On the Relationship between Reachability Problems in Timed and - PowerPoint PPT Presentation

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? No we cannot!

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ?

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? ( 1 , c 1 �→ 6 , c 2 �→ 0 )

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? ( 1 , c 1 �→ 6 , c 2 �→ 0 ) → ( 2 , c 1 �→ 6 , c 2 �→ 10 )

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? ( 1 , c 1 �→ 6 , c 2 �→ 0 ) → ( 2 , c 1 �→ 6 , c 2 �→ 10 ) → ( 3 , c 1 �→ 6 , c 2 �→ 8 )

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? ( 1 , c 1 �→ 6 , c 2 �→ 0 ) → ( 2 , c 1 �→ 6 , c 2 �→ 10 ) → ( 3 , c 1 �→ 6 , c 2 �→ 8 ) → ( 2 , c 1 �→ 5 , c 2 �→ 8 )

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? ( 1 , c 1 �→ 6 , c 2 �→ 0 ) → ( 2 , c 1 �→ 6 , c 2 �→ 10 ) → ( 3 , c 1 �→ 6 , c 2 �→ 8 ) → ( 2 , c 1 �→ 5 , c 2 �→ 8 ) → · · ·

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? ( 1 , c 1 �→ 6 , c 2 �→ 0 ) → ( 2 , c 1 �→ 6 , c 2 �→ 10 ) → ( 3 , c 1 �→ 6 , c 2 �→ 8 ) → ( 2 , c 1 �→ 5 , c 2 �→ 8 ) → · · · → ( 2 , c 1 �→ 1 , c 2 �→ 0 )

Reachability in Counter Automata 3 1 2 4 Can we reach ( 4 , c 1 �→ 0 , c 2 �→ 0 ) starting in ( 1 , c 1 �→ 6 , c 2 �→ 0 ) ? ( 1 , c 1 �→ 6 , c 2 �→ 0 ) → ( 2 , c 1 �→ 6 , c 2 �→ 10 ) → ( 3 , c 1 �→ 6 , c 2 �→ 8 ) → ( 2 , c 1 �→ 5 , c 2 �→ 8 ) → · · · → ( 2 , c 1 �→ 1 , c 2 �→ 0 ) → �

Reachability in Counter Automata • Reachability in counter automata is undecidable already for two counters [Minsky, 1961] • Reachability is NP-complete for one counter [H., Kreutzer, O., W., 2009] • Reachability is NL OG S PACE -complete for one counter with numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]

Reachability in Counter Automata • Reachability in counter automata is undecidable already for two counters [Minsky, 1961] • Reachability is NP-complete for one counter [H., Kreutzer, O., W., 2009] • Reachability is NL OG S PACE -complete for one counter with numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]

Reachability in Counter Automata • Reachability in counter automata is undecidable already for two counters [Minsky, 1961] • Reachability is NP-complete for one counter [H., Kreutzer, O., W., 2009] • Reachability is NL OG S PACE -complete for one counter with numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]

Reachability in Counter Automata • Reachability in counter automata is undecidable already for two counters [Minsky, 1961] • Reachability is NP-complete for one counter [H., Kreutzer, O., W., 2009] • Reachability is NL OG S PACE -complete for one counter with numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]

This talk: Can we naturally relate reachability problems in timed and counter automata?

This talk: Can we naturally relate reachability problems in timed and counter automata?

Bounded Counter Automata

Bounded Counter Automata • Counters are constrained to take values from bounded intervals from N • Zero tests can be discarded • Can be viewed as strongly-bounded VASS as defined by [Memmim, Roucairol, 1980]

Bounded Counter Automata • Counters are constrained to take values from bounded intervals from N • Zero tests can be discarded • Can be viewed as strongly-bounded VASS as defined by [Memmim, Roucairol, 1980]

Bounded Counter Automata • Counters are constrained to take values from bounded intervals from N • Zero tests can be discarded • Can be viewed as strongly-bounded VASS as defined by [Memmim, Roucairol, 1980]

Bounded Counter Automata • Reachability is trivially decidable and in PS PACE • Reachability with one counter is NP-hard and in PS PACE [Bouyer et al ., 2008] • Reachability with one counter inter-reducible with model considered by Demri and Gascon where counter ranges over Z and sign tests are allowed

Bounded Counter Automata • Reachability is trivially decidable and in PS PACE • Reachability with one counter is NP-hard and in PS PACE [Bouyer et al ., 2008] • Reachability with one counter inter-reducible with model considered by Demri and Gascon where counter ranges over Z and sign tests are allowed

Bounded Counter Automata • Reachability is trivially decidable and in PS PACE • Reachability with one counter is NP-hard and in PS PACE [Bouyer et al ., 2008] • Reachability with one counter inter-reducible with model considered by Demri and Gascon where counter ranges over Z and sign tests are allowed

Bounded Counter Automata • Reachability is trivially decidable and in PS PACE • Reachability with one counter is NP-hard and in PS PACE [Bouyer et al ., 2008] • Reachability with one counter inter-reducible with model considered by Demri and Gascon where counter ranges over Z and sign tests are allowed

Remainder of this talk Relating Reachability in Timed and Bounded Counter Automata with respect to logspace reductions: Reach. in n -clock TA, n ≥ 3 ⇐ ⇒ Reach. in bounded 2-CA Reach. in 2-clock TA ⇐ ⇒ Reach. in bounded 1-CA Reach. in 1-clock TA ⇐ ⇒ Reach. in bounded 0-CA

Remainder of this talk Relating Reachability in Timed and Bounded Counter Automata with respect to logspace reductions: Reach. in n -clock TA, n ≥ 3 ⇐ ⇒ Reach. in bounded 2-CA Reach. in 2-clock TA ⇐ ⇒ Reach. in bounded 1-CA Reach. in 1-clock TA ⇐ ⇒ Reach. in bounded 0-CA

Remainder of this talk Relating Reachability in Timed and Bounded Counter Automata with respect to logspace reductions: Reach. in n -clock TA, n ≥ 3 ⇐ ⇒ Reach. in bounded 2-CA Reach. in 2-clock TA ⇐ ⇒ Reach. in bounded 1-CA Reach. in 1-clock TA ⇐ ⇒ Reach. in bounded 0-CA

Remainder of this talk Relating Reachability in Timed and Bounded Counter Automata with respect to logspace reductions: Reach. in n -clock TA, n ≥ 3 ⇐ ⇒ Reach. in bounded 2-CA Reach. in 2-clock TA ⇐ ⇒ Reach. in bounded 1-CA Reach. in 1-clock TA ⇐ ⇒ Reach. in bounded 0-CA

Remainder of this talk Relating Reachability in Timed and Bounded Counter Automata with respect to logspace reductions: Reach. in n -clock TA, n ≥ 3 ⇐ ⇒ Reach. in bounded 2-CA Reach. in 2-clock TA ⇐ ⇒ Reach. in bounded 1-CA Reach. in 1-clock TA ⇐ ⇒ Reach. in bounded 0-CA

Bounded Two-Counter Automata and n -Clock Timed Automata

Bounded Two-Counter Automata and n -Clock Timed Automata bounded n -counter automata ⇓ bounded two-counter automata ⇓ n -clock timed automata, n ≥ 3 ⇓ bounded ( 2 n + 2 ) -counter automata

Bounded Two-Counter Automata and n -Clock Timed Automata bounded n -counter automata ⇓ bounded two-counter automata ⇓ n -clock timed automata, n ≥ 3 ⇓ bounded ( 2 n + 2 ) -counter automata

make bounds equal

make bounds equal

make bounds equal

make bounds equal

encode additional counters into second counter

encode additional counters into second counter

encode additional counters into second counter

“reserve” temporary storage on first counter

move “higher” counter values to temporary storage

block “upper” bits and simulate operation

move temporarily stored counters back

Bounded Two-Counter Automata and n -Clock Timed Automata bounded n -counter automata ⇓ bounded two-counter automata ⇓ n -clock timed automata, n ≥ 3 ⇓ bounded ( 2 n + 2 ) -counter automata

Bounded Two-Counter Automata and n -Clock Timed Automata bounded n -counter automata ⇓ bounded two-counter automata ⇓ n -clock timed automata, n ≥ 3 ⇓ bounded ( 2 n + 2 ) -counter automata

Simulating Bounded Two-Counter Automata with Timed Automata Main idea: • Assume uniform bound b on two counters • Store values of counters in difference of clock values • If x = b then x − y represents value of the first counter and x − z the value of the second counter • Replace in- and decrements by gadgets

Simulating Bounded Two-Counter Automata with Timed Automata Main idea: • Assume uniform bound b on two counters • Store values of counters in difference of clock values • If x = b then x − y represents value of the first counter and x − z the value of the second counter • Replace in- and decrements by gadgets

Simulating Bounded Two-Counter Automata with Timed Automata Main idea: • Assume uniform bound b on two counters • Store values of counters in difference of clock values • If x = b then x − y represents value of the first counter and x − z the value of the second counter • Replace in- and decrements by gadgets

Simulating Bounded Two-Counter Automata with Timed Automata Main idea: • Assume uniform bound b on two counters • Store values of counters in difference of clock values • If x = b then x − y represents value of the first counter and x − z the value of the second counter • Replace in- and decrements by gadgets

Simulating Bounded Two-Counter Automata with Timed Automata Main idea: • Assume uniform bound b on two counters • Store values of counters in difference of clock values • If x = b then x − y represents value of the first counter and x − z the value of the second counter • Replace in- and decrements by gadgets

Simulating Bounded Two-Counter Automata with Timed Automata

Simulating Bounded Two-Counter Automata with Timed Automata

Bounded Two-Counter Automata and n -Clock Timed Automata bounded n -counter automata ⇓ bounded two-counter automata ⇓ n -clock timed automata, n ≥ 3 ⇓ bounded ( 2 n + 2 ) -counter automata

Bounded Two-Counter Automata and n -Clock Timed Automata bounded n -counter automata ⇓ bounded two-counter automata ⇓ n -clock timed automata, n ≥ 3 ⇓ bounded ( 2 n + 2 ) -counter automata

Simulating n -Clock Timed Automata with Bounded Counter Automata • Main idea: simulate region abstraction on the counters

Simulating n -Clock Timed Automata with Bounded Counter Automata • Main idea: simulate region abstraction on the counters