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Satisfiability and State-Transition Systems: An AI Perspective

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Title: Satisfiability and State-Transition Systems: An AI Perspective


1
Satisfiability and State-Transition Systems An
AI Perspective
  • Henry Kautz
  • University of Washington

2
Introduction
  • Both the AI and CADE/CAV communities have long
    been concerned with reasoning about
    state-transition systems
  • AI Planning
  • CADE/CAV Hardware and software verification
  • Recently propositional satisfiability testing has
    turned out to be surprisingly powerful tool
  • Planning SATPLAN (Kautz Selman)
  • Verification Bounded model checking (Clarke),
    Debugging relational specifications (Jackson)

3
Shift in KRR
  • Traditional approach specialized languages /
    specialized reasoning algorithms
  • New direction
  • Compile combinatorial reasoning problems into a
    common propositional form (SAT)
  • Apply new, highly efficient general search
    engines

SAT Encoding
Combinatorial Task
SAT Solver
Decoder
4
Advantages
  • Rapid evolution of fast solvers
  • 1990 100 variable hard SAT problems
  • 2000 100,000 variables
  • Sharing of algorithms and implementations from
    different fields of computer science
  • AI, theory, CAD, OR, CADE, CAV,
  • Competitions - Germany 91 / China 96 /
    DIMACS-93/97/98
  • JAR Special Issues SAT 2000
  • RISC vs CISC
  • Can compile control knowledge into encodings

5
OUTLINE
  • 1. Planning ? Model Checking
  • 2. Planning as Satisfiability
  • 3. SAT Petri Nets Randomization Blackbox
  • 4. State of the Art
  • 5. Using Domain-Specific Control Knowledge
  • 6. Learning Domain-Specific Control Knowledge
  • GOAL Overview of recent advances in planning
    that may (or may not!) be relevant to the CADE
    community!

6
1. Planning ? Model Checking
7
The AI Planning Problem
  • Given a world description, set of primitive
    actions, and goal description (utility function),
    synthesize a control program to achieve those
    goals (maximize utility)
  • most general case covers huge area of computer
    science, OR, economics
  • program synthesis, control theory, decision
    theory, optimization

8
STRIPS Style Planning
  • Classic work in AI has concentrated on STRIPS
    style planning (state space)
  • Open loop no sensing
  • Deterministic actions
  • Sequential (straight line) plans
  • SHAKEY THE ROBOT (Fikes Nilsson 1971)
  • Terminology
  • Fluent a time varying proposition, e.g.
    on(A,B)
  • State complete truth assignment to a set of
    fluents
  • Goal partial truth assignment (set of states)
  • Action a partial function State State
  • specified by Operator schemas

9
Operator Schemas
  • Each yields set of primitive actions, when
    instantiated over a given finite set of objects
    (constants)
  • Pickup(x, y)
  • precondition on(x,y), clear(x), handempty
  • delete on(x,y), clear(x), handempty
  • add holding(x), clear(y)
  • Plan A (shortest) sequence of actions that
    transforms the initial state into a goal state
  • E.g. Pickup(A,B) Putdown(A,C)

10
Parallelism
  • Useful extension parallel composition of
    primitive actions
  • Only allowed when all orderings are well defined
    and equivalent no shared pre / effects
  • (act1 act2)(s) act2(act1(s))
    act1(act2(s))
  • Can dramatically reduce size of search space
  • Easy to serialize
  • Distinguish
  • number of actions in a plan sequential length
  • number of sequentially composition operators in a
    plan parallel length, horizon
  • (a1 a2) (a3 a4 a5) a6
  • - sequential length 6, parallel length 3

11
Some Applications of STRIPS-Style Planning
  • Autonomous systems
  • Deep Space One Remote Agent (Williams Nayak
    1997)
  • Natural language understanding
  • TRAINS (Allen 1998)
  • Internet agents
  • Rodney (Etzioni 1994)
  • Manufacturing
  • Supply chain management (Crawford 1998)

12
Abdundance of Negative Complexity Results
  • Unbounded STRIPS planning PSPACE-complete
  • Exponentially long solutions
  • (Bylander 1991 Backstrom 1993)
  • Bounded STRIPS planning NP-complete
  • Is there a solution of (sequential/parallel)
    length N?
  • (Chenoweth 1991 Gupta and Nau 1992)
  • Domain-specific planning may depend on whether
    solutions must be the shortest such plan
  • Blocks world
  • Shortest plan NP-hard
  • Approximately shortest plan NP-hard
  • (Selman 1994)
  • Plan of length 2 x number blocks Linear time

13
Approaches to AI Planning
  • Three main paradigms
  • Forward-chaining heuristic search over state
    space
  • original STRIPS system
  • recent resurgence TLPlan, FF,
  • Causal link Planning
  • search in plan space
  • Much work in 1990s (UCPOP, NONLIN, ), little
    now
  • Constraint based planning
  • view planning as solving a large set of
    constraints
  • constraints specify relationships between actions
    and their preconditions / effects
  • SATPLAN (Kautz Selman), Graphplan (Blum
    Furst)

14
Relationship to Model Checking
  • Model checking determine whether a formula in
    temporal logic evaluates to true in a Kripke
    structure described by a finite state machine
  • FSM may be represented explicitly or symbolically
  • STRIPS planning special case where
  • Finite state matchine (transition relation)
    specified by STRIPS operators
  • Very compact
  • Expressive can translate many other
    representations of FSMs into STRIPS with little
    or no blowup

15
Relationship, continued
  • Formula to be checked is of the form
  • exists path . eventually . GOAL
  • Reachability
  • Distinctions between linear / branching temporal
    logics not important
  • Difference
  • Concentration on finding shortest plans
  • Emphasis on efficiently finding single witness
    (plan) as opposed to verifying a property holds
    in all states
  • NP vs co-NP

16
Why Not Use OBDDs?
  • Size of OBDD explodes for typical AI benchmark
    domains
  • Overkill need not / cannot check all states,
    even if they are represented symbolically!

O(2n2) states
(But see recent work by M. Velosa on using OBDDs
for non-deterministic variant of STRIPS)
17
Verification using SAT
  • Similar phenomena occur in some verification
    domains
  • Hardware multipliers
  • Has led to interest in using SAT techniques for
    verification and bug finding
  • Bounded fixed horizon
  • Under certain conditions can prove that only
    considering a fixed horizon is adequate
  • Empirically, most bugs found with small bounds
  • E. Clarke Bounded Model Checking
  • LTL specifications, FSM in SMV language
  • D. Jackson Nitpick
  • Debugging relational specifications in Z

18
2. Planning as Satisfiability
19
Planning as Satisfiability
  • SAT encodings are designed so that plans
    correspond to satisfying assignments
  • Use recent efficient satisfiability procedures
    (systematic and stochastic) to solve
  • Evaluation performance on benchmark instances

20
SATPLAN
instantiated propositional clauses
instantiate
axiom schemas
problem description
length
SAT engine(s)
interpret
satisfying model
plan
21
SAT Encodings
  • Target Propositional conjunctive normal form
  • Sets of clauses specified by axiom schemas
  • Create model by hand
  • Compile STRIPS operators
  • Discrete time, modeled by integers
  • upper bound on number of time steps
  • predicates indexed by time at which fluent holds
    / action begins
  • each action takes 1 time step
  • many actions may occur at the same step
  • fly(Plane, City1, City2, i) É at(Plane, City2, i
    1)

22
Solution to a Planning Problem
  • A solution is specified by any model (satisfying
    truth assignment) of the conjunction of the
    axioms describing the initial state, goal state,
    and operators
  • Easy to convert back to a STRIPS-style plan

23
Complete SAT Algorithms
  • Davis-Putnam-Loveland-Logeman (DPLL)
  • Depth-first backtrack search on partial truth
    assignments
  • Basis of nearly all practical complete SAT
    algorithms
  • Exception Stahlmarks method
  • Key to efficiency good variable choice at branch
    points
  • 1961 unit propagation, pure literal rule
  • 1993 - explosion of improved heuristics and
    implementations
  • MOMs heuristic
  • satz (Chu Min Li) lookhead to maximize rate of
    creation of binary clauses
  • Dependency directed backtracking derive new
    clauses during search rel_sat (Bayardo), GRASP
    (di Silva)
  • See SATLIB 1998 / Hoos Stutzle

24
Incomplete SAT Algorithms
  • GSAT and Walksat (Kautz, Selman Cohen 1993)
  • Randomized local search over space of complete
    truth assignments
  • Heuristic function flip variables to minimize
    number of unsatisfied clauses
  • Noisy random walk moves to escape local minima
  • Provably solves 2CNF, empirically successful on a
    broad class of problems
  • random CNF, graph coloring, circuit synthesis
    encodings (DIMACS 1993, 1997)

25
Planning Benchmark Test Set
  • Extension of Graphplan benchmark set
  • logistics - transportation domain, ranging up to
  • 14 time slots, unlimited parallelism
  • 2,165 possible actions per time slot
  • optimal solutions containing 74 primitive actions
  • 22000 legal states (60,000 Boolean variables)
  • Problems of this size not previously handled by
    any domain-independent planning system

26
Initial SATPLAN Results
problem horizon / actions Graphplan naïve SAT encoding hand SAT encoding
rocket-b 7 / 30 9 min 16 min 41 sec
log-a 11 / 47 13 min 58 min 1.2 min
log-b 13 / 54 32 min 1.3 min
log-c 13 / 63 1.7 min
log-d 14 / 74 3.5 min
SAT solver Walksat (local search) indicates no
solution found after 24 hours
27
How SATPLAN Spent its Time
problem instantiation walksat DPLL satz
rocket-b 41 sec 0.04 sec 1.8 sec 0.3 sec
log-a 1.2 min 2.2 sec 1.7 min
log-b 1.3 min 3.4 sec 0.6 sec
log-c 1.7 min 2.1 sec 4.3 sec
log-d 3.5 min 7.2 sec 1.8 hours
Hand created SAT encodings indicates no
solution found after 24 hours
28
3. SAT Petri Nets Randomization Blackbox
29
Automating Encodings
  • While SATPLAN proved the feasibility of planning
    using satisfiability, modeling the transition
    function was problematic
  • Direct naïve encoding of STRIPS operators as
    axiom schemas gave poor performance
  • Handcrafted encodings gave good performance, but
    were labor intensive to create
  • similar issues arise in work in verification
    division of labor between user and model checker!
  • GOAL fully automatic generation and solution of
    planning problems from STRIPS specifications

30
Graphplan
  • Graphplan (Blum Furst 1995)
  • Set new paradigm for planning
  • Like SATPLAN...
  • Two phases instantiation of propositional
    structure, followed by search
  • Unlike SATPLAN...
  • Efficient instantiation algorithm based on
    Petri-net type reachability analysis
  • Employs specialized search engine
  • Neither approach best for all domains
  • Can we combine advantages of both?

31
Blackbox
Plan Graph
Petri Net Analysis
STRIPS
Translator
CNF
Simplifier
General SAT engines
Solution
CNF
32
Component 1 Petri-Net Analysis
  • Graphplan instantiates a plan graph in a
    forward direction, pruning (some) unreachable
    nodes
  • plan graph ? unfolded Petri net (McMillian 1992)
  • Polynomial-time propagation of mutual-exclusion
    relationships between nodes
  • Incomplete must be followed by search to
    determine if all goals can be simultaneously
    reached

33
Growing the Plan Graph
facts
facts
actions
actions
P0
34
Growing the Plan Graph
facts
facts
actions
actions
P0
A1
P2
Q2
B1
R2
35
Growing the Plan Graph
facts
facts
actions
actions
P0
A1
P2
Q2
C3
B1
R2
36
Growing the Plan Graph
facts
facts
actions
actions
P0
A1
P2
Q2
C3
B1
R2
37
Growing the Plan Graph
facts
facts
actions
actions
P0
A1
P2
Q2
C3
B1
R2
38
Growing the Plan Graph
facts
facts
actions
actions
P0
A1
P2
Q2
B1
R2
39
Component 2 Translation
facts
facts
actions
actions
P0
A1
P2
Q2
B1
R2
Action implies preconditions A1 ? P0 , B1 ?
P0 Mutual exclusion ? A1 ? ?B1 , ? P2 ? ?
Q2 Initial facts hold at time 0 Goals holds at
time n
40
Component 3 Simplification
  • Generated wff can be further simplified by more
    general consistency propagation techniques
  • unit propagation is Wff inconsistant by
    resolution against unit clauses?
  • O(n)
  • failed literal rule is Wff P inconsistant
    by unit propagation?
  • O(n2)
  • binary failed literal rule is Wff P V Q
    inconsistant by unit propagation?
  • O(n3)
  • General simplification techniques complement
    Petri net analysis

41
Effective of Simplification
42
Component 3 Randomized Systematic Solvers
43
Background
  • Combinatorial search methods often exhibit
  • a remarkable variability in performance. It is
  • common to observe significant differences
  • between
  • different heuristics
  • same heuristic on different instances
  • different runs of same heuristic with different
    random seeds

44
How SATPLAN Spent its Time
problem instantiation walksat DPLL satz
rocket-b 41 sec 0.04 sec 1.8 sec 0.3 sec
log-a 1.2 min 2.2 sec 1.7 min
log-b 1.3 min 3.4 sec 0.6 sec
log-c 1.7 min 2.1 sec 4.3 sec
log-d 3.5 min 7.2 sec 1.8 hours
Hand created SAT encodings indicates no
solution found after 24 hours
45
Preview of Strategy
  • Well put variability / unpredictability to our
    advantage via randomization / averaging.

46
Cost Distributions
  • Consider distribution of running times of
    backtrack search on a large set of equivalent
    problem instances
  • renumber variables
  • change random seed used to break ties
  • Observation (Gomes 1996) distributions often
    have heavy tails
  • infinite variance
  • mean increases without limit
  • probability of long runs decays by power law
    (Pareto-Levy), rather than exponentially (Normal)

47
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48
Heavy Tails
  • Bad scaling of systematic solvers can be caused
    by heavy tailed distributions
  • Deterministic algorithms get stuck on particular
    instances
  • but that same instance might be easy for a
    different deterministic algorithm!
  • Expected (mean) solution time increases without
    limit over large distributions
  • Log-log plot of distribution of running times
    approximately linear

49
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50
Heavy-Tailed Distributions
  • infinite variance infinite mean
  • Introduced by Pareto in the 1920s
  • probabilistic curiosity
  • Mandelbrot established the use of heavy-tailed
    distributions to model real-world fractal
    phenomena
  • stock-market, Internet traffic delays, weather
  • New discovery good model for backtrack search
    algorithms
  • formal statement of folk wisdom of theorem
    proving community

51
Randomized Restarts
  • Solution randomize the systematic solver
  • Add noise to the heuristic branching (variable
    choice) function
  • Cutoff and restart search after a fixed number of
    backtracks
  • Provably Eliminates heavy tails
  • In practice rapid restarts with low cutoff can
    dramatically improve performance
  • (Gomes, Kautz, and Selman 1997, 1998)
  • Related analysis Luby Zuckerman 1993 Alt
    Karp 1996

52
Rapid Restart on LOG.D
Note Log Scale Exponential speedup!
53
  • Overall insight
  • Randomized tie-breaking with
  • rapid restarts can boost
  • systematic search algorithms
  • Speed-up demonstrated in many versions of
    Davis-Putnam
  • basic DPLL, satz, rel_sat,
  • Related analysis Luby Zuckerman 1993 Alt
    Karp 1996

54
Blackbox Results
problem naïve SAT encoding hand SAT encoding blackbox walksat blackbox satz-rand
rocket-b 16 min 41 sec 2.5 sec 4.9 sec
log-a 58 min 1.2 min 7.4 sec 5.2 sec
log-b 1.3 min 1.7 min 7.1 sec
log-c 1.7 min 15 min 9.3 sec
log-d 3.5 min 52 sec
Naïve/Hand SAT solver Walksat (local search)
indicates no solution found after 24 hours
55
4. State of the Art
56
Which Strategies Work Best?
  • Causal-link planning
  • lt5 primitive actions in solutions
  • Works best if few interactions between goals
  • Constraint-based planning
  • Graphplan, SATPLAN, descendents
  • 100 primitive actions in solutions
  • Moderate time horizon lt30 time steps
  • Handles interacting goals well
  • 1995 1999 Constraint-based approaches dominate
  • AIPS 1996, AIPS 1998

57
Graph Search vs. SAT
SATPLAN
Blackbox with solver schedule
Time
Graphplan
Problem size / complexity
Caveat on some domains SAT approach can exhaust
memory even though direct graph search is easy
58
Resurgence of A Search
  • In most of 1980 1990s forward chaining A
    search was considered a non-starter for planning
  • Voices in the wilderness
  • TLPlan (Bacchus) hand-tuned heuristic function
    could make approach feasible
  • LRTA (Geffner) can automatically derive good
    heuristic functions
  • Surprise AIPS-2000 planning competition
    dominated by A planners!
  • What happened?

59
Solution Length vs Hardness
  • Key issue relationship between solution length
    and problem hardness
  • RECALL In many domains, finding solutions that
    minimize the number of time steps is NP-hard,
    while finding an arbitrary solution is in P
  • Put all the blocks on the table first
  • Deliver packages one at a time
  • Long solutions minimize goal interactions, so
    little or no backtracking required by
    forward-chaining search
  • AIPS-2000 Planning Competition did not consider
    plan length criteria!

60
Non-Optimal Planning
61
Optimal-Length Planning
62
Which Works Best, Continued
  • Constraint-based planning
  • Short parallel solutions desired
  • Many interactions between goals
  • SAT translation a win for larger problems where
    time is dominated by search (as opposed to
    instantiation and Petri net analysis)
  • Forward-chaining search
  • Long sequential solutions okay
  • Few interactions between goals
  • Much recent progress in domain-independent
    planning
  • but further scaling to large real-world problems
    requires domain-dependent techniques!

63
5. Using Domain-Specific Control Knowledge
64
Kinds of Domain-Specific Knowledge
  • Invariants true in every state
  • A truck is only in one location
  • Implicit constraints on optimal plans
  • Do not remove a package from its destination
    location
  • Simplifying assumptions
  • Do not unload a package from an airplane, if the
    airplane is not at the packages destination city
  • eliminates connecting flights

65
Expressing Knowledge
  • Such information is traditionally incorporated in
    the planning algorithm itself
  • Instead use additional declarative axioms
  • (Bacchus 1995 Kautz 1998 Huang, Kautz, Selman
    1999)
  • Problem instance operator axioms initial and
    goal axioms control axioms
  • Control knowledge constraints on search and
    solution spaces
  • Independent of any search engine strategy

66
Axiomatic Form
  • State Invariant
  • at(truck,loc1,i) loc1 ¹ loc2 É Ø
    at(truck,loc2,i)
  • Optimality
  • at(pkg,loc,i) Ø at(pkg,loc,i1) iltj É Ø
    at(pkg,loc,j)
  • Simplifying Assumption
  • incity(airport,city) at(pkg,loc,goal) Ø
    incity(airport,city) É Ø unload(pkg,plane,airp
    ort)

67
Adding Control Knowledge
Problem Specification Axioms
Domain-specific Control Axioms
Instantiated Clauses
As control knowledge increases, Core shrinks!
SAT Simplifier
SAT Core
SAT Engine
68
Effect of Domain Knowledge
problem walksat walksat Kx DPLL DPLL Kx
rocket-b 0.04 sec 0.04 sec 1.8 sec 0.13 sec
log-a 2.2 sec 0.11 sec 1.8 min
log-b 3.4 sec 0.08 sec 11 sec
log-c 2.1 sec 0.12 sec 7.8 min
log-d 7.2 sec 1.1 sec
Hand created SAT encodings indicates no
solution found after 24 hours
69
6. Learning Domain-Specific Control Knowledge
70
Learning Control Rules
  • Axiomatizing domain-specific control knowledge by
    hand is a time consuming art
  • Certain kinds of knowledge can be efficiently
    deduced
  • simple classes of invariants (Fox Long
    Gerevini Schubert)
  • Can more powerful control knowledge be
    automatically learned, by watching planner solve
    small instances?

71
Form of Rules
  • We will learn two kinds of control rules,
    specified as temporal logic programs
  • (Huang, Selman, Kautz 2000)
  • Select rule conditions under which an action
    must be performed at the current time instance
  • Reject rule conditions under which an action
    must not be performed at the current time
    instance
  • incity(airport,city) GOAL(at(pkg,loc)) Ø
    incity(airport,city) É Ø unload(pkg,plane,airpo
    rt)

72
Training Examples
  • Blackbox initially solves a few small problem
    instances
  • Each instance yields
  • POSITIVE training examples states at which
    actions occur in the solution
  • NEGATIVE training examples states at which an
    action does NOT occur, even though its
    preconditions hold in that state
  • Note that this data is very noisy!

73
Rule Induction
  • Rules are induced using a version of Quinlans
    FOIL inductive logic programming algorithm
  • Generates rules one literal at time
  • Select rules maximize coverage of positive
    examples, but do not cover negative examples
  • Reject rules maximize coverage of negative
    examples, but do not cover positive examples
  • Prune rules that are inconsistent with any of the
    problem instances
  • For details, see Learning Declarative Control
    Rules for Constraint-Based Planning, Huang,
    Selman, Kautz, ICML 2000

74
Logical Status of Induced Rules
  • Some of the learned rules could in principle be
    deduced from the domain operators together with a
    bound on the length on the plan
  • Reject rules for unnecessary actions
  • But in general rules are not deductive
    consequences
  • Could rule out some feasible solutions
  • In worst case could rule out all solutions to
    some instances
  • not a problem in practice such rules are usually
    quickly pruned in the training phase

75
Effect of Learning
problem horizon blackbox learning blackbox
grid-a 13 21 4.8
grid-b 18 74 16.6
gripper-3 15 gt7200 7.2
gripper-4 19 gt7200 260
log-d 14 15.8 5.7
log-e 15 3522 291
mystery-10 8 gt7200 47.2
mystery-13 8 161 12.2
AIPS-98 competition benchmarks
76
Summary
  • Close connections between much work in AI
    Planning and CADE/CAV work on model checking
  • Remarkable recent success of general
    satisfiability testing programs on hard benchmark
    problems
  • Success of Blackbox and Graphplan in combining
    ideas from planning and verification suggest many
    more synergies exist
  • Techniques for learning and applying domain
    specific control knowledge dramatically boost
    performance for planning could ideas also be
    applied to verification?
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