Transformation of Timed Automata into Mixed Integer Linear Programs Sebastian Panek

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Transformation ofTimed Automata into Mixed
Integer Linear ProgramsSebastian Panek
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Overview

• Motivation
• First modeling approach
• Syntax
• Semantic
• MILP-Formulation
• Formulation of scheduling problems
• Further work

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Motivation What?

• What do we want to do?
• Model an (optimization) problem as Linearly
  Priced Timed Automata (LPTA)
• Transform it into a Mixed-Integer Linear Program
  (MILP)
• Solve it using MILP algorithms
• Compare the results to other approaches in terms
  of computational effort and usability

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Motivation How?

• We have some experience in building optimization
  models of hybrid systems and scheduling problems
  (Engell, Stursberg, Sand)
• TA are simpler than hybrid automata, but there
  have been some open questions
• how to formulate parallel compositions?
• how to formulate synchronization?
• how to formulate continuous time?
• how to exploit the simpler structure of TA
• how to solve MILP models of TA faster?

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Motivation Why TA?

• TA allow modeling complex behaviors quite easily
  (simple syntax)
• Parallel compositions of TA allow the user
  specifying decomposed parts of a system
  separately
• There exist graphical editors and languages for
  the modeling of TA
• Powerful analysis tools are available

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Motivation Why MILP?

• MILP is appropriate for problems in many
  application domains
• A well-investigated MILP theory is known for many
  years
• Many different MILP solution algorithms and
  heuristics are available
• Powerful free and commercial MILP solvers have
  been developed
• Modeling and debugging is very difficult!

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First approach to the MILP formulation of TA

• LPTA (costs on locations and transitions)
• Continuous time in the MILP
• Finite set of time points at which transitions
  may occur
• Networks of TA are possible
• Bidirectional synchronization using labels
• More complex clock constraints for invariants and
  guards are supported (arbitrary polyhedra)

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Syntax

• Syntax of LPTA
• Additionally
• Since MILP models are static, the automaton cant
  simply stop in the final state
• Insert self-loops in all locations
• Those loops allow the TA to do nothing without
  additional costs

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Semanticstransitions

• Bounded time horizon (an upper bound for all
  clock valuations must exist)
• Finite number of time points n in the MILP
  implies finite number of transitions 2n in the TA
• Each time point in the MILP corresponds with
• 1 delay transition
• 1 discrete transition

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Semanticssynchronization

• Semantic for synchronization
• Synchronized transitions are taken if there are
  at least two automata waiting for the same label
• (bidirectional synchronization)

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MILP formulationvariables

• Example LPTA (Larsen et al.)
• MILP model with n4 time points
• Real variables for clock vectors
• Binary variables for
  locations and transitions

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MILP formulationconstraints on binary variables

• Real variables for time delays
• Binary product variables for all combinations of
  locations and outgoing transitions
• At every k the automaton is in one location
• At every k one transition is taken

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MILP formulationcomputing products

• Real product variables of clocks and locations
• Real product variables of clocks and transitions

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MILP formulationclock constraints

• Use polyhedral description to express clock
  constrains, i.e.
• Enforce invariants for locations
• Enforce guards for transitions

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MILP formulationevolution

• Delay transitions
• Discrete transitions
• Clock resets on transitions

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MILP formulationobjective function

• Define start state and final states fixing
  corresponding variables
• Define products for the objective function
• Objective function minmize costs over all runs

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TA formulation of scheduling problems

• According to O. Maler and A. Fehnker
• Use additional constraints to assert exclusive
  allocation of resources
• Example 2 jobs and 2 resources

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Further work

• Improve the MILP formulation
• Test it on large scale models
• Implement other types of TA (i.e. Uppaal-TA)
• Build a parser which accepts common TA
  description languages and generates MILP code
  automatically
• Compare the different approaches
• TA model, symbolic solution
• MILP model, MILP solution,
• TA model, MILP solution