A theory-based decision heuristic for DPLL(T)

Dan Goldwasser Ofer Strichman Shai Fine Haifa university TechnionIBM-HRL

DPLL

Decide

BCPAnalyze conflict

Backtrack

SAT

UNSAT

full assignment

partial assignment

conflict

DPLL(T)

Decide

BCP

Deduction Add Clauses

Analyze conflict

Backtrack

SAT

UNSAT

full assignment

partial assignment

conflict

T-propagation / T-conflict

Theory propagation

Matters for efficiency, not correctness. Depending on the theory, the best strategy can be:

No T-implications

One T-implication at a time

All possible T-implications (“exhaustive theory-

propagation”).

Cheap-to-compute T-implications

…

In the case of Linear Real Arithmetic (LRA) … None.

We will see:

The potential of theory propagation

Why doesn’t it work today

How can it be approximated efficiently

Speculations: can the theory lead the way ?

Outline

A geometric interpretation Let H be a finite set of hyperplanes in d dimensions. Let n

= |H|

An arrangement of H, denoted A(H), is a partition of Rd.

An arrangement in d=2:

# cells · O(nd)

A geometric interpretation

Consider a consistent partial assignment of size r. e.g. assignment to (l1,l2,l3), hence r =3.

How many such T-implications are there ?

2l

3l

l1current partial

assignment

(1,0,0)

r = 3

l4

l5

T-Implied

A geometric interpretation

Consider a consistent partial assignment of size r .

Theorem 1: O((n ¢ log r) /r) of the remaining constraints intersect the cell [HW87] with high probability (1 - 1/rc).

Some example numbers: r = 3, ~47% of the remaining constraints are implied. r = 12, ~70% of the remaining constraints are implied. r = 60, ~90% of the remaining constraints are implied.

[HW87] D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Comput. Geom., 2:127- 151, 1987.

Assigned vs. implied in practice

Two benchmarks. Measured averages at T-consistent points

Theory propagation for LRA

Let l1, l2, l3 be asserted. Is l4 (or :l4) T-implied ?

Two techniques for finding T-implications.

1.“Plunging”: check satisfiability of (l1 Æ l2 Æ l3 Æ l4) and of (l1 Æ l2 Æ l3 Æ :l4)

Requires solving a linear system.

Too expensive in practice (see e.g. [DdM06]).

[DdM06] Integrating simplex with DPLL(T), Dutertre and De Moura, SRI-CSL-06-01

Theory propagation for LRA

Let l1, l2, l3 be asserted. Is l4 (or :l4) T-implied ?

Two techniques for finding T-implications.

2. Check if all vertices on the same side of l4

There is an exponential number of vertices.

Too expensive in practice.

Approximating theory propagation

Problem 1: How can we use conjectured information without losing soundness ?

Problem 2: how can we find (cheaply) good conjectures i.e., conjectured T-implications

Problem 1: how to use conjectures ? We use conjectured implications just to bias

decisions.

SAT chooses a variable to decide, we conjecture its value. SAT’s heuristics are T-ignorant.

Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

l4

In this case we conjecture :l4

Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

l4

In this case we conjecture nothing

Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

l4

In this case we (falsely) conjecture l4

Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

Too expensive in practice

Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

Here we always conjecture a T-implication.

l4

Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

Here we always conjecture a T-implication.

l4

Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

Here we always conjecture a T-implication.

l4

Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

The idea: use the assignment maintained by Simplex. It’s for free.

l4

Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

The idea: use the assignment maintained by Simplex. It’s for free.

Competitive SMT solvers Do not activate (general) Simplex after each

assignment They only update the assignment according to the

‘simple’ constraints (e.g. “x < c”).

Problem 2: conjecturing T-implications

Several possibilities:

is T-inconsistent

is T-consistent doesn’t satisfy it

is T-consistent satisfies it

22%

Problem 2: conjecturing T-implications Our hope: is ‘close’ to the polygon. Therefore it can be successful in guessing

implications.

Even if l4 is not T-implied, can guide the search.

l4

Results

Some results for the 200 benchmarks from SMT-COMP’07

Implementation on top of ArgoLib

Each column refers to a different strategy of choosing the value.

0-pivot vs. MinisatMiniSat

The bigger picture # of cells is exponential in d rather than exponential in n

nd rather than 2n

In the SMT-LIB benchmark set, on average n = 10 d.

A reversed lazy approach ?

Current SAT-based ‘lazy’ approaches Search the Boolean domain check assignment in the

theory domain A ‘reversed lazy approach’:

Search the theory domain check assignment in the Boolean domain

T-solver

SAT

Summary

We studied LRA from the perspective of computational geometry.

We showed efficient (approximated) theory propagation.

We showed how approximated information can be used safely.

Future research: How can we let the theory lead the search ?