SLIDE 1 On the Relationship between Reachability Problems in Timed and Counter Automata
Christoph Haase1,2 Joël Ouaknine2 James Worrell2
1now at LSV, CNRS & ENS de Cachan, France 2Department of Computer Science, University of Oxford, UK
Reachability Problems ’12 — September 18, 2012
SLIDE 2
Introduction
SLIDE 3 Timed Automata
- Comprise a finite-state controller with a finite number of clocks
ranging of R≥0
- Along transitions clocks can be compared to constants and
reset
- Constants are encoded in binary
SLIDE 4 Timed Automata
- Comprise a finite-state controller with a finite number of clocks
ranging of R≥0
- Along transitions clocks can be compared to constants and
reset
- Constants are encoded in binary
SLIDE 5 Timed Automata
- Comprise a finite-state controller with a finite number of clocks
ranging of R≥0
- Along transitions clocks can be compared to constants and
reset
- Constants are encoded in binary
SLIDE 6 Timed Automata
- Comprise a finite-state controller with a finite number of clocks
ranging of R≥0
- Along transitions clocks can be compared to constants and
reset
- Constants are encoded in binary
SLIDE 7 Timed Automata
- Comprise a finite-state controller with a finite number of clocks
ranging of R≥0
- Along transitions clocks can be compared to constants and
reset
- Constants are encoded in binary
SLIDE 8 Timed Automata
- Comprise a finite-state controller with a finite number of clocks
ranging of R≥0
- Along transitions clocks can be compared to constants and
reset
- Constants are encoded in binary
SLIDE 9
Reachability in Timed Automata
SLIDE 10
Reachability in Timed Automata
SLIDE 11
Reachability in Timed Automata
1 2 3 4
SLIDE 12 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2)
SLIDE 13 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2)
SLIDE 14 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) Yes we can!
SLIDE 15 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2)
SLIDE 16 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2)
SLIDE 17 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0)
SLIDE 18 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1)
SLIDE 19 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1) → (3, x → 0, y → 1)
SLIDE 20 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1) → (3, x → 0, y → 1) (3, x → 0.5, y → 1.5)
SLIDE 21 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1) → (3, x → 0, y → 1) (3, x → 0.5, y → 1.5) → (2, x → 0.5, y → 1.5)
SLIDE 22 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1) → (3, x → 0, y → 1) (3, x → 0.5, y → 1.5) → (2, x → 0.5, y → 1.5) (2, x → 1, y → 2)
SLIDE 23 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1) → (3, x → 0, y → 1) (3, x → 0.5, y → 1.5) → (2, x → 0.5, y → 1.5) (2, x → 1, y → 2) → · · ·
SLIDE 24 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1) → (3, x → 0, y → 1) (3, x → 0.5, y → 1.5) → (2, x → 0.5, y → 1.5) (2, x → 1, y → 2) → · · · → (2, x → 1, y → 23)
SLIDE 25 Reachability in Timed Automata
1 2 3 4
Can we reach (4, x → 1, y → 23) starting in (1, x → 4, y → 2) (1, x → 4, y → 2) → (2, x → 0, y → 0) (2, x → 1, y → 1) → (3, x → 0, y → 1) (3, x → 0.5, y → 1.5) → (2, x → 0.5, y → 1.5) (2, x → 1, y → 2) → · · · → (2, x → 1, y → 23)→ (4, x → 1, y → 23)
SLIDE 26 Reachability in Timed Automata
- General reachability problem is PSPACE-complete [Alur, Dill
1994]
- Reachability is PSPACE-complete for 3 clocks, or for an
unbounded number of clocks and constants from {0, 1} [Courcoubetis, Yannakakis, 1992]
- Reachability is NLOGSPACE-complete for one clock and
NP-hard for two clocks [Laroussinie, Markey, Schnoebelen, 2004]
SLIDE 27 Reachability in Timed Automata
- General reachability problem is PSPACE-complete [Alur, Dill
1994]
- Reachability is PSPACE-complete for 3 clocks, or for an
unbounded number of clocks and constants from {0, 1} [Courcoubetis, Yannakakis, 1992]
- Reachability is NLOGSPACE-complete for one clock and
NP-hard for two clocks [Laroussinie, Markey, Schnoebelen, 2004]
SLIDE 28 Reachability in Timed Automata
- General reachability problem is PSPACE-complete [Alur, Dill
1994]
- Reachability is PSPACE-complete for 3 clocks, or for an
unbounded number of clocks and constants from {0, 1} [Courcoubetis, Yannakakis, 1992]
- Reachability is NLOGSPACE-complete for one clock and
NP-hard for two clocks [Laroussinie, Markey, Schnoebelen, 2004]
SLIDE 29 Reachability in Timed Automata
- General reachability problem is PSPACE-complete [Alur, Dill
1994]
- Reachability is PSPACE-complete for 3 clocks, or for an
unbounded number of clocks and constants from {0, 1} [Courcoubetis, Yannakakis, 1992]
- Reachability is NLOGSPACE-complete for one clock and
NP-hard for two clocks [Laroussinie, Markey, Schnoebelen, 2004]
SLIDE 30 Reachability in Timed Automata
- General reachability problem is PSPACE-complete [Alur, Dill
1994]
- Reachability is PSPACE-complete for 3 clocks, or for an
unbounded number of clocks and constants from {0, 1} [Courcoubetis, Yannakakis, 1992]
- Reachability is NLOGSPACE-complete for one clock and
NP-hard for two clocks [Laroussinie, Markey, Schnoebelen, 2004]
SLIDE 31 Counter Automata
- Comprise a finite-state controller with a finite number of
counters ranging of N
- Along transitions counters can be incremented, decremented or
tested for zero
- Constants are encoded in binary
SLIDE 32 Counter Automata
- Comprise a finite-state controller with a finite number of
counters ranging of N
- Along transitions counters can be incremented, decremented or
tested for zero
- Constants are encoded in binary
SLIDE 33 Counter Automata
- Comprise a finite-state controller with a finite number of
counters ranging of N
- Along transitions counters can be incremented, decremented or
tested for zero
- Constants are encoded in binary
SLIDE 34 Counter Automata
- Comprise a finite-state controller with a finite number of
counters ranging of N
- Along transitions counters can be incremented, decremented or
tested for zero
- Constants are encoded in binary
SLIDE 35 Counter Automata
- Comprise a finite-state controller with a finite number of
counters ranging of N
- Along transitions counters can be incremented, decremented or
tested for zero
- Constants are encoded in binary
SLIDE 36 Counter Automata
- Comprise a finite-state controller with a finite number of
counters ranging of N
- Along transitions counters can be incremented, decremented or
tested for zero
- Constants are encoded in binary
SLIDE 37 Counter Automata
- Comprise a finite-state controller with a finite number of
counters ranging of N
- Along transitions counters can be incremented, decremented or
tested for zero
- Constants are encoded in binary
SLIDE 38
Reachability in Counter Automata
SLIDE 39
Reachability in Counter Automata
SLIDE 40
Reachability in Counter Automata
1 2 3 4
SLIDE 41 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)?
SLIDE 42 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)?
SLIDE 43 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? No we cannot!
SLIDE 44 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)?
SLIDE 45 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? (1, c1 → 6, c2 → 0)
SLIDE 46 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? (1, c1 → 6, c2 → 0) → (2, c1 → 6, c2 → 10)
SLIDE 47 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? (1, c1 → 6, c2 → 0) → (2, c1 → 6, c2 → 10) → (3, c1 → 6, c2 → 8)
SLIDE 48 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? (1, c1 → 6, c2 → 0) → (2, c1 → 6, c2 → 10) → (3, c1 → 6, c2 → 8) → (2, c1 → 5, c2 → 8)
SLIDE 49 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? (1, c1 → 6, c2 → 0) → (2, c1 → 6, c2 → 10) → (3, c1 → 6, c2 → 8) → (2, c1 → 5, c2 → 8) → · · ·
SLIDE 50 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? (1, c1 → 6, c2 → 0) → (2, c1 → 6, c2 → 10) → (3, c1 → 6, c2 → 8) → (2, c1 → 5, c2 → 8) → · · · → (2, c1 → 1, c2 → 0)
SLIDE 51 Reachability in Counter Automata
1 2 3 4
Can we reach (4, c1 → 0, c2 → 0) starting in (1, c1 → 6, c2 → 0)? (1, c1 → 6, c2 → 0) → (2, c1 → 6, c2 → 10) → (3, c1 → 6, c2 → 8) → (2, c1 → 5, c2 → 8) → · · · → (2, c1 → 1, c2 → 0) →
SLIDE 52 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 NLOGSPACE-complete for one counter with
numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]
SLIDE 53 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 NLOGSPACE-complete for one counter with
numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]
SLIDE 54 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 NLOGSPACE-complete for one counter with
numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]
SLIDE 55 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 NLOGSPACE-complete for one counter with
numbers encoded in unary [Lafourcade, Lugiez, Treinen, 2004]
SLIDE 56 This talk: Can we naturally relate reachability problems in timed and counter automata?
SLIDE 57 This talk: Can we naturally relate reachability problems in timed and counter automata?
SLIDE 58
Bounded Counter Automata
SLIDE 59 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]
SLIDE 60 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]
SLIDE 61 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]
SLIDE 62 Bounded Counter Automata
- Reachability is trivially decidable and in PSPACE
- Reachability with one counter is NP-hard and in PSPACE
[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
SLIDE 63 Bounded Counter Automata
- Reachability is trivially decidable and in PSPACE
- Reachability with one counter is NP-hard and in PSPACE
[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
SLIDE 64 Bounded Counter Automata
- Reachability is trivially decidable and in PSPACE
- Reachability with one counter is NP-hard and in PSPACE
[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
SLIDE 65 Bounded Counter Automata
- Reachability is trivially decidable and in PSPACE
- Reachability with one counter is NP-hard and in PSPACE
[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
SLIDE 66 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
⇐ ⇒
SLIDE 67 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
⇐ ⇒
SLIDE 68 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
⇐ ⇒
SLIDE 69 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
⇐ ⇒
SLIDE 70 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
⇐ ⇒
SLIDE 71
Bounded Two-Counter Automata and n-Clock Timed Automata
SLIDE 72 Bounded Two-Counter Automata and n-Clock Timed Automata
bounded n-counter automata ⇓ bounded two-counter automata ⇓ n-clock timed automata, n ≥ 3 ⇓ bounded (2n + 2)-counter automata
SLIDE 73 Bounded Two-Counter Automata and n-Clock Timed Automata
bounded n-counter automata ⇓ bounded two-counter automata ⇓ n-clock timed automata, n ≥ 3 ⇓ bounded (2n + 2)-counter automata
SLIDE 74
SLIDE 75 make bounds equal
SLIDE 76 make bounds equal
SLIDE 77 make bounds equal
SLIDE 78 make bounds equal
SLIDE 79
SLIDE 80 encode additional counters into second counter
SLIDE 81 encode additional counters into second counter
SLIDE 82 encode additional counters into second counter
SLIDE 83 “reserve” temporary storage on first counter
SLIDE 84
SLIDE 85 move “higher” counter values to temporary storage
SLIDE 86 block “upper” bits and simulate operation
SLIDE 87 move temporarily stored counters back
SLIDE 88 Bounded Two-Counter Automata and n-Clock Timed Automata
bounded n-counter automata ⇓ bounded two-counter automata ⇓ n-clock timed automata, n ≥ 3 ⇓ bounded (2n + 2)-counter automata
SLIDE 89 Bounded Two-Counter Automata and n-Clock Timed Automata
bounded n-counter automata ⇓ bounded two-counter automata ⇓ n-clock timed automata, n ≥ 3 ⇓ bounded (2n + 2)-counter automata
SLIDE 90 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
SLIDE 91 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
SLIDE 92 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
SLIDE 93 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
SLIDE 94 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
SLIDE 95
Simulating Bounded Two-Counter Automata with Timed Automata
SLIDE 96
Simulating Bounded Two-Counter Automata with Timed Automata
SLIDE 97 Bounded Two-Counter Automata and n-Clock Timed Automata
bounded n-counter automata ⇓ bounded two-counter automata ⇓ n-clock timed automata, n ≥ 3 ⇓ bounded (2n + 2)-counter automata
SLIDE 98 Bounded Two-Counter Automata and n-Clock Timed Automata
bounded n-counter automata ⇓ bounded two-counter automata ⇓ n-clock timed automata, n ≥ 3 ⇓ bounded (2n + 2)-counter automata
SLIDE 99 Simulating n-Clock Timed Automata with Bounded Counter Automata
- Main idea: simulate region abstraction on the counters
SLIDE 100 Simulating n-Clock Timed Automata with Bounded Counter Automata
- Main idea: simulate region abstraction on the counters
SLIDE 101 Simulating n-Clock Timed Automata with Bounded Counter Automata
- Main idea: simulate region abstraction on the counters
- Region abstraction treats two configurations as equivalent if
(a) their control locations are the same (b) the integral parts of each clock with a value below the maximum constant are the same (c) the relative orders of the fractional parts of the values of the clocks are the same (d) the clocks with fractional part 0 are the same
SLIDE 102 Simulating n-Clock Timed Automata with Bounded Counter Automata
- Main idea: simulate region abstraction on the counters
- Region abstraction treats two configurations as equivalent if
(a) their control locations are the same (b) the integral parts of each clock with a value below the maximum constant are the same (c) the relative orders of the fractional parts of the values of the clocks are the same (d) the clocks with fractional part 0 are the same
SLIDE 103 Simulating n-Clock Timed Automata with Bounded Counter Automata
- Main idea: simulate region abstraction on the counters
- Region abstraction treats two configurations as equivalent if
(a) their control locations are the same (b) the integral parts of each clock with a value below the maximum constant are the same (c) the relative orders of the fractional parts of the values of the clocks are the same (d) the clocks with fractional part 0 are the same
SLIDE 104 Simulating n-Clock Timed Automata with Bounded Counter Automata
- Main idea: simulate region abstraction on the counters
- Region abstraction treats two configurations as equivalent if
(a) their control locations are the same (b) the integral parts of each clock with a value below the maximum constant are the same (c) the relative orders of the fractional parts of the values of the clocks are the same (d) the clocks with fractional part 0 are the same
SLIDE 105 Simulating n-Clock Timed Automata with Bounded Counter Automata
- Main idea: simulate region abstraction on the counters
- Region abstraction treats two configurations as equivalent if
(a) their control locations are the same (b) the integral parts of each clock with a value below the maximum constant are the same (c) the relative orders of the fractional parts of the values of the clocks are the same (d) the clocks with fractional part 0 are the same
SLIDE 106 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
SLIDE 107 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
elapse of time
SLIDE 108 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
elapse of time
SLIDE 109 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
elapse of time
SLIDE 110 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
elapse of time
SLIDE 111 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
elapse of time
SLIDE 112 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
elapse of time
SLIDE 113 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
elapse of time
SLIDE 114 Simulating n-Clock Timed Automata with Bounded Counter Automata
... ... ... ... ...
1 1 1 1 1 1 1 relative order of clocks integral part of clocks
...
clock reset analogously
SLIDE 115 Theorem Reachability in k-clock timed automata with k ≥ 3 is logarithmic- space inter-reducible with reachability in bounded two-counter automata.
SLIDE 116 Theorem Reachability in k-clock timed automata with k ≥ 3 is logarithmic- space inter-reducible with reachability in bounded two-counter automata. Corollary Reachability in bounded k-counter automata is PSPACE-complete for k ≥ 2.
SLIDE 117
Bounded One-Counter Automata and Two-Clock Timed Automata
SLIDE 118 Two-Clock Timed Automata to Bounded One-Counter Automata
Given a timed automaton with x-constants {0, 1, 5} and y-constants {0, 1, 3}
SLIDE 119 Two-Clock Timed Automata to Bounded One-Counter Automata
Given a timed automaton with x-constants {0, 1, 5} and y-constants {0, 1, 3}
SLIDE 120 Two-Clock Timed Automata to Bounded One-Counter Automata
Given a timed automaton with x-constants {0, 1, 5} and y-constants {0, 1, 3} regions of the automaton
SLIDE 121 Two-Clock Timed Automata to Bounded One-Counter Automata
Given a timed automaton with x-constants {0, 1, 5} and y-constants {0, 1, 3}
SLIDE 122 Two-Clock Timed Automata to Bounded One-Counter Automata
Given a timed automaton with x-constants {0, 1, 5} and y-constants {0, 1, 3} regions and clock difference zones of the automaton
SLIDE 123 Two-Clock Timed Automata to Bounded One-Counter Automata
Elapse of time x ∈ (2, 3), y ∈ [0, 0]
SLIDE 124 Two-Clock Timed Automata to Bounded One-Counter Automata
Elapse of time x ∈ (2, 3), y ∈ (0, 1)
SLIDE 125 Two-Clock Timed Automata to Bounded One-Counter Automata
Elapse of time x ∈ (2, 3), y ∈ [1, 1]
SLIDE 126 Two-Clock Timed Automata to Bounded One-Counter Automata
Regions and clock difference zones are too coarse to fully capture reachability properties
SLIDE 127 Two-Clock Timed Automata to Bounded One-Counter Automata
Regions and clock difference zones are too coarse to fully capture reachability properties
SLIDE 128 Two-Clock Timed Automata to Bounded One-Counter Automata
Regions and clock difference zones are too coarse to fully capture reachability properties
SLIDE 129 Two-Clock Timed Automata to Bounded One-Counter Automata
Regions and clock difference zones are too coarse to fully capture reachability properties
SLIDE 130 Two-Clock Timed Automata to Bounded One-Counter Automata
Regions and clock difference zones are too coarse to fully capture reachability properties use counter in order to store difference between x and y
SLIDE 131 Two-Clock Timed Automata to Bounded One-Counter Automata
Counter encodes the difference between clocks when it is an integral value...
SLIDE 132 Two-Clock Timed Automata to Bounded One-Counter Automata
...and two consecutive integers if the difference between clocks lies in between those integral values
SLIDE 133 Two-Clock Timed Automata to Bounded One-Counter Automata
Suppose we wish to reset clock y only
SLIDE 134
Two-Clock Timed Automata to Bounded One-Counter Automata
SLIDE 135 Two-Clock Timed Automata to Bounded One-Counter Automata
- Resulting counter must be smaller than z + yu
SLIDE 136 Two-Clock Timed Automata to Bounded One-Counter Automata
- Resulting counter must be smaller than z + yu
- Resulting counter must be above xl
SLIDE 137 Two-Clock Timed Automata to Bounded One-Counter Automata
- Resulting counter must be smaller than z + yu
- Resulting counter must be above xl
SLIDE 138 Two-Clock Timed Automata to Bounded One-Counter Automata
- Resulting counter must be smaller than z + yu
- Resulting counter must be above xl
add yu to the counter, non-deterministically decrement counter and then check it is greater than xl
SLIDE 139
Two-Clock Timed Automata to Bounded One-Counter Automata
SLIDE 140
Two-Clock Timed Automata to Bounded One-Counter Automata
SLIDE 141 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below n + yu
SLIDE 142 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below n + yu
- counter must be above n + yl
SLIDE 143 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below n + yu
- counter must be above n + yl
SLIDE 144 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below n + yu
- counter must be above n + yl
add value from the interval [yl, yu] to the counter (requires gadget)
SLIDE 145
Two-Clock Timed Automata to Bounded One-Counter Automata
SLIDE 146
Two-Clock Timed Automata to Bounded One-Counter Automata
SLIDE 147 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below yl
- counter must be above yl
SLIDE 148 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below yl
- counter must be above yl
SLIDE 149 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below yl
- counter must be above yl
SLIDE 150 Two-Clock Timed Automata to Bounded One-Counter Automata
- counter must be below yl
- counter must be above yl
connect to a gadget which non-deterministically decrements the counter and then verifies that it is in (−yu, −yl)
SLIDE 151 Two-Clock Timed Automata to Bounded One-Counter Automata
Remaining polynomially many cases follow analogously
SLIDE 152 Bounded One-Counter Automata to Two-Clock Timed Automata to
- Other direction follows straightforwardly by encoding counter as
the difference of two clocks, similar to the case with two counters
SLIDE 153 Bounded One-Counter Automata to Two-Clock Timed Automata to
- Other direction follows straightforwardly by encoding counter as
the difference of two clocks, similar to the case with two counters Theorem Reachability in two-clock timed automata is logarithmic-space inter-reducible with reachability in bounded one-counter automata.
SLIDE 154
Answering the Pólya Question
SLIDE 155 George Pólya (1887-1985)
“If there is a problem you can’t solve, then there is an easier problem you can solve: find it.”
SLIDE 156
One control location, one self-loop
SLIDE 157 One control location, one self-loop
Reachability is NP-hard if the number of edges is unbounded and numbers are encoded in binary
SLIDE 158 One control location, one self-loop
Given a bound and a target, reachability is clearly decidable in polynomial time
SLIDE 159 One control location, one self-loop
Given a bound and a target, reachability is clearly decidable in polynomial time √
SLIDE 160 One control location, two self-loops
Given a bound a target and the Parikh image of a reaching run, reachability is decidable in polynomial time
SLIDE 161 One control location, two self-loops
Given a bound a target and the Parikh image of a reaching run, reachability is decidable in polynomial time √
SLIDE 162 Reachability via Lattice Paths
Idea: transform the reachability question into a question about the existence of lattice path in a convex polygon
SLIDE 163
Reachability via Lattice Paths
SLIDE 164
Reachability via Lattice Paths
SLIDE 165 Reachability via Lattice Paths
We get stuck since bound is too tight
SLIDE 166
Reachability via Lattice Paths
SLIDE 167
Reachability via Lattice Paths
SLIDE 168
Reachability via Lattice Paths
SLIDE 169
Reachability via Lattice Paths
SLIDE 170 Reachability via Lattice Paths
There exists a lattice path reaching a particular point (x, y) if, and
- nly if, the number of lattice points in the polygon is at least
x + y + 1
SLIDE 171 Implications for Two-Clock Timed Automata
Bézout automaton introduced in [Naves, 2006]
SLIDE 172
Implications for Two-Clock Timed Automata
SLIDE 173 One control location, three self-loops
Let the bound be 20 and the target be 12
SLIDE 174 One control location, three self-loops
Let the bound be 20 and the target be 12 Example of a reaching run where red=7, green=-11 and blue=17
SLIDE 175 Conclusion
This talk showed
- a relationship between reachability problems in timed and
bounded counter automata with respect to the resources available
- equivalence between two major problems that have been
stated as open
- a simple class of bounded one-counter automata for which
reachability is open
SLIDE 176 Conclusion
This talk showed
- a relationship between reachability problems in timed and
bounded counter automata with respect to the resources available
- equivalence between two major problems that have been
stated as open
- a simple class of bounded one-counter automata for which
reachability is open
SLIDE 177 Conclusion
This talk showed
- a relationship between reachability problems in timed and
bounded counter automata with respect to the resources available
- equivalence between two major problems that have been
stated as open
- a simple class of bounded one-counter automata for which
reachability is open
SLIDE 178 Conclusion
This talk showed
- a relationship between reachability problems in timed and
bounded counter automata with respect to the resources available
- equivalence between two major problems that have been
stated as open
- a simple class of bounded one-counter automata for which
reachability is open
