Classical and quantum algorithms for Boolean satisfiability

1
Classical and quantum algorithms for Boolean
satisfiability

• Ashley Montanaro

2
Talk structure

• Intro to Boolean satisfiability (SAT)
• Classical algorithms
• Quantum algorithms
• Query complexity lower-bound results

3
What is SAT?

• The problem of finding an assignment to a set of
  variables that satisfies a given Boolean logical
  expression E
• For exampleE (a v b) (a v b) (a v c)
  (c v b)
• has satisfying assignment a b c TRUE
• But if we change the last clause, thusE (a v
  b) (a v b) (a v c) (c v a)
• this formula is not satisfiable
• There are obviously 2n possible assignments to
  the n variables, so exhaustive search takes time
  O(2n)

4
Why is SAT important?

• Its NP-complete
• if we can solve SAT quickly, we can solve
  anything in NP quickly (Cooks theorem, 1971)
• Many and varied applications in itself
• theorem proving
• hardware design
• machine vision
• ...
• In fact, any problem where there exist
  constraints that have to be satisfied!

5
Some restricted versions of SAT

• We generally consider the case where the
  expression E is in CNF, i.e. is made up of
  clauses of ORs linked by ANDs
• (a v b v ...) (c v d v ...) ...
• Thus its hard to find a satisfying assignment,
  but easy to find an unsatisfying one DNF is the
  opposite
• Other variants
• Horn-SAT clauses with all but 1 negation
• MAX-SAT find the maximum number of satisfied
  clauses
• NAESAT all literals in a clause not allowed to
  be TRUE
• ...

6
k-SAT

• If the maximum number of variables in each clause
  is k, we call the problem k-SAT
• 1-SAT is simple E a b ...
• and can be solved in time O(n)
• 2-SAT is also straightforward
• can be solved in time O(n2) using a simple random
  walk algorithm
• 3-SAT is NP-complete
• eek!

7
Classical algorithms for SAT

• Davis-Putnam
• Depth-first search
• Random walk algorithms
• Greedy local search
• ... many others ...

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The Davis-Putnam algorithm (1960)

• Uses the fact that clauses like
• (a v b v c) and (a v b v c)
• can be simplified to
• (a v b)
• This simplification process is called resolution
  
• Algorithm keep on resolving until you find a
  contradiction, otherwise output satisfiable
• Impractical for real-world instances (exponential
  memory usage normally required)

9
DPLL algorithm

• Davis, Logemann, Loveland (1962)
• Basic idea depth-first search with backtracking
  on the tree of possible assignments
• This idea is common to many modern SAT
  algorithms
• Still exponential time in worst case, but lower
  memory usage

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Example solving(a v b) (a v c) (a v b)
(a v c)
a
0
1
b
b
0
0
1
1
c
c
c
c
0
0
0
0
1
1
1
1
ü
ü
û
û
û
û
û
û
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Example solving(a v b) (a v c) (a v b)
(a v c)
a
0
1
b
b
0
0
1
1
c
c
û
û
0
0
1
1
ü
ü
û
û
12
Random walk algorithms

• Schöning developed (1999) a simple randomised
  algorithm for 3-SAT
• start with a random assignment to all variables
• find which clauses are not satisfied by the
  assignment
• flip one of the variables which features in that
  clause
• repeat until satisfying assignment found (or 3n
  steps have elapsed)
• This simple algorithm has worst-case time
  complexity of O(1.34n)
• and its (almost) the best known algorithm for
  3-SAT

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Example solving(a v b) (a v c) (a v b)
(a v c)
110
111
010
011
100
101
000
001
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Example solving(a v b) (a v c) (a v b)
(a v c)
110
111
010
011
100
101
000
001
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How does it work?

• Its almost a simple random walk on the hypercube
  whose vertices are labelled by the assignments
• Apart from the crucial step
• flip one of the variables which features in that
  clause
• This turns it into a walk on a directed graph
  with the same topology
• We can use the theory of Markov chains to
  determine its probability of success, and hence
  its expected running time

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The directed graph of(a v b) (a v c) (a v
b) (a v c)
110
111
010
011
100
101
000
001
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Turning the random walk into a quantum walk

• Is it possible to convert Schönings algorithm
  into a quantum walk in a straightforward way?
• No! The algorithm performs a walk on a directed
  graph with sinks (the satisfying assignments)
• It turns out that quantum walks cannot be defined
  easily on such graphs
• If we remove the directedness, we end up with
  simple unstructured search

18
Greedy local search (GSAT)

• Selman, Levesque, Mitchell (1992)
• Similar to random walk, but only accept changes
  that improve the number of satisfied clauses
• (but sometimes accept changes that dont, to
  avoid local minima)
• Worse than the simple random walk in a worst-case
  scenario
• finds it too easy to get stuck in local minima

19
Classical upper bounds for k-SAT
(m is the number of clauses note that the
algorithms for cases k3,4,5 are randomised)
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Quantum algorithms for SAT

• Unstructured search
• Multi-level unstructured search
• Hoggs algorithm
• Adiabatic evolution

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Unstructured search

• Dont use any knowledge of the problems
  structure just pass in an assignment and ask
  does this satisfy the expression?
• Well-known that you can find a satisfying
  assignment in O(1.42n) tests of satisfiability
  using Grovers algorithm
• The other quantum algorithms given here dont do
  much better...

22
Multi-level unstructured search

• Idea perform a Grover search on a subset of the
  variables, then nest another search within the
  subspace of those variables that satisfies the
  expression
• for 3-SAT, optimal nesting level is 2/3 of the
  variables
• can think of it as a natural quantum analogue of
  the DPLL algorithm
• Results in an average case O(1.27n) query
  complexity for 3-SAT
• worse than the square root of the best classical
  algorithm
• could this be because expressions are very
  sensitive to the values of all the variables they
  contain?
• Due to Cerf, Grover Williams1.

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Multi-level search example

• Lets solve (a v b) (a v c) (a v b) (a v
  c)
• First, search in the space of (a, b) ie. find
  the satisfying assignments to (a v b) (a v b)
• This will give us a superposition 0a1bgt
  1a1bgt
• Now search for a satisfying assignment to the
  original expression in this space
• ending up with a (correct) superposition
  1a1b0cgt1a1b1cgt

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Hoggs algorithm1

• Works in a similar way to Grovers algorithm
• in fact, Grovers algorithm is a special case of
  it
• Starts with a superposition over all assignments,
  then combines phase rotations Pt (based on the
  number of conflicts in a given assignment) with
  mixing matrices Mt
• jendgt MnPn...M1P1gt
• These matrices are heuristically parametrised,
  and change over the course of the algorithm,
  becoming closer to the identity

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Hoggs algorithm (2)

• Phase matrix (problem-dependent)Pii eip K
  c(i)
• where K changes throughout the run and c(i) is
  the number of conflicts in assignment i
• compare Grover phase oracle Pii -(-1f(i) )
• Mixing matrix (problem-independent)M Hxn T
  Hxn Tii eip L w(i)
• where L changes throughout the run and w(i) is
  the Hamming weight of the binary string i
• compare Grover diffusion Tii -(-1(di1))
• Values Mab in mixing matrix are only dependent on
  distance(a, b)
• Values Pii in phase matrix are only dependent on
  number of conflicts in assignment i

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Performance of Hoggs algorithm

• 1-SAT can be solved in 1 step with this algorithm
• the number of conflicts in a 1-SAT assignment is
  the same as its distance from the solution
• so we can choose our mixing matrix cleverly to
  destroy those assignments with gt0 conflicts
• For k-SAT, the number of conflicts provides a
  rapidly worsening estimate of the distance we
  have to use heuristics to try to adjust the
  estimate
• No rigorous worst-case analysis done, but
  simulation on (small) hard random instances of
  3-SAT suggests an average case query complexity
  of O(1.05n)

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Adiabatic evolution

• Uses the quantum adiabatic theorem
• Idea start in the ground state of a known
  Hamiltonian, and continuously evolve to the
  unknown ground state of a solution Hamiltonian
• The solution Hamiltonian is set up so its lowest
  energy eigenstate is the state with no conflicts
  (ie. the solution)
• No rigorous analysis of its power has been made,
  but its known that problem instances exist that
  take exponential time (e.g. van Dam et al1)
• these rely on a very large local minimum, and a
  hard-to-find global minimum
• Due to Farhi et al2

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Lower bounds for these algorithms

• Proving lower bounds on time complexity is a bit
  tricky
• One way we can do it for quantum algorithms is to
  consider query complexity
• All of the algorithms mentioned here use oracles
  black boxes which give us the answer to a
  question
• If we can put a bound on the minimum number of
  calls to these oracles, this gives us an idea of
  the time complexity of the algorithms

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Oracle models in SAT (1)

• These quantum algorithms use (implicitly or
  otherwise) the following oracles
• Black box
• Grovers algorithm, multi-level Grover search

f(x,E)
1 if x satisfies E 0 if x doesnt
x
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Oracle models in SAT (2)

• Conflict counting
• Hoggs algorithm, adiabatic algorithm

f(x, E)
The number of clauses in E that x doesnt satisfy
x
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Oracle models in SAT (3)

• Another obvious oracle is clause satisfaction
• not used by any algorithms so far...

f(x,c,E)
1 if x satisfies clause c of E 0 if x doesnt
x
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Lower bounds for oracle models

• We consider bounds in the number of calls to
  these oracles aka query complexity
• Adversary method used
• consider multiple instances of the problem i.e.
  multiple oracles that are somehow close but
  different
• show a limit on the amount any two instances can
  be distinguished with one oracle call
• work out how many oracle calls are needed to
  distinguish them all
• Several different formulations of the method
• all known formulations have been shown to be
  equivalent1

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Geometric adversary method

• Summed over a set of N oracles, consider the
  largest possible overlap xGgt of an input xgt
  with the good states ie. ones for which the
  oracle returns 1
• intuitively, the best value of xgt to input for
  any instance of the problem will produce the
  largest overlap
• Can show that T2 N / ? xGgt2
• proof omitted ?

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Lower bounds for oracle models (2)

• Unstructured search is well-known to have a lower
  bound of W(2n/2) queries
• This implies that the multi-level search should
  have the same worst-case lower bound, as it uses
  the same oracle
• To put a bound on the other oracle models, we
  pick instances of SAT such that they essentially
  reduce down to unstructured search i.e. so that
  the more powerful oracles are no help to us

35
Lower bound for the conflict counting oracle

• We consider a set of 2n instances of SAT, each of
  which has a single and different satisfying
  assignment
• Each instance has n clauses, varying in length
  from 1 to n variables
• Set the clauses up so none of them overlap
  i.e. cause conflicts with more than one
  assignment
• The number of conflicts will then be 1 for every
  assignment, bar the satisfying assignment the
  oracle becomes no more powerful than unstructured
  search
• So we can show the minimum query complexity is
  W(2n/2)

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Example expression used
a (a v b) (a v b v c)
000 001 010 011 100 101 110 111
Each assignment satisfies all but one clause
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Lower bound for the clause satisfaction oracle

• A similar approach. But this time, we need more
  clauses
• Consider a set of 2n expressions which have
  different, unique satisfying assignments. Each
  expression has 2n clauses, and each clause of
  each expression contains all n variables
• Can then show a bound of W(2n/2) queries,
  extensible to W(sqrt(m)), where m is the number
  of clauses
• Considerably weaker! We need exponential input
  size to show an exponential lower bound

38
Example expression used
(a v b v c) (a v b v c) (a v b v c) (a v
b v c) ( a v b v c) ( a v b v c) (
a v b v c)
000 001 010 011 100 101 110 111
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Do these results extend to k-SAT?

• No! van Dam et al1 have shown that, for 3-SAT, an
  algorithm using the conflict counting oracle can
  recover the input in O(n3) calls to the oracle
• Ive extended this to k-SAT to show that the
  input can be recovered in O(nk) calls
• Idea behind this once you know about the number
  of conflicts in all the assignments of Hamming
  weight k or less, you can work out the number of
  conflicts for all other assignments without
  needing to call the oracle again

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Conclusion

• SAT has been known for 50 years, but classical
  algorithms to solve it are still improving
• Quantum algorithms havent beaten the performance
  of classical ones by much if at all
• Thinking about the oracle models we use
  implicitly or otherwise gives us clues to how
  we should develop quantum algorithms
• It looks like no algorithm can solve SAT quickly
  without looking inside the clauses
• Its also clear that we cant prove any lower
  bounds for k-SAT using these restricted oracle
  methods