration Caluclus

Example - Energy Consumption

Working consumes energy Idle conserves energy Property: The energy lasts long

Example - Job Completion

Job needs a fixed duration of work to finish Property: The job is finished soon

Ody Discounted DC October 19, 2015 2 / 12

2 4 6 8 10 1 2 3 4 working idle time

work

Motivation

Discounting in Duration Caluclus

Example - Energy Consumption

Working consumes energy Idle conserves energy Property: The energy lasts long

Example - Job Completion

Job needs a fixed duration of work to finish Property: The job is finished soon We want properties over durations

Ody Discounted DC October 19, 2015 2 / 12

2 4 6 8 10 1 2 3 4 working idle time

work

Background and Intuition

Discounting in LTL and CTL

Truth value ... ... is a real number from interval [0, 1] ... represents quality of a system or our satisfaction with the system Fdφ The earlier a good state is reached the better

2 4 6 8 10 0.2 0.4 0.6 0.8 1 f0(t) = dt, d = 0.7 f1(t) = dt, d = 0.9 time truth

Ody Discounted DC October 19, 2015 3 / 12

Background and Intuition

Discounting in LTL and CTL

Truth value ... ... is a real number from interval [0, 1] ... represents quality of a system or our satisfaction with the system Fdφ The earlier a good state is reached the better Gdφ The later a bad state is reached the better

2 4 6 8 10 0.2 0.4 0.6 0.8 1 f0(t) = dt, d = 0.7 f1(t) = dt, d = 0.9 f2(t) = 1 − dt, d = 0.7 time truth

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Discounting - Background

Introduction in economics [Sam37] Introduced into temporal logics in [DAFH+04] (Discounted CTL) Definition of discounted LTL [ABK14]

Ody Discounted DC October 19, 2015 4 / 12

Background and Intuition

Discounting in LTL and CTL

Example

F0.7P on path

starting point P

π :

Discount is the basis Time of satisfaction is the exponent

Ody Discounted DC October 19, 2015 5 / 12

Background and Intuition

Discounting in LTL and CTL

Example

F0.7P on path

starting point P

π :

Discount is the basis Time of satisfaction is the exponent Truth value is IFdP, π = sup

0≤t

{0.7t · IP, π[t..]} = 0.73 · IP, π[3..] = 0.343 · IP, π[3..]

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Discounted Duration Calculus

Syntax

φ ::= ♦dφ |

dS ≥ c | dS > c | ¬φ | φ ∨ φ

S ::= P | ¬S | S ∧ S . P is a Boolean proposition ♦φ There is a neighbouring interval satisfying φ

♦φ φ

S ≥ c

State expression S holds for at least c

S S

Following abbreviations can be expressed ⌈P ∨ ¬Q⌉ State expression P ∨¬Q holds in this interval

P ∨ ¬Q

φ φ is satisfied on all neighbouring intervals

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Discounted Duration Calculus

Semantics

I♦dφ(tr, [k, m]) = sup

l≥m

{dl−m · Iφ(tr, [m, l])} I

S ≥ c(tr, [k, m]) =

• if

m

t=k S(t) dt < c

1

• therwise

I¬φ(tr, [k, m]) = 1 − Iφ(tr, [k, m]) Iφ0 ∨ φ1(tr, [k, m]) = max{Iφ0(tr, [k, m]), Iφ1(tr, [k, m])}

Ody Discounted DC October 19, 2015 7 / 12

Example

Statements over durations d work < 3

2 4 6 8 10 1 2 3 4 working idle time

work

Id work < 3(tr, [k, m]) = I¬♦d¬

• work < 3(tr, [k, m])

Ody Discounted DC October 19, 2015 8 / 12

Example

Statements over durations d work < 3

2 4 6 8 10 1 2 3 4 working idle time

work

Id work < 3(tr, [k, m]) = I¬♦d¬

• work < 3(tr, [k, m])

= 1 − sup

l≥m

{dl−m · I¬

• work < 3(tr, [m, l])}

Ody Discounted DC October 19, 2015 8 / 12

Example

Statements over durations d work < 3

2 4 6 8 10 1 2 3 4 working idle time

work

Id work < 3(tr, [k, m]) = I¬♦d¬

• work < 3(tr, [k, m])

= 1 − sup

l≥m

{dl−m · I¬

• work < 3(tr, [m, l])}

= 1 − sup

l≥m

{dl−m · (1 − I

• work < 3(tr, [m, l]))}

Ody Discounted DC October 19, 2015 8 / 12

Example

Statements over durations d work < 3

2 4 6 8 10 1 2 3 4 working idle time

work

Id work < 3(tr, [k, m]) = I¬♦d¬

• work < 3(tr, [k, m])

= 1 − sup

l≥m

{dl−m · I¬

• work < 3(tr, [m, l])}

= 1 − sup

l≥m

{dl−m · (1 − I

• work < 3(tr, [m, l]))}

with d = 0.7, m = 0 = 1 − 0.76 = 0.88

Ody Discounted DC October 19, 2015 8 / 12

Questions of Decidability

The formulas we consider

In negation normal form No nested modalities and All modalities are discounted

Threshold Satisfiability Sketch

We can decide ∃tr. Iφ, tr ∼ v with ∼∈ {<, >, ≥, ≤} Example ∃tr. Iφ, tr ≥ v with φ ≡ ♦0.7 work ≥ c

2 4 6 8 10 0.2 0.4 0.6 0.8 1 f (t) = 0.7t v δ = logd v time truth

Truthvalue of ♦dφ when φ is satisfied at time t

Ody Discounted DC October 19, 2015 9 / 12

Questions of Decidability

The formulas we consider

In negation normal form No nested modalities and All modalities are discounted

Threshold Satisfiability Sketch

We can decide ∃tr. Iφ, tr ∼ v with ∼∈ {<, >, ≥, ≤} Example ∃tr. Iφ, tr ≥ v with φ ≡ ♦0.7 work ≥ c Transform φ into a time-bounded linear hybrid automaton Aφ (reach- ability is decidable) [BDG+11] Aφ has a location reachable iff

work ≥ c is satisfied in at most

δ time

2 4 6 8 10 0.2 0.4 0.6 0.8 1 f (t) = 0.7t v δ = logd v time truth

Truthvalue of ♦dφ when φ is satisfied at time t

Ody Discounted DC October 19, 2015 9 / 12

Questions of Decidability

Treshhold Satisfiability

Example

∃tr. I♦0.7 work ≥ c, tr ≥ v

Threshold Satisfiability Sketch

2 4 6 8 10 0.2 0.4 0.6 0.8 1 f (t) = 0.7t v δ = logd v time truth

y ≤ 0 y := 0 W ˙ x := 1 ¬W ˙ x := 0 final x ≥ c ∧ y ≤ δ x ≥ c ∧ y ≤ δ Ody Discounted DC October 19, 2015 10 / 12

Questions of Decidability

Model Checking Sketch

Let M be a timed automaton with only clock constraints of the form x ∼ c, i.e. no comparisons of clocks Model checking is ∀tr ∈ M. Iφ, tr ≥ v Equivalently: ¬∃tr ∈ M. Iφ, tr < v Check ∃tr ∈ M. Iφ, tr < v on M ⊗ Aφ A witnessing trace constitutes a counter example

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Conclusion

Gave several examples to show usefulness of our logic Some meaningful questions are decidable Nested modalities pose a challenge I believe I have a procedure for approximate threshold satisfiability

Nested Modalities

Service should be online soon, and then run for a long time ♦d0d1⌈S⌉ Future Formal proofs of decidability Implementation and case studies?

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Shaull Almagor, Udi Boker, and Orna Kupferman. Discounting in LTL. In Tools and Algorithms for the Construction and Analysis of Systems, pages 424–439. Springer, 2014. Thomas Brihaye, Laurent Doyen, Gilles Geeraerts, Joël Ouaknine, Jean-Fran ¸cois Raskin, and James Worrell. On reachability for hybrid automata over bounded time. In Automata, Languages and Programming, pages 416–427. Springer, 2011. Luca De Alfaro, Marco Faella, Thomas A Henzinger, Rupak Majumdar, and Mariëlle Stoelinga. Model checking discounted temporal properties. Springer, 2004. Paul A Samuelson. A note on measurement of utility. The Review of Economic Studies, 4(2):155–161, 1937.

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Decidability

Nested Modalities

I think we are able to decide approximate threshold satisfiability, i.e. ∃tr ∈ M. Iφ, tr ∼ v ± ǫ

Example

Consider ♦d0d1ψ We evaluate the formula at some timepoint δi The satisfaction value is dδi · Id1ψ, tr Then the comparison is satisfied iff Id1ψ, tr[δi..] ≥ v dδi If we pick enough points δi at which to try satisfying d1ψ then we should come within ǫ distance of v, if that is possible at all As the discounting formula is exponential hopefully we do not need too many δi..?

Ody Discounted DC October 19, 2015 12 / 12