CS 367: Model-Based Reasoning Lecture 5 (01/29/2002) Gautam Biswas

Today’s Lecture Last Lecture: Concept of Blocking Non deterministic Automata Operations on Automata Accessible, Coaccessible parts, Composition Operators (Product, Parallel) Today’s Lecture: Parallel Composition Observer Automata State Space Refinement Automata with Input and Output Analysis of Discrete Event Systems

Composition of Automata Two kinds Product:   completely synchronous Parallel:   synchronous

Product Composition

Example: Product Composition b

Parallel Composition

Parallel Composition: Example b Two automata are synchronized on common events, E 1  E 2 One automaton can execute a private event without participation of other automata (E 1 \ E 2 )  (E 2 \ E 1 ) If E 1 = E 2 then parallel composition reduces to product If E 1  E 2 =  then G 1  G 2 is the concurrent behavior of G 1 and G 2 – called the shuffle of G 1 and G 2 G 1  G 2 = G 2  G 1 (G 1  G 2 )  G 3 = G 1  (G 2  G 3 )

Example of Parallel Composition Dining Philosophers (Multiple users sharing common resources) Philosophers: (i) think, (ii) eat Forks: (i) available, (ii) used Incomplete model Contains two deadlock states -- controller added to disallow deadlock

What is control? Control – selecting right input to system to achieve desired behavior: r(t) – reference signal – describes desired behavior

Complexity of Parallel Composition k component system, each component model has m states. If the event sets of each automata are distinct, then model of complete system has m k states (exponential growth) – curse of dimensionality How do we handle complexity? Incremental analysis Symbolic representations ( “Symbolic Model Checking: States and Beyond,” Burch, et al., Information and Computation, vol. 98, pp , 1998.

Observer Automata Conversion of non deterministic FSA to deterministic FSA : language preserving transformation Converted deterministic automaton is called the observer: Partition the set of events: Observer contains no unobservable events

Observer -- Example Note: G obs is deterministic

Observer : Algorithm

State Space Refinement Task: Comparison of two languages Refinement by Product: How to make this comparison computationally efficient?

Moore and Mealy Automata Moore automata: automata with state outputs Mealy: input/output automata FSA  Regular Languages

Analysis of Discrete Event Systems Safety and Blocking Properties Safety: avoiding undesirable states, or undesirable sequence of events for a composed automaton – “legal” or “admissible” language Determine if state y is reached from state x : perform accessible operation on automaton with x as initial state, look for y in result Determine if substring possible in automaton: “execute” substring for all accessible states Parallel composition complexity: Accessible, Coaccessible algorithms are linear in size of automaton Blocking Properties:

State Estimation Unobserved events:  events can be attributed to: (i) absence of sensors, (ii) event occurred remotely, not communicated, (iii) fault events Genuine unobservable events:

Daignostics Determine whether certain events with certainty: fault events Build new automata like observer, but attach “labels” to the states of G diag To build Attach N label to states that can be reached from x 0 by unobservable strings Attach Y label to states that can be reached from x 0 by unobservable strings that contain at least one occurrence of e d If state z can be reached both with and without executing e d then create two entries in the initial state set of G diag : zN and zY.

Diagnoser Automata