Beating Brute Force Search for Formula SAT and QBF SAT

Rahul SanthanamUniversity of Edinburgh

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

Motivation

• When can we beat brute-force search for NP-hard problems?

• In practice, we do need to solve SAT and QBF SAT instances in various contexts (planning, verification etc.)– What can we say formally about algorithms

solving for these problems?

Satisfiability Variants

• k-SAT: Satisfiability of k-CNFs• CNF SAT: Satisfiability of formulae in conjunctive

normal form• Formula SAT: Satisfiability of arbitrary Boolean

formulae• Circuit SAT: Satisfiability of arbitrary Boolean circuits

– Constant-depth Circuit SAT: Satisfiability of constant-depth circuits

• Versions of above where variables can be existentially or universally quantified (PSPACE-complete)

Satisfiability Variants

k-SAT

CNF SAT

Formula SAT Constant-depth Circuit SAT

Circuit SAT

Algorithms for Satisfiability

• m: input size; n: number of variables• Brute-force algorithm runs in time 2npoly(m) • We are interested in algorithms running in

time 2n-f(n)poly(m), for f(n) asymptotically as large as possible– We call f the savings of the algorithm

Main Algorithmic Paradigms

• DLL: Search for a solution by iteratively setting variables, and backtracking if a full assignment does not yield a solution

• Random Walk: Start with an arbitrary assignment, and iteratively modify it to satisfy random unsatisfied clauses

Algorithms for Satisfiability: State of the Art

• 3-SAT: ~ 1.3n [R06]• k-SAT: Savings Ω(n/k) [PPSZ98, S99]• CNF SAT: Savings Ω(n/log(m/n)) [S05]• Constant-depth Circuit SAT: Savings Ω(n) for m

= O(n) [CIP09]• Formula SAT: ?• Circuit SAT: ?• QBF SAT: ?

Our New Upper Bounds

• Theorem 1: Formula SAT can be solved in time 2n-Ω(n) on formulae of linear size

• Theorem 2: QBF SAT can be solved in time 2n-

Ω(n/log(n)) on bounded-read formulae (i.e., each variable occurring bounded number of times)

• Theorem 3: QBF SAT can be solved in time 2n-Ω(n) on “structured” bounded-read formulae

Barriers to Improved SAT Algorithms

• Exponential-Time Hypothesis (ETH) formulated by [IP99,IPZ01]: 3-SAT cannot be solved in time 2o(n)

– Under ETH, we cannot achieve savings n-o(n) on k-SAT or Formula SAT/Circuit SAT on linear size

• [W09] shows that savings of ω(log(n)) for Circuit SAT for superpoly m would imply NEXP not in SIZE(poly) (and analogously for Formula SAT)

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

The Upper Bound for Formula SAT

• Theorem 1 (restated): There is a constant k > 2 for which there is a deterministic algorithm solving Formula SAT with savings n/ck on formulae of size cn

• Note: if we could achieve k = 0.1, then by [W09] we would have new formula size lower bounds for ENP

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

The Algorithm

• Search(φ)– Simplify φ according to simplification rules– If φ ↔ 1, return “yes” and halt– If φ ↔ 0, return “no”– Let x be the variable with max no of occurrences– Search (φ|x=0)

– Search (φ|x=1)– Return “no”

Simplification Rules

• 1 Λ φ → φ• 1 V φ → 1• 0 Λ φ → 0• 0 V φ → φ• x V φ → x V φ|x=0

• x Λ φ → x Λ φ|x=1

1-simplification rules

0-simplification rules

Variable simplification rules

An Example

(x V y) Λ (x V (x Λ y’ Λ z))

An Example

(x V y) Λ (x V (0 Λ y’ Λ z))(applying variable simpl. rule)

An Example

(x V y) Λ (x V 0)(applying 0-simpl. rule)

An Example

(x V y) Λ x(applying 0-simpl. rule)

An Example

(x V y) Λ x

(0 V y) Λ 0

1st recursive call

An Example

(x V y) Λ x

y Λ 0

An Example

(x V y) Λ x

0

An Example

(x V y) Λ x

0 (1 V y) Λ 1

2nd recursive call

An Example

(x V y) Λ x

0 1 Λ 1

An Example

(x V y) Λ x

0 1

Success!

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

Analyzing Recursion Tree Size

• Typically done by solving a recurrence on m and n

• Instead, we derive inspiration from the method of random restrictions (though our algorithm itself is deterministic)

• A random restriction is a probability distribution on partial assignments to variables

Pure Random Restrictions

• Let 0 < p < 1 be a parameter. Given n input variables, choose each one independently to be 1 w.p. (1-p)/2, 0 w.p. (1-p)/2 and free w.p p

• Note that the choice of which variables to set is uniform, as well as the choice of which value to set a given variable to

Formula Size Lower Bounds via Pure Random Restrictions

• [S61] proved that when a formula of size m is hit by a random restriction with parameter p, expected size of simplified formula is O(p1.5m)– Implies Parity requires formulae of size Ω(n1.5), by

choosing p = O(1/n)• [H98] proved optimal result: expected size of

simplified formula is O(p2m)– Implies best known formula size lower bound of

n3-o(1) for a function in P

Formula Size Lower Bounds via Pure Random Restrictions

• [S61] proved that when a formula of size m is hit by a random restriction with parameter p, expected size of simplified formula is O(p1.5m)

• [H98] proved optimal result: expected size of simplified formula is O(p2m)

• Note that for either result, if m = O(n), there is constant p such that expected size of simplified formula << pn

Adaptively Random Restrictions

• Choice of which variable to set next is not uniform– Indeed, in our algorithm, setting of variables is

deterministic, according to number of occurrences• Choice of value, however, is uniformly random• Greedy a.r.r: Variables are set sequentially in

decreasing order of no. of occurrences• [S61] and [H98] results hold also for (1-p)n -

step greedy a.r.r

Random Restrictions and Recursion Tree Size: Basic Idea

• The simplified formulae at depth d of the recursion tree correspond to d-step greedy a.r.r

• Lemma: After (1-p)n steps of greedy a.r.r, size of simplified formula << pn with prob. 1-2-Ω(n) (strong concentration version of Subbotovskaya’s result)

• This implies non-trivial bound on size of recursion tree

Why a Concentration Bound Helps

. . . .

(1-p)n

Good node: Simplified formula at node has size < pn/2Bad node: Simplified formula has size >= pn/2

Why a Concentration Bound Helps

. . . .

(1-p)n

Say we could show that fraction of bad nodes at depth (1-p)n is at most q. Then size of decision tree is at most 2n-pn/2 + q2n, which is 2n-Ω(n) if q=2-Ω(n)

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

Beating Brute Force Search for Exact Count

• Count(φ;n)– Simplify φ according to simplification rules– If φ ↔ 1, return 2n

– If φ ↔ 0, return 0– Let x be the variable with max no of occurrences– Return Count(φ|x=0;n-1) + Count(φ|x=1;n-1)

• Analysis same as before, giving same runtime

Detour: Decision Trees

x1

x2

x3

0

0

0 1Φ = x1 Λ x2 Λ x3

Average Case Lower Bounds for Formula Size

• Recursion tree of Search algorithm yields decision tree for function computed by input formula

• Proof of Theorem 1 shows that formulae of linear size have decision trees of size 2n-Ω(n)

Average Case Lower Bounds for Formula Size

• Proof of Theorem 1 shows that formulae of linear size have decision trees of size 2n-Ω(n)

• Let advantage of a decision tree T on a function f be Pr(T=f) – Pr(T≠f)

• Lemma: Any decision tree of size s has advantage at most s/2n on Parity

• Corollary : Any formula of linear size has advantage 2-Ω(n) on Parity

Analysis First, Algorithm Afterwards

• Could other random restriction results be used to get new upper bounds?

• Hastad has a famous result showing that constant-depth circuits simplify under (pure) random restrictions

• From this, we “extract” a randomized algorithm solving Constant-depth Circuit SAT with savings Ω(n1/(d+1)), where d is depth

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

A New Upper Bound for QBF SAT

• Theorem 2 (re-stated): There is an algorithm running in time 2n-Ω(n/log(n)) solving QBF SAT on bounded-read formulae

Proof Idea for Theorem 2

• We would like to use random restriction method again, but we have no control over order in which variables are to be set

• Let k be an upper bound on number of occurrences for any variable

• By fixing all but t variables, our new formula will have size at most kt– But how does this help?

Proof Idea for Theorem 2

• Idea: Memoization• When t<n/(5k log(n)), simplified formula has size at

most n/(5 log(n)), and hence can be represented by < n/4 bits

• We can pre-compute answers to all such small QBF SAT questions in time 2n/2 and store them in random-access memory

• Now, given an instance φ, we need only do exhaustive search over first n-t variables, replacing the rest of the search by a memory access

Structured Instances

• Theorem 2 only gives Ω(n/log(n)) savings for bounded-read formulae

• Can we get Ω(n) savings?• A set S of instances is structured if every

instance of length n in S has a description of size o(n) from which it can be recovered efficiently

• Eg., set of all sparse graphs is structured

Linear Savings for Structured Instances

• Theorem 3 (re-stated): There is an algorithm for QBF SAT which has savings Ω(n) on any set of structured bounded-read formulae

• Proof Idea: Formula obtained by fixing the first εn quantified variables of a QBF is also “somewhat structured”

• In the memoization phase, we don’t need to store answers to all small formulae, but only for reasonably structured ones

Plan of the Talk

• Introduction• A New Upper Bound for Formula SAT– The Algorithm– The Analysis– Other Applications of Technique

• New Upper Bounds for QBF SAT• Future Directions

Future Directions

• Using the method of random restrictions in other settings or to get better parameters

• More connections between upper bounds and lower bounds

• Better upper bounds for QBF SAT

Thank You!