Lower bounds for reachability in VASS in fixed dimension Wojciech - PowerPoint PPT Presentation

Lower bounds for reachability in VASS in fixed dimension Wojciech Czerwi ń ski Jerome Leroux S ł awomir Lasota University of Bordeaux University of Warsaw Agata Dubiak Ranko Lazic Filip Mazowiecki Ł ukasz Orlikowski University of Warsaw MPI Saarbruecken University of Warwick Infinity’20, online, 2020-07-07 1

Plan 1. Vector addition systems with states ( VASS ) and the reachability problem 2. Lower bounds in small fixed dimensions 3. Lower bounds in large fixed dimensions 2

Many faces of vector addition systems with states = • vector addition systems with states VASS • counter programs without 0-tests : [Hopcroft, Pansiot ’79] : • Petri nets [Petri 1962] : x y • VAS [Karp, Miller ’69] • multiset rewriting p q • … z 3

Counter programs without zero tests counters are nonnegative integer variables initially all equal zero a sequence of commands of the form: abort if x < n except for the very last command which is of the form: otherwise abort Example: initially: x’ = x = y = 0 no zero tests finally: x' = 0 x = 100 y = 200 4

Loop programs 5

Minsky machines the conditional jump of Minsky machines is simulated by counter program with zero tests : 6

Reachability (and coverability) Reachability problem : given a counter program without zero tests configuration reachability can it terminate (execute its halt command)? Coverability problem: given a counter program without zero tests with trivial halt command control-location reachability can it terminate (reach its halt command)? 7

Complexity of reachability (and coverability) … 2 } 2 2 O (n) 2 Time/space needed is at least F 3 (n) = coverability reachability Tower [Czerwi ń ski, L., Lazic, Leroux, Mazowiecki ’19] EXPSPACE lower see Jerome Leroux’s EXPSPACE bound invited talk on Thursday [Lipton ’76] [Lipton ’76] Ackermann [Leroux, Schmitz ’15, ’19] decidable EXPSPACE upper [Sacerdote, Tenney ‘77] [Mayr ’81] bound [Rackoff ’78] [Kosaraju ’82] [Lambert ’92] [Leroux ’09] Time/space needed is at most F 𝜕 (n) 8

2. Lower bounds for reachability in small fixed dimensions dimension = number of counters: 9

Reachability in dimension 1 and 2 encoding of integers unary binary NP * NL * dimension 1 [Haase, Kreutzer, Ouaknine, [Valiant, Peterson ’75] Worrell ’09] PSPACE * NL * dimension 2 [Blondin, Finkel, Goeller, [Englert, Lazic, Totzke ’16] Haase, McKenzie ’15] * complete shortest run has shortest run has polynomial length exponential length effectively flattable in dimension 2 Upper bounds similar to dimension 2 for every fixed dimension? …at least for flat counter programs? 10

Flat = no nested loops 11

Shortest runs in small dimensions [Czerwi ń ski, L., Lazic, Leroux, Mazowiecki ’20] unary flat unary binary poly * poly * exp * dim 1 poly * poly * exp * dim 2 ? exp * exp * dim 3 2-exp * dim 4 dim 5 … * upper bound * lower bound Key ingredient: computing exactly large numbers 12

Exponential shortest run in unary flat dim 3 objective: halts iff y divisible by x <= y halts iff x = (n+1) ・ y iff all multiplications exact iff all inner loops iterated maximally iff (n+1) ・ y divisible by 1, 2, …, n program size O (n 2 ), shortest run 2 O (nc) 13

Complexity lower bounds for reachability What about coverability? [Czerwi ń ski, L., Lazic, Leroux, Mazowiecki ’19, ’20] [Dubiak ’20] unary flat unary binary NL * NL * NP * dim 1 NL NL PSPACE NL * NL * PSPACE * dim 2 NL NL PSPACE dim 3 ? ? NL NL PSPACE dim 4 NL NL PSPACE NP * dim 5 also for binary flat NL NL PSPACE NP * … NL NL PSPACE NP * PSPACE * EXPSPACE * dim 13 NL NL PSPACE NP * EXPSPACE * 2-EXPSPACE * dim 14 NL NL PSPACE NP * 2-EXPSPACE * 3-EXPSPACE * dim 15 NL NL PSPACE … … … … * complete * hard 14

3. Lower bounds for reachability in large fixed dimensions 15

Parametric lower bound for reachability [Czerwi ń ski, L., Lazic, Leroux, Mazowiecki ’19] [Czerwi ń ski, L., Orlikowski ’??] unary flat unary binary dim 13 PSPACE * dim 14 EXPSPACE * dim 15 2-EXPSPACE * … … Minsky machine M of size n … n } with counters bounded by 2 O (n+h) Counter program of size O (n+h) 2 h+1 2 with O (h) = h+13 counters O ( n + l o g h ) i m p r o v e d r e d u Counter program of size O (n+log h) c t i o n with O (log h) counters 16

Parametric lower bound for reachability [Czerwi ń ski, L., Lazic, Leroux, Mazowiecki ’19] [Czerwi ń ski, L., Orlikowski ’??] unary flat unary binary dim 13 PSPACE * dim 14 EXPSPACE * dim 15 2-EXPSPACE * … … … n } 2 2 O (d) d-13 counters needs space at least 2 Reachability problem for programs of size n with d … n } 2 2 2 (d-13)/3 2 O (d) 2 improved lower bound: 17

Exponential amplifier x → x’ : starting with x>0 and x’ = 0, computes x’ exponentially larger than x (if x = 0 at the end) if so, also all other counters are forcedly 0 at the end 18

Composing exponential amplifiers unary flat unary binary dim 13 PSPACE * dim 14 EXPSPACE * dim 15 2-EXPSPACE * x0 { … … → x0 } x0 → x1 relay-race x2 { x1 x1 → x2 } x2 → x3 x3 M simulate M using x3 halt if x0, x1, x2, x3 = 0 19

x0 { Relay-race } objective: decrease the number of counters to logarithmic x2 { x1 } x4 { x3 } x6 { x5 } x7 halt if x0, x1, x2, x3, x4, x5, x5, x7 = 0 20

x0 { Counter recycling? } x2 { x0 = 0 x1 x0 } x4 { x3 } x6 { x5 } x7 halt if x0, x1, x2, x3, x4, x5, x5, x7 = 0 21

x0 { Counter recycling? } x0 { x1 x0 = 0 } x1 = 0 x0 { x1 x0 = 0 x1 = 0 } x0 { x0 = 0 x1 x1 = 0 x0 = 0 } 22

y0 { x0 { Supervisors } x0 + x1 = } x2 { x1 } y1 = x2 + x3 y2 { x4 { x3 } x4 + x5 = } x6 { x5 } = x6 + x7 y3 x7 halt if x0, x1, x2, x3, x4, x5, x5, x7, y0, y1, y2, y3, … = 0 23

y0 { x0 { Supervisors } x0 + x1 = } x2 { x1 } y1 = x2 + x0 y2 { x1 { x0 } x1 + x2 = } x0 { x2 } = x0 + x1 y3 x1 halt if x0, x1, x2, y0, y1, y2, y3, … = 0 24

z0 { y0 { Supersupervisors x0 { } } x2 { x1 } y1 } y2 { x1 { x0 } } x0 { x2 z1 } y0 x1 halt if x0, x1, x2, and so on… y0, y1, y2, z0, z1, … = 0 25

Summary unary flat unary binary unary flat unary binary dim 1 poly * poly * exp * NL * NL * NP * dim 1 dim 2 poly * poly * exp * ? NL * NL * PSPACE * exp * exp * dim 2 dim 3 ? dim 3 dim 4 2-exp * ? dim 4 dim 5 dim 5 NP * … * upper bound … NP * * lower bound dim 13 NP * PSPACE * EXPSPACE * I recruit for a dim 14 NP * EXPSPACE * 2-EXPSPACE * fully-funded dim 15 NP * 2-EXPSPACE * 3-EXPSPACE * PhD position in … … … … the NCN grant * complete … n } * hard 2 2 2 O (d) 2 Reachability problem for d- thank you! dimensional VASS of size n requires space at least 26