CSCI1600: Embedded and Real Time Software Lecture 11: Modeling IV: Concurrency Steven Reiss, Fall 2015

Statecharts  What is missing  Timing information for real time analysis  Performance evaluation  Synchronization information between tasks  How to model synchronization properties of a system  To understand what is happening  To reason about those properties  To prove theorems (e.g. no deadlock)

Petri Nets  Created in 1962 by Carl Petri  Mathematical tool for studying communication between automata  With a graphical representation  With a mathematical underpinning  Later tools (CSP, CCS, …) are more language oriented  Data flow languages are a generalization of these  Utility of Petri Nets  Modeling and studying synchronization  Modeling and studying asynchronous behaviors  Modeling and studying resource sharing

Petri Net Usage  Good fit for embedded systems  Model communciation in FSAs  Some more modern equivalents  They remain widely used  Communication protocols  Manufacturing systems  Sequence controllers  Software systems  Mathematical framework can be used for  Performance evaluation (not the best notation for this)  Scheduling problems (okay, but not preferred)  Reliability, safety, boundedness, liveness

Motivation  Transitions are as important as states  Both involve labels, actions, etc.  Represent both transitions and states as nodes  With different notations  Use arcs just to represent connections  States actually represent preconditions to a transition  Result of a transition is a set of postconditions  Represented by a state

Petri Net Components  Bipartite graph  Places  Labeled circles  Represent states  Transitions  Labeled bars  Represent transitions  Arcs  P -> T : preconditions to a transition  T -> P : postconditions to a transition  Can be integer-weighted (multiple instances of same arc)  No P -> P, T -> T arcs

Petri Net Tokens  Each place can hold 0 or more tokens  Dots in the circle  Indicate precondition is true  State is active  Mapping of places to #tokens  Is a marking

Dynamics of Petri Nets  Petri nets are interesting for their dynamic behavior  Not their static description  Tokens move around according to fixed rules  Enabling rule : when a transition can fire  Firing rule : what happens when a transition fires

Enabling Rule  A transition T is enabled if each input place P of T contains at least the number of tokens equal to the weight of the arc connecting P to T

Firing Rule  An enabled transition may or may not fire depending on the interpretation  Only one at a time, fairness, nondeterminism, priorities  The firing of an enabled transition T removes from each input place P the number of tokens equal to the weight of the directed arc from P to T  It also deposits in each output place P’ the number of tokens equal to the weight of the directed arc connecting T to P.

Petri Net Extensions  Can add negation (absence of a token)  Can limit the number of tokens on a place  Effectively an output condition on a transition  Pure : no self loops  No place is input and output of a single transition  Can always convert a Petri net into a pure one  Choosing transitions to fire  Nondeterministic, one at a time  Nondeterministic, set of non-conflicting transitions  Can add priorities

Example: Mutli-Robot System  Two robot arms can pick and place items in common workspace  Only one can access the workspace at a time  Modeling mutual exclusion  There is a buffer with limited space for products  One robots adds the the buffer  The other removes from it  Modeling producer consumer  Robots can also work outside the workspace

Robot Model  P1, P4 : Robot performing tasks outside the work space  P2, P5: Robot waiting for access to the work space  P3, P6: Robot performing task in the common workspace  P7: Mutual exclusion  P8: Empty position in buffer  P9: Full position in buffer

Example Dining Philosphers

Petri Nets Reachability  Reachability  Can a Petri net reach a specific state  What might you be interested in  Reach = there exists a sequence of firings from M 0 to M x  The set of markings reachable from M 0 is the reachability set R(M 0 )

Petri Net Properties  A Petri Net is k-bounded if the number of tokens in any place P is always <= k for every reachable marking  Why is this important  Might want different k’s for different states  A Petri Net is safe if it is 1-bounded  A Petri Net is conservative if the number of tokens is conserved  Does not require the same number of tokens all the time  Tokens have different weights in different states

Petri Net Properties  A Petri Net is live if it is deadlock free  No reachable state that has no subsequent states  But could deadlock only part of the Petri Net  From any stare reachable from M0, it is possible to fire any transition in the net  Sufficient, not necessary (why)  Can quantify various degrees of liveness

Petri Net Properties  A Petri Net is reversable if for every marking M in R(M 0 ), M 0 is reachable from M  Error recovery  Could use a different state other them M 0

Proving Properties  Petri nets are inherently finite  Construct a tree of all possible markings  Flag states that are duplicates  Can handle unbounded token counts  Use ω to represent infinity, ω + n = ω, ω – n = ω  Can detect from the tree when a count can be arbitrary – then replace with ω  Overall tree is finite

Proving Properties  Given the finite coverability tree  Proving properties is easy  Just check the property holds in each tree node  Can be done mechanically

Homework  Read 2.4