Around State Equation Piotr Hofman University of Warsaw
Outline 1 Petri Nets. 2 The continuous reachability and the state equation. 3 Qcover and Icover. 4 Extensions of Petri Nets and corresponding state equations: reset, 1 test for zero, 2 data (unordered, ordered). 3
Petri Nets. T 1 Places. P 1 P 2 Transitions. T 2 P 3 P 4
Petri Nets. T 1 Places. P 1 P 2 Transitions. T 2 P 3 P 4 Tokens, a Configuration.
Petri Nets. T 1 Places. P 1 P 2 Transitions. T 2 P 3 P 4 Tokens, a Configuration. Firing a transition.
Petri Nets. T 1 Places. P 1 P 2 Transitions. T 2 P 3 P 4 Tokens, a Configuration. Firing a transition.
Petri Nets. T 1 Places. P 1 P 2 Transitions. T 2 P 3 P 4 Tokens, a Configuration. Firing a transition.
Description of the net, three matrices. T 1 P 1 P 2 1 0 0 1 Pre ( N ) = T 2 0 1 P 3 P 4 0 0 0 1 1 0 Post ( N ) = 1 0 0 1 ∆ = Post ( N ) − Pre ( N ) − 1 1 − 1 1 1 − 1 0 1
Reachability. Reachability is hard.
Reachability. Reachability is hard. We consider its relaxations.
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation).
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative.
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative. 2 As transitions are non blocking, the effect of the path depends only on the occurrences of transitions.
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative. 2 As transitions are non blocking, the effect of the path depends only on the occurrences of transitions. 3 Integer programming.
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative. 2 As transitions are non blocking, the effect of the path depends only on the occurrences of transitions. 3 Integer programming. Continuous reachability.
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative. 2 As transitions are non blocking, the effect of the path depends only on the occurrences of transitions. 3 Integer programming. Continuous reachability. 1 We consider a model where a fraction of a transition may be fired (the number of tokens gets rational).
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative. 2 As transitions are non blocking, the effect of the path depends only on the occurrences of transitions. 3 Integer programming. Continuous reachability. 1 We consider a model where a fraction of a transition may be fired (the number of tokens gets rational). 2 Reachability may be captured by the fragment of the existential fragment of FO ( Q , + , < ) .
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative. 2 As transitions are non blocking, the effect of the path depends only on the occurrences of transitions. 3 Integer programming. Continuous reachability. 1 We consider a model where a fraction of a transition may be fired (the number of tokens gets rational). 2 Reachability may be captured by the fragment of the existential fragment of FO ( Q , + , < ) . 3 We denote the fragment by FO poly 0 ( Q , + , < ) (by M. Blondin, C. Haase, LICS 2017).
Reachability. Reachability is hard. We consider its relaxations. Integer reachability (state equation). 1 We consider a model where the number of tokens may get negative. 2 As transitions are non blocking, the effect of the path depends only on the occurrences of transitions. 3 Integer programming. Continuous reachability. 1 We consider a model where a fraction of a transition may be fired (the number of tokens gets rational). 2 Reachability may be captured by the fragment of the existential fragment of FO ( Q , + , < ) . 3 We denote the fragment by FO poly 0 ( Q , + , < ) (by M. Blondin, C. Haase, LICS 2017). 4 A key part is the state equation solved over Q + .
Reachability. Reachability is hard.We consider its relaxations. f - target configuration. i - initial configuration. Integer reachability (state equation). x ∈ N | T | f − i = ∆ � x where �
Reachability. Reachability is hard.We consider its relaxations. f - target configuration. i - initial configuration. Integer reachability (state equation). x ∈ N | T | f − i = ∆ � x where � and an equivalent form: Can f − i be expressed as � i ∆[ j i ] where ∆[ j i ] are columns of ∆?
Reachability. Reachability is hard.We consider its relaxations. f - target configuration. i - initial configuration. Integer reachability (state equation). x ∈ N | T | f − i = ∆ � x where � and an equivalent form: Can f − i be expressed as � i ∆[ j i ] where ∆[ j i ] are columns of ∆? Continuous reachability, the main equation. x ∈ Q + | T | f − i = ∆ � x where �
Reachability. Reachability is hard.We consider its relaxations. f - target configuration. i - initial configuration. Integer reachability (state equation). x ∈ N | T | f − i = ∆ � x where � and an equivalent form: Can f − i be expressed as � i ∆[ j i ] where ∆[ j i ] are columns of ∆? Continuous reachability, the main equation. x ∈ Q + | T | f − i = ∆ � x where � and an equivalent form: Can f − i be expressed as � i a i ∆[ j i ] where a i ∈ Q + and ∆[ j i ] are columns of ∆?
Qcover + Icover 1 Make a standard backward coverability algorithm. 2 For every new element in the pre ( base ) check if it can be covered from the initial configuration. if NO then drop it else add it to the base. 3 covered in the continuous reachability sense (Qcover), integer reachability sense (Icover). 4 Use SMT-solver to solve linear programs / integer programs.
Different extensions of Petri Nets New types of arc Reset + affine. Test for zero (backward coverability algorithm does not work). Coloured tokens /data 1 Unordered data Petri nets. The backward coverability algorithm terminates. 2 Ordered data Petri nets. The backward coverability algorithm terminates. 3 Tuples of unordered data. The backward coverability algorithm does not terminate (coverability is undecidable).
State equation in the case of the reset nets. 1 A place gets value 0 if it is reset. 2 We consider integer runs. 3 Integer reachability is NP − complete , result by S. Halfon and Ch. Haase, RP 2014. Affine Filip explained it.
Test for zero( unpublished results form June 2018). Description. 1 A special test for zero arc. 2 A transition can be fired if the number of tokens in the place equals zero.
Test for zero( unpublished results form June 2018). Description. 1 A special test for zero arc. 2 A transition can be fired if the number of tokens in the place equals zero. 2 places may be tested for zero + finite automaton. Integer reachability is undecidable, by S. Lasota.
Test for zero( unpublished results form June 2018). Description. 1 A special test for zero arc. 2 A transition can be fired if the number of tokens in the place equals zero. 2 places may be tested for zero + finite automaton. Integer reachability is undecidable, by S. Lasota. 1 place can be tested for zero. Integer reachability is NP − complete by P. Hofman, D. Knop, M. Pilipczuk, Akshay S., M. Wrochna.
Test for zero( unpublished results form June 2018). Description. 1 A special test for zero arc. 2 A transition can be fired if the number of tokens in the place equals zero. 2 places may be tested for zero + finite automaton. Integer reachability is undecidable, by S. Lasota. 1 place can be tested for zero. Integer reachability is NP − complete by P. Hofman, D. Knop, M. Pilipczuk, Akshay S., M. Wrochna. 1 place can be tested for zero. Continuous reachability is P − complete by P. Hofman, Akshay S.
Test for zero( unpublished results form June 2018). Description. 1 A special test for zero arc. 2 A transition can be fired if the number of tokens in the place equals zero. 2 places may be tested for zero + finite automaton. Integer reachability is undecidable, by S. Lasota. 1 place can be tested for zero. Integer reachability is NP − complete by P. Hofman, D. Knop, M. Pilipczuk, Akshay S., M. Wrochna. 1 place can be tested for zero. Continuous reachability is P − complete by P. Hofman, Akshay S.
Continuous reachability + one test for zero Abstract Petri nets 1 The finite set of places, say d . 2 The finite set of formulas over 2 d free variables x 1 , x 2 . . . x 2 d . 3 The set of transitions is a set of all true valuations of a formula. 4 The set of configurations is Q + d .
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