Symbolic Model Checking without BDDs
Armin Biere1 Alessandro Cimatti2 Yunshan Zhu1
January 4, 1999 CMU-CS-99-101
Edmund Clarke1
School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213
Submitted for TACAS99 Science Department, Carnegie M
Lecture 2: Symbolic Model Checking With SAT
Edmund M. Clarke, Jr.
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
(Joint work over several years with: A. Biere, A. Cimatti, Y. Zhu,
A. Gupta, J. Kukula, D. Kroening, O. Strichman)
Proving Theorems Automatically,
Semi-Automatically, and Interactively
with TPS
Peter B. Andrews
http:/gtps.math.cmu.edu/tps.html
Developers of TPS:
Peter B. Andrews
Eve Longini Cohen
Dale A. Miller, Ph.D. 1983
Frank Pfenning, Ph.D. 1987
Sunil Issar, Ph.D.
Heuristics for
Efficient SAT Solving
As implemented in GRASP, Chaff and GSAT.
Formulation of famous problems as SAT: k-Coloring (1/2)
The K-Coloring problem:
Given an undirected graph G(V,E) and a natural number k, is
there an assignment color:
Formulatio
Model Checking with the
Partial Order Reduction
Edmund M. Clarke, Jr.
Computer Science Department
Carnegie Mellon University
Pittsburgh, PA 15213
1
Asynchronous Computation
The interleaving model for asynchronous systems allows
concurrent events to be ord
Lecture1: Symbolic Model Checking with BDDs
Edmund M. Clarke, Jr.
Computer Science Department
Carnegie Mellon University
Pittsburgh, PA 15213
Temporal Logic Model Checking
Specication Language: A propositional temporal logic.
Verication Procedure: Exhaust
Lecture 0: Computation Tree Logics
Model of Computation Computation Tree Logics The Logic CTL Path Formulas and State Formulas CTL and LTL Expressive Power of Logics
1
Model of Computation
a b State Transition Graph or Kripke Model
b c
c
a b
b c
c
a b
c
15-820A, Spring 2003
Solutions to Homework 1
1
Coloring a Graph with k-colors
The goal of this homework is to gain familiarity with SAT. We will encode an
interesting graph problem into a SAT problem and use modern SAT solvers
like GRASP and Cha to solve
15-820-a
Assignment 4
Partial Order Reduction
Due Mar. 19, 2003
1
LTL and Stuttering Equivalence
An LTL formula A f is invariant under stuttering if and only if for each pair of
paths and such that st ,
|= f if and only if |= f.
We denote the subset of t
15-820-a
Assignment 2
Computation Tree Logics
Due Feb. 26, 2003
1
CTL*
Show
2
CTL Operators
Show
A
3
U
E
U
EG
Buchi Automata
A non-deterministic B chi automaton (NBA)
u
is a ve-tuple
where is the alphabet, a nite set of states,
the set of initial stat
15-820-a
Assignment 5
Verication of ANSI-C with PVS
Due Apr. 30, 2003
1
Find the Minimum
1. Write a function in ANSIC that nds the minumum number in an array.
The size of the array is passed as a parameter.
2. Translate your ANSIC code into PVS language,
15-820-a
Assignment 3
Using SMV
Due March 5th, 2003
1
Informal Description
Following is a description of an elevator controller. In this exercise you will specify, implement, and check this controller using the SMV model checker.
The elevator spans three