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Classical and quantum algorithms for Boolean

satisfiability

- Ashley Montanaro

Talk structure

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

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)

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!

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 - ...

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!

Classical algorithms for SAT

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

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)

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

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

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

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

Example solving(a v b) (a v c) (a v b)

(a v c)

110

111

010

011

100

101

000

001

Example solving(a v b) (a v c) (a v b)

(a v c)

110

111

010

011

100

101

000

001

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

The directed graph of(a v b) (a v c) (a v

b) (a v c)

110

111

010

011

100

101

000

001

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

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

Classical upper bounds for k-SAT

(m is the number of clauses note that the

algorithms for cases k3,4,5 are randomised)

Quantum algorithms for SAT

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

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...

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.

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

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

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

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)

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

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

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

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

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

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

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 ?

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

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)

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

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

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

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

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

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