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 After a request, Eventually we get a grant Access to shared Resource Ody Discounted DC October 19, 2015 1 / 12
Motivation Discounting in Temporal Logics ‘Eventually properties’ are common Booting Eventually the system is ready After a request, Eventually we get a grant Access to shared Resource 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 working idle � work 4 3 Example - Energy Consumption 2 Working consumes energy 1 Idle conserves energy 2 4 6 8 10 time Property: The energy lasts long Ody Discounted DC October 19, 2015 2 / 12
Motivation Discounting in Duration Caluclus working idle � work 4 3 Example - Energy Consumption 2 Working consumes energy 1 Idle conserves energy 2 4 6 8 10 time 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
Motivation Discounting in Duration Caluclus working idle � work 4 3 Example - Energy Consumption 2 Working consumes energy 1 Idle conserves energy 2 4 6 8 10 time 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
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 F d φ The earlier a good state is reached the better 1 truth 0 . 8 f 1 ( t ) = d t , d = 0 . 9 0 . 6 0 . 4 f 0 ( t ) = d t , d = 0 . 7 0 . 2 2 4 6 8 10 time 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 F d φ The earlier a good state is reached the better G d φ The later a bad state is reached the better 1 truth f 2 ( t ) = 1 − d t , d = 0 . 7 0 . 8 f 1 ( t ) = d t , d = 0 . 9 0 . 6 0 . 4 f 0 ( t ) = d t , d = 0 . 7 0 . 2 2 4 6 8 10 time Ody Discounted DC October 19, 2015 3 / 12
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 F 0 . 7 P on path π : starting P point 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 F 0 . 7 P on path π : starting P point Discount is the basis Time of satisfaction is the exponent Truth value is { 0 . 7 t · I � P , π [ t .. ] � } I � F d P , π � = sup 0 ≤ t = 0 . 7 3 · I � P , π [3 .. ] � = 0 . 343 · I � P , π [3 .. ] � Ody Discounted DC October 19, 2015 5 / 12
Discounted Duration Calculus Syntax � d S ≥ c | � d S > c | ¬ φ | φ ∨ φ φ ::= ♦ d φ | S ::= P | ¬ S | S ∧ S . P is a Boolean proposition ♦ φ φ There is a neighbouring interval satisfying φ ♦ φ � S ≥ c S S State expression S holds for at least c Following abbreviations can be expressed P ∨ ¬ Q ⌈ P ∨ ¬ Q ⌉ State expression P ∨¬ Q holds in this interval � φ φ is satisfied on all neighbouring intervals Ody Discounted DC October 19, 2015 6 / 12
Discounted Duration Calculus Semantics { d l − m · I � φ � ( tr , [ m , l ]) } I � ♦ d φ � ( tr , [ k , m ]) = sup l ≥ m � m � � S ≥ c � ( tr , [ k , m ]) = 0 if t = k S ( t ) d t < c I � 1 otherwise I � ¬ φ � ( tr , [ k , m ]) = 1 − I � φ � ( tr , [ k , m ]) I � φ 0 ∨ φ 1 � ( tr , [ k , m ]) = max {I � φ 0 � ( tr , [ k , m ]) , I � φ 1 � ( tr , [ k , m ]) } Ody Discounted DC October 19, 2015 7 / 12
Example working idle � work 4 3 Statements over durations � d � work < 3 2 1 2 4 6 8 10 time I � � d � work < 3 � ( tr , [ k , m ]) = I � ¬ ♦ d ¬ � work < 3 � ( tr , [ k , m ]) Ody Discounted DC October 19, 2015 8 / 12
Example working idle � work 4 3 Statements over durations � d � work < 3 2 1 2 4 6 8 10 time I � � d � work < 3 � ( tr , [ k , m ]) = I � ¬ ♦ d ¬ � work < 3 � ( tr , [ k , m ]) { d l − m · I � ¬ � = 1 − sup work < 3 � ( tr , [ m , l ]) } l ≥ m Ody Discounted DC October 19, 2015 8 / 12
Example working idle � work 4 3 Statements over durations � d � work < 3 2 1 2 4 6 8 10 time I � � d � work < 3 � ( tr , [ k , m ]) = I � ¬ ♦ d ¬ � work < 3 � ( tr , [ k , m ]) { d l − m · I � ¬ � = 1 − sup work < 3 � ( tr , [ m , l ]) } l ≥ m { d l − m · (1 − I � � = 1 − sup work < 3 � ( tr , [ m , l ])) } l ≥ m Ody Discounted DC October 19, 2015 8 / 12
Example working idle � work 4 3 Statements over durations � d � work < 3 2 1 2 4 6 8 10 time I � � d � work < 3 � ( tr , [ k , m ]) = I � ¬ ♦ d ¬ � work < 3 � ( tr , [ k , m ]) { d l − m · I � ¬ � = 1 − sup work < 3 � ( tr , [ m , l ]) } l ≥ m { d l − m · (1 − I � � = 1 − sup work < 3 � ( tr , [ m , l ])) } l ≥ m = 1 − 0 . 7 6 = 0 . 88 with d = 0 . 7 , m = 0 Ody Discounted DC October 19, 2015 8 / 12
Questions of Decidability The formulas we consider In negation normal form All modalities are discounted No nested modalities and Threshold Satisfiability Sketch We can decide ∃ tr . I � φ, tr � ∼ v with 1 truth ∼∈ { <, >, ≥ , ≤} 0 . 8 Example ∃ tr . I � φ, tr � ≥ v with 0 . 6 φ ≡ ♦ 0 . 7 � work ≥ c 0 . 4 v f ( t ) = 0 . 7 t 0 . 2 2 4 6 8 10 δ = log d v time Truthvalue of ♦ d φ 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 All modalities are discounted No nested modalities and Threshold Satisfiability Sketch We can decide ∃ tr . I � φ, tr � ∼ v with 1 truth ∼∈ { <, >, ≥ , ≤} 0 . 8 Example ∃ tr . I � φ, tr � ≥ v with 0 . 6 φ ≡ ♦ 0 . 7 � work ≥ c 0 . 4 v f ( t ) = 0 . 7 t Transform φ into a time-bounded 0 . 2 linear hybrid automaton A φ (reach- ability is decidable) [BDG + 11] 2 4 6 8 10 δ = log d v time A φ has a location reachable iff � work ≥ c is satisfied in at most Truthvalue of ♦ d φ when φ is satisfied at time t δ time 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 1 truth W x := 1 ˙ 0 . 8 x ≥ c ∧ y ≤ δ 0 . 6 y := 0 y ≤ 0 final 0 . 4 v f ( t ) = 0 . 7 t 0 . 2 x ≥ c ∧ y ≤ δ ¬ W x := 0 ˙ 2 4 6 8 10 δ = log d v time 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 . I � φ, tr � ≥ v Equivalently: ¬∃ tr ∈ M . I � φ, tr � < v Check ∃ tr ∈ M . I � φ, tr � < v on M ⊗ A φ A witnessing trace constitutes a counter example Ody Discounted DC October 19, 2015 11 / 12
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 ♦ d 0 � d 1 ⌈ S ⌉ Service should be online soon, and then run for a long time Future Formal proofs of decidability Implementation and case studies? Ody Discounted DC October 19, 2015 12 / 12
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. Ody Discounted DC October 19, 2015 12 / 12
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