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Exploration of Large State Spaces

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Title: Exploration of Large State Spaces


1
Exploration of Large State Spaces
  • Armando Tacchella
  • Lab - Software Engineering
  • DIST Università di Genova

2
Scenario
  • Applications
  • Formal verification
  • Planning
  • Issues
  • Is there a bug in the design?
  • Is there a plan to reach the goal?

3
Formal verification
  • Modulo 4 counter
  • Bug it is not possible to reach s00 starting
    from s01 or s10
  • The bug can be discovered, e.g., by trying to
    reach s00 either from s01 or s10

4
Why formal verification?
Presented at DAC2001 by Bob Bentley, Intel Corp.
5
Planning
  • Blocks world
  • A block can be
  • on top of another block
  • on top of the table
  • Blocks can be moved from a source to a
    destination
  • The goal is to rebuild the tower upside-down
  • The plan is the sequence of moves to the goal

1
2
3
4
5
6
Common model
  • Set of states (configurations)
  • Transitions between states
  • Set of initial states
  • Set of final states
  • Is there a path from some initial state to some
    final state?
  • Solving a reachability problem on a graph

7
Reachability
  • Graph representation
  • each node is a state
  • each arc is a transition
  • One ore more sources (initial states)
  • One ore more targets (final states)
  • Reachability can be solved with standard graph
    algorithms
  • Optimization on the path length can be done
    using, e.g., Djikstra algorithm

8
Representing states
  • States are encoded using vectors of boolean
    variables
  • State variable x x1, ... ,xN
  • A state is an assignment of boolean values 0,1
    to a state variable
  • State s v1, ... ,vN where vi ? 0,1

9
How large is the state space?
  • 2N states (and 22N transitions) at most
  • In real sized problems N is easily gt100
  • How large is 2100?
  • Consider that 2100ns 31012yr
  • Classical graph representations may not be
    feasible in practice!

10
Symbolic encoding
  • Use boolean formulas to encode
  • Initial states I(x)
  • Transitions T(x, x)
  • Final states F(x)
  • Given two states s,t
  • I(s) 1 exactly when s is an initial state
  • T(s,t) 1 exactly when there is a transition
    between s and t
  • F(s) 1 exactly when s is a final state

11
A glimpse into Boolean logic...
  • Every variable (x1, x2, ...) is a formula
  • If F and G are formulas
  • F is a formula (negation of F)
  • FG (disjunction), FG (conjunction), F?G
    (implication) are formulas
  • Consider the following abbreviations

12
Symbolic encoding (example)
Counter modulo N ? 2N nodes
TN ? O(N2) symbols
13
Bounded symbolic reachability
  • Reaching a final state from an initial one with a
    path of length at most k (nodes)
  • If R(s1, ... ,sk)1 then the sequence s1, ... ,sk
    has the following properties (i ? 1, ... ,k)
  • I(s1)1
  • T(si,si1)1 for all si
  • F(si)1 for some si

14
Symbolic reachability (example)
R(x1,x2,x3) 0 for all values of x1,x2,x3 ? s00
is unreachable from s10
15
Solving symbolic reachability
  • Symbolic encondings enable handling of large
    state spaces
  • Bounded symbolic reachability amounts to finding
    s1, ... ,sk s.t. R(s1, ... ,sk)1
  • Decide whether the boolean formula R is
    satisfiable or not (a.k.a. SAT problem)
  • There is no free lunch SAT is NP-hard!
  • Is this a limitation?

16
A glimpse into complexity...
  • Two resources TIME (omitted) and SPACE
  • P polynomial, EXP exponential
  • N non-deterministic
  • co complement of

Symbolic reachability and Q-SAT
Bounded symbolic reachability and SAT
Reachability
NP
co-NP
P
PSPACE
EXP
17
Solving SAT preliminaries
  • Formulas in Conjunctive Normal Form
  • The formula is a set (conjunction) of clauses
  • Each clause is a set (disjunction) of literals
  • A literal is a variable or the negation of a
    variable
  • Given any formula F it is always possible to
    produce F in CNF s.t. F is satisfiable exactly
    when F is satisfiable and Fpoly(F)

18
Formulas and CNF (example)
?
?
?
T4(x,x) in CNF
T4(x,x)
19
Solving SAT search algorithm
  • Search(F)Simplify(F)if F? return 1if ??F
    return 0l ? ChooseLiteral(F)if Search(F?l)
    then return 1else return Search(F?-l)
  • Simplify(F)while ?l l?F do for each C?F
    l?C F F/C for each C?F -l?C F
    F/C?C/-lend

20
Search process (example)
21
Solving SAT in practice
  • The performance of the search algorithm
    critically depends on
  • the particular ChooseLiteral heuristic
  • the amount of simplification performed
  • the smartness of the backtracking schema
  • No silver bullet, but state-of-the-art SAT
    solvers can solve industrial scale problems with
    thousands of variables!

22
Research issues
  • Bounded symbolic reachability via SAT
  • performs very well on bug-finding
  • when the error trace is short, or
  • the diameter of the search space is small
  • Nevertheless
  • since there can be up to 2N states, it may not be
    feasible for general symbolic reachability, and
  • it can become impractical even for error traces
    of reasonable lengths

23
Research issues (ctd.)
  • Tools for reasoning with boolean formulas
  • are routinely used in reasearch and industry
  • reach good performance and capacity standards
  • Nevertheless
  • most of them is special purpose (disposable code)
  • they are difficult (if not impossible) to
    integrate into existing systems
  • most often they are unsupported, undocumented,
    not robust enough for time/safety/money-critical
    applications

24
Lab core research
  • Encodings for (bounded) symbolic reachability
    exploiting quantified Boolean formulas
  • compact and (possibly) effective, but
  • challenging solving Q-SAT is PSPACE-hard!
  • A toolkit for reasoning with Boolean formulas
  • handles quantified Boolean formulas
  • features a component-based architecture
  • Integrates several services, e.g., enumeration of
    assignments, logic minimization,
  • is reasonably efficient w.r.t. special purpose
    tools

25
Formal verification projects
  • FIRB Knowledge Level Automated Software
    Engineering ( ends in 2005)
  • PRIN Advanced Reasoning Systems for the
    representation and Formal Verification of Complex
    Systems (ends in 2004)
  • INTEL SAT Solvers for Symbolic Model Checking
    and Formal Verification (2001-2003)

26
Planning projects
  • ASI-DOVES Enabling On-board Autonomy A platform
    for the Development of Verified Software (ends in
    2004)
  • ASI-SACSO Safety Critical Software for planning
    space robotics (ends in 2004)
  • ASI-GMES Un Sistema Innovativo per la gestione
    di Costellazioni di Satelliti e la sua
    Applicazione alla Tutela Ambientale (proposta)
  • RoboCare Sistema multi-agente con componenti
    fisse e robotiche mobili intelligenti (fine nel
    2005)

27
FIRB
  • Knowledge Level
  • Automated Software
  • Engineering

4 Milioni di Euro
28
FIRB (objectives)
  • A Knowledge Level Automated Software Engineering
    methodology,
  • A requirement actor and goal oriented framework
  • Theories and techniques for the code analysis
  • A concept demonstrator prototype, integrating the
    developed techniques
  • The application of the prototype to a case study

29
FIRB (activities)
  • Development of a methodology based on the
    goal/actors paradigm
  • Automated Reasoning for validation and
    verification of software (QBF, BMC, SAT...)
  • Automated Planning for software development
    automation
  • Natural language processing for documentation
    analysis
  • Analysis and Testing of systems based on the
    goal/actors paradigm

30
Lab activies on FIRB
  • Development of a planning language for the
    goal/actor framework
  • Study and development of planning techniques
    based on SAT
  • Study and development of planning techniques
    based on QBF
  • Development of a Tool for formal verification

31
Ricerca tesisti per FIRB ?
  • Buone conoscenze di
  • Informatica di base (algoritmi e strutture dati)
  • Linguaggi C/C standard
  • Lingua Inglese
  • Disponibiltà
  • A lavorare sodo in un team giovane e in crescita
  • A trascorrere periodi a Trento durante la tesi
  • Ad iniziare la tesi a Settembre/Ottobre 2003
  • Programma
  • Formazione iniziale a Genova durante la tesi
  • Completemento attività presso ITC/IRST di Trento
    con contratto di collaborazione annuale
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