Artificial Intelligence and Lisp #11
Satisfiability Solvers (SAT Techniques)
Constraint Propagation
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Remember from lecture #9: Details of proof
[a? [b? [c?
[c?
[b? [c?
[c?
red green]
blue white]]
white red]
blue green]]]
white
-a v -b v -c v red
-a v -b v c v green
-a v b v -c v blue
-a v b v c v white
a v -b v -c v white
a v -b v c v red
a v b v -c v blue
a v b v c v green
-white v -red
-white v -green
-white v -blue
(and more...)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
-red
-a v -b v -c
a v -b v c
-green
-a v -b v c
a v b v c
-blue
-a v b v -c
a v b v -c
-b v c
b v -c
(-b v c) & (b v -c)
(b -> c) & (c -> b)
b <-> c
Propositional vs Predicative Representation
•
Propositional: each literal is a proposition symbol or its
negation. Simple resolution operator.
•
Predicative: literals are formed using a predicate symbol
and its arguments, which may be formed using function
symbols, variables, and constant symbols. Resolution is
generalized to include unification (substitution of
subexpressions for variables)
•
Predicative representation is often necessary (rabbit
offspring example!)
Propositional vs Predicative Representation
•
Propositional: each literal is a proposition symbol or its
negation. Simple resolution operator.
•
Predicative: literals are formed using a predicate symbol
and its arguments, which may be formed using function
symbols, variables, and constant symbols. Resolution is
generalized to include unification (substitution of
subexpressions for variables)
•
Predicative representation is often necessary (rabbit
offspring example!)
•
Idea in SAT solvers: work with very large numbers of
propositional clauses (even millions). Use computational
brute force with very efficient algorithms.
Boolean Satisfiability Problem (SAT)
•
Given: a set of propositional clauses
•
Question: does there exist an assignment of
truthvalues to the proposition symbols whereby all
the clauses are true?
•
In other words, does there exist an assignment whereby
at least one of the literals in each clause is true?
•
Two main methods: (1) The DPLL Procedure: search the
space of truthvalue assignments in a systematic,
depth-first way
•
(2) Stochastic Local Search: Pick one assignment
randomly, then change the value of one proposition
symbol at a time
A Very Simple Example
a,
a,
-a
a,
a,
-b, c
b, d
-c
-d
a, c, d
c, d
a, d
d
a
False
Consider each clause as a set of literals, and the empty set as False
Completeness theorem
•
If there does not exist an assignment of truth values
whereby all given clauses are made true, then the
resolution operator can produce the empty clause
•
If there exists an assignment of truth values whereby all
given clauses are made true, then the resolution
operator can not produce the empty clause
•
Proof for the second part: Consider a case where there is
such an assignment, and assume that there is a resolution
sequence leading to False. One can easily see that if two
clauses are true with a given assignment, then any clause
that is obtained by resolving them is also true with that
assignment. By induction, this holds for all clauses in the
resolution sequence. However, the last step before arriving to
False must resolve two unit clauses e.g. a and -a. This is
impossible.
Proving that A |- c
•
Suppose that A is a set of clauses and c is a clause
•
Obtain A' by adding the negation of c to A
•
Try to prove that A' leads to False. If you succeed then
you have also proved that A leads to c
•
One way of proving that A leads to False is to show that
there does not exist any assignment of truth values
whereby all clauses in A' are true
•
Proving that such an assignment does not exist can be
done by making an exhaustive search for it; if the
exhaustive search fails then the assignment does not
exist.
Example
•
Given: a v b, a → c, c → d, b → e
•
Try to prove: d v e
•
Rewrite as: {a, b}, {-a, c}, {-c, d}, {-b, e}, {-d}, {-e}
•
Resolution obtains: {c, b}, {d, b}, {d, e}, {e}, {}
•
Another way of doing the resolutions: {-c}, {-a}, {b}, {e},
{}; this does resolutions using a unit preference strategy
•
In the first resolution sequence we obtained the desired
conclusion just before we obtained False i.e. {}
•
In the second resolution sequence we did not have the
desired conclusion as an intermediate result.
Summing up so far
•
In order to prove A |- c, form A' by adding the negation of
c to A, and prove that A' leads to a contradiction (False,
empty clause)
•
A' leads to a contradiction if and only if there does not
exist any assignment of truth values to proposition
symbols whereby all clauses in A' are true
•
SAT is a method for determining exactly that
•
Notice that it may not be necessary to know the exact
assignment, just to know whether it exists!
The DPLL Procedure
•
DPLL = Davis-Putnam-Logemann-Loveland, after the inventors of
the basic procedure (long ago)
•
The procedure obtains its strengths by the various “plug-in”
methods that have been developed for it
•
Basic algorithm for a given set A of clauses:
•
Pick one of the prop symbols, e.g. p, and construct one
modification of A for each of the two truth-values. Obtain A[p] by
removing all clauses containing p and by removing -p in all clauses
where they occur. Similarly for A[-p].
•
Repeat the same operation with another prop symbol, obtaining
e.g. A[p,-q], and proceed recursively obtaining a search tree.
•
If a branch obtains a descendant of A containing both {a} and {-a}
for some literal a, then close that branch, i.e. do not expand the
tree further from that point. This is called a conflict.
•
If you can find a branch where all the prop symbols have a value,
then you have found an assignment. If all branches become closed
then no assignment can exist whereby all clauses are true.
Implementation considerations for DPLL
•
Do a (modified) depth-first search, not a breadth-first search
of the tree of possible assignments
•
Implement iteratively rather than using recursion
•
Literals in unit clauses are immediately set to true (as a
preprocessing step and during the computation), except if
you have a conflict (in which case close that branch)
•
Proposition symbols that only occur positively in all the given
clauses are immediately set to true and those clauses are
removed. Conversely for those prop symbols that only occur
negatively
•
Decision strategies must be used; a large number of these
have been developed. A decision strategy gives rules for
which proposition symbol to consider next, and which value to
choose for it
Example, revisited
•
Given: a v b, a → c, c → d, b → e
•
Try to prove: d v e
•
Rewrite as: {a, b}, {-a, c}, {-c, d}, {-b, e}, {-d}, {-e}
•
Select d and e false since they occur in unit clauses,
obtaining {a, b}, {-a, c}, {-c}, {-b}
•
Select c and b false for the same reason, obtaining
{a}, {-a} which is a conflict
•
The current branch must be closed, but it is the only one,
so no assignment can be found
•
Important: the actual implementation does not rewrite the
set of clauses, it just keeps track of the current, partial
assignment of truth-values to proposition symbols
Examples of Decision Strategies
•
Maximum Occurrence in clauses of Minimum Size
(MOMS) as a goodness measure for selecting prop
symbol
•
Dynamic Largest Individual Sum (DLIS): choose the
literal occurring the most frequently in the clauses at
hand
•
Variable State Independent Decaying Sum (VSIDS):
keep track of the 'weight' of each literal, allow it to 'decay'
i.e. it is gradually reduced over time, but if a literal is
used for closing a branch then it is 'boosted' (its value is
increased) for use elsewhere in the search tree
•
There are also several other decision strategies.
The Watched Literals Scheme
•
The purpose of this scheme is to reduce the amount of bookkeeping for
truth-value assignments after backup in an iterative implementation of
DPLL. It is used in most contemporary SAT systems.
•
Maintain two literals in each clause, called its watched literals. These
must be unassigned or true in the current assignment
•
Maintenance when a literal a is set to False: (1) In every clause
has a as a watched literal, find another one to watch
•
(2) In every clause containing -a, make that one of its two watched
literals, replacing one of the present ones
•
Corresponding operation when a is set to True i.e. -a to False
•
Then, as long as you descend recursively into the tree, to check
whether a clause is satisfied you merely check its two watched literals
•
Also, when you backtrack, it is not necessary to update the watched
literals; they will self-adjust after a while. Until then the computation is
less efficient, but the overall efficiency is much improved.
that
Clause learning and Randomized Restart
•
Modify the basic algorithm as follows. When you arrive to a
conflict then analyze the situation and identify what clauses
contributed to the conflict. Extract one or more additional
“learned” clauses that are added to the given ones. Also,
identify the level in the search tree that one has to return to.
Proceed from there.
•
Randomized restart: restart the search process from the root
of the search tree, but retain the learned clauses.
•
The purpose of these techniques is to let the process “learn”
more direct ways of arriving to the desired result in the sense
of the closing of a branch in the search tree
Backtrack Strategies
•
Conflict-directed backjumping, Fast backjumping: Suppose at
one point in the tree you have assigned True to a prop symbol
a, and further down in the tree you have a particular locality
condition (details omitted). These strategies allow you to
close the entire subtree under a = True and to switch to the
assignment a = False (or to backup further if that assignment
has also been closed.
•
Randomized restart can be thought of as kind of backtrack
strategy.
State of the Art for SAT Solvers
•
SAT solvers have been strikingly successful, both within
Artificial Intelligence and in other areas
•
In principle, they provide an “engine” for combinatorial
reasoning and search which is able to handle very large
sets of clauses
•
A theorem-proving program that draws conclusions using
e.g. the resolution operator is another example of such
an “engine” or “platform”
•
A SAT solver uses a more primitive representation, but it
has the advantage of a number of very efficient
implementation techniques
From predicative to propositional
representation
•
Predicative: use predicates, functions, constants,
variables, quantifiers
•
Propositional: none of those, just 'atomic' proposition
symbols
•
Conversion of a predicative formulation to a propositional
one, at the expense of large increase in size:
•
For every predicate and every combination of constantsymbol arguments, introduce a proposition symbol
•
For every predicate with variables in some of the
arguments, generate all possible instances
•
For every function, do the same with its arguments and
introduce new constant symbols
Example of Propositional Conversion
•
Given: P(a,b), -P(a,c), P(c,c)
•
Construct proposition symbols Paa, Pab, Pac, Pba,...
•
Add also P(f(a),c) as given
•
Add constant symbols Fa, Fb, Fc and proposition
symbols PFaa, PFab, ... obtaining altogether 36 prop
symbols
•
In the general case we must also add constant symbols
FFa, FFb, and FFFa, etc. which means we obtain an
infinite set of proposition symbols, which means it is not
amenable to SAT
•
Type restrictions may prevent recursive use of functions,
for example in (the-bodypart Groucho lefthand)
Requirements for Propositional Conversion
•
Finite number of 'objects' in the domain
•
Individual names for all the objects
•
No function symbols, or restrictions on their use so that
arbitrarily large terms can not be formed
•
The application of a function on arguments shall result in
a new 'object'. No rules such as
•
•
father(Groucho) = father(Bronx)
(or at least, the existence of such rules make things
much more difficult)
Statistical Local Search Techniques
•
Approach: pick one assignment randomly, then change the
value of one proposition symbol at a time
•
A number of techniques of this kind exist. We only consider
one, called GSAT (G for Greedy)
•
Basic idea in GSAT: Start with an randomly chosen
assignment. Calculate, for each proposition symbol, the
increase or decrease in the number of clauses that become
true if the value of that prop symbol is reversed. Pick the one
that gives the best increase.
•
Repeat this process until a satisfying assignment has been
found or a maximum number (max-flips) has been reached
•
If max-flips has been reached, then try another randomly
chosen assignment. Repeat until success or until a maximum
number (max-tries) has been reached.
What Happens in Practice with GSAT
•
When it was new, it outperformed even the best backtracking
algorithms on several types of test problems
•
The process does not very often get stuck in local minima
•
Very often, it gets to a 'plateau' of assignments of roughly
equal value, moves around there, and then moves to a lower
'plateau'
•
Such a plateau often seems to have fairly well-defined 'exits'
to lower plateaus, and the problem for the search is to find
those. Otherwise the search may spend quite long time within
one plateau
•
One method for this is to let a certain, random proportion of
moves go upward when the search finds itself in a plateau.
This was first proposed by the Walksat strategy.
Phase Transition in Random k-SAT
•
In order to study the computational properties of statistical
SAT techniques, it is common to focus on e.g. 3-SAT
problems where all clauses contain exactly 3 literals.
Randomized sets of clauses are generated for experiments.
•
Each 3-SAT problem is characterized by the number of
clauses and the number of prop symbols. Consider the ratio
between these.
•
Now, in each problem, identify the computation time during
the random search, and the outcome (satisfiable or not).
•
Observation 1: The computation time has a distinct maximum
at a ratio around 4.26. This maximum becomes more sharp
when the size of the problem increases.
•
Observation 2: The same value is the point of switch from
mostly satisfiable to mostly unsatisfiable, and this also
becomes more distinct for larger problems.
Phase Transition Problems in General
•
The observation of phase transition behavior in 3-SAT
has counterparts for several other kinds of randomsearch techniques
•
On the mathematical level, this is related to the process
of annealing in material science; the search can be
viewed as “simulated annealing”
•
There is interesting research on using this in the context
of entirely new types of computing hardware
Next Topic: Constraint Programming
SEND
MORE
MONEY
Expressed in logic programming:
Example:
+
sendmore(Digits) :Digits = [S,E,N,D,M,O,R,Y], % Create variables
Digits :: [0..9],
% Associate domains
S #\= 0,
% Constraint
M #\= 0,
alldifferent(Digits),
% all the elements
% must take different values
% Additional constraint:
1000*S + 100*E + 10*N + D
+ 1000*M + 100*O + 10*R + E
#= 10000*M + 1000*O + 100*N + 10*E + Y,
.... (start the search)
Constraint Programming
•
In general, a constraint programming problem specifies:
•
A set of “variables”
•
A domain of possible values for each variable
•
A set of constraints (“relations”) on these variables
•
An assignment of values to the variables that satisfies the
restrictions is a solution to the constraint programming
problem
•
Constraint programming is a programming technique where a
programming language contains a facility for stating and
solving constraint programming problems
•
Logic programming was the original host language for
constraint programming. In this case we talk of constraint
logic programming.
Constraint Programming, SAT, host languages
•
SAT is a constraint problem
•
SAT solvers are not typical examples of constraint
programming, since they are not tied to a host programming
language
•
A tight integration of constraint programming in its host
language requires it to use as much as possible the data
structures, declarations, and operators on data that are
provided by the host language
•
Therefore, constraint programming is the most easily hosted
by languages with an interpretive character, e.g. functional
programming languages (including Lisp and Scheme) and
even Java, besides logic programming languages.
•
However, Constraint programming packages exist even for
C++. (See Wikipedia article on Constraint Programming).
Constraint Programming and SAT Solvers
•
In spite of that difference, Constraint Programming is related
to SAT solvers in the sense that many of the efficiency
improving techniques of SAT solvers have counterparts in
constraint programming:
•
Both systematic search and local random search are used for
constraint programming
•
Backjumping and random restart techniques are used in
constraint programming as well
•
A local inconsistency in a CP problem is an assignment of
values to some of the variables that does not violate any of
the constraints, but which can not be extended with an
assignment to one more variable, so it can not be part of a
solution – it requires backup in the case of depth-first search.
This corresponds to a conflict in a SAT solver.
Arc consistency
•
Arc consistency is an important kind of local consistency
•
Given a constraint, a value for a variable in the constraint has
support there iff there exist values for the other variables in
the constraint such that the constraint is satisfied.
•
Example: the value 0 for the variable x does not have support
in the constraint x * y = 100, but the value 5 for the variable x
does have support
•
A constraint is arc consistent if every value in the domain of
every variable in the constraint has support
•
The pruning operation removes unsupported values from the
domains of variables. Each constraint can be made arc
consistent in this way.
Systematic Search + Pruning
•
A very common framework in constraint programming is
to do a systematic search for variable assignments, but
with additional operations in each step of the search
•
Pruning is one such operation that can be interleaved
with the search. There are different varieties of pruning
•
More generally, search is interleaved with constraint
propagation, i.e. using some of the constraints for
modifying the assignment at hand
•
There is a tradeoff between how much time to spend on
pruning or other constraint propagation, and how much
to spend on just proceeding with the search.
Global constraints
•
A “global constraint” (better called a generic constraint) is
a constraint affects all, or many of the variables, and that
can be used in many applications
•
Constraint programming packages typically provide
support for a number of global constraints
•
The alldifferent constraint (all variables, or a given set of
variables must have different values) is a widely used
global constraint
•
The global cardinality constraint (gcc) is a generalization
which provides a lower and an upper bound on the
number of variables having the same value
Distributed Constraint Programming
•
Problem: given a set of communicating agents each of
which has its own set of constraints for the same (or
overlapping) sets of variables, find an assignment of
values to the variables that satisfies all the constraints in
all the agents.
Conclusion
•
SAT solvers, constraint programming packages, and
resolution theorem-proving programs all provide
“engines” or “platforms” for combinatorial reasoning and
search which can be used in various applications, both
for specific applications and as parts of larger systems,
and both within AI and outside
•
The concept of an “engine” or “platform” is an important
one. A relational database system is an example of a
“platform” in mainstream computing
•
In Artificial Intelligence there is also a number of
examples of more specialized “engines”, for example for
action planning (remember partial-order planning!), and
for diagnostic reasoning (covered in a later course).