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### A propositional world

Ofer Strichman

Joint work with

Randal Bryant and Sanjit Seshia

School of Computer Science, Carnegie Mellon University

Integrated decision procedures in Theorem-Provers

Deciding a combination of theories is the key for automation

in Theorem Provers:

Boolean operators, Bit-vector, Sets, Linear-Arithmetic,

Uninterpreted functions, More …

Bit-Vector

operators

Linear Arithmetic

Uninterpreted functions

f(f(x)-f(y)) != f(z) & y <=x + 2 | b & 3 > 10

Normally, each theory is solved with its own decision procedure and

the results are combined (Shostak, Nelson..).

Integrated decision procedures in Theorem-Provers

All of these theories, except linear arithmetic, have known

efficient direct reductions to propositional logic.

Thus, reducing linear arithmetic to propositional logic will:

1. Enable integration of theories in the propositional logic level.

2. Potentially be faster than known techniques.

Linear Arithmetic and its sub-theories

2x –3y +5z < 0

5x + 2w 2

- Some useful methods for solving a conjunction of linear
- arithmetic expressions:
- Simplex, Elliptic curve
- Variable Elimination Methods (Hodes, Fourier-Motzkin,..)
- Shostak’sloop residues
- Separation theory: Bellman / Pratt ...
- ...

3

1

z

y

-1

A decision procedure for separation theorySeparation predicates have the form x > y + c

where x,y are real variables, and c is a constant

Pratt [73] (/Bellman[57]):

Given a set of conjuncted separation predicates

1. Construct the `inequality graph’

2. is satisfiable iff there is no cycle with non-negative accumulated weight

: (x > z +3 z>y –1 y > x+1)

Handling disjunctions through case splitting

- All previously mentioned algorithms handle disjunctions
- by splitting the formula.
- This can be thought of as a two stage process:
- Convert formula to Disjunctive Normal Form (DNF)
- Solve each clause separately, until satisfying one of them.

(A common improvement: split ‘when needed’)

Case splitting is frequently the bottleneck of the procedure

So what can be done against case-splitting ?

Answer: Split the domain, not the formula.

Given a formula , this transformation can be done if

’ s.t. |= |=’, and ’ is decidable under a finite domain.

- When is this possible?
- enjoys the ‘Small model property’, or
- Tailor-made reduction

SAT vs. infinite-state decision procedures

With finite instantiation (e.g. SAT), we split the domain.

Infinite state decision procedures split the formula.

So what’s the big difference ?

2. Learning.

3. Guidance (prioritizing internal steps)

SAT vs. infinite-state decision proceduresThree mechanisms, crucial for efficient decision making:

SAT has a significant advantage in all three.

0

1

.

(x y)

.

.

Backtrack

y

1

0

Pruned!

SAT vs. infinite-state decision procedures (1/4)1. Pruning

SAT: each clause c prunes

up to 2|v|-|c| states.

|v|=1000, |c| =2

Pruning 2998states

Others: ? (stops when finds a satisfiable clause)

SAT vs. infinite-state decision procedures (2/4)

2. Learning

SAT: Partial assignments that lead to a conflict are recorded and

hence not repeated.

Others: (depends on decision procedure)

- Adding proved sub-goals as antecedents to new sub-goals

- …

SAT vs. infinite-state decision procedures (3/4)

3. Guidance (prioritizing internal steps)

Consider 1 2, where 1is unsat and hard, and 2is sat and easy.

With proper guidance, a theorem prover should start from 2.

Guidance requires efficient estimation:

- How hard it is to solve each sub-formula?

- To what extent will it simplify the rest of the proof?

SAT vs. infinite-state decision procedures (4/4)

3. Guidance (cont’d)

“..To what extent will it simplify the rest of the proof?”

SAT: Guidance through decision heuristics (e.g. DLIS).

(x y z)

(x v)

(~x ~z)

Estimating simplification by counting literals

in each phase

Others: Expression ordering, ...

Example: Equality Logic with Uninterpreted Functions (1/3)

Equality Logic with Uninterpreted Functions:

(Uninterpreted functions are reducible to equality logic. Thus, we

can concentrate on equality logic)

Traditional infinite-state decision procedure:

Congruence Closurewith case splitting.

Example: Equality Logic (2/3)

- Since 1998, several groups devised finite-state decision procedures
- for this theory:
- Goel et. al. (CAV’98) – Boolean encoding and BDDs
- Bryant et. al.(CAV’99) – Positive-equality + finite instantiation
- Pnueli et. al. (CAV’99) – Small domains instantiation
- Bryant et. al.(CAV’00) – Boolean encoding with explicit constraints

exz

exy

z

y

eyz

Example: Equality Logic (3/3)Bryant et. al. (CAV’00): Add transitivity constraints to the formula.

Let (x=y, y=z, x=z) be the equality predicates in .

1. Construct the equality graph.

2. Impose transitivity on cycles:

exy + eyz +exz 2

The resulting formula is propositional BDDs , SAT, etc.

This work

Extends the results of Bryant et.al. to a Boolean combination of:

- Separation predicates:
- Separation predicates for integers:
- Linear arithmetic:
- Integer linear arithmetic:

Done

Usability

Separation predicates:

“Most verification conditions involving inequalities are

separation predicates” [Pratt, 1973]:

Array bounds checks, tests on index variables, timing constraints,

worst execution time analysis, etc.

Linear arithmetic: All of the above + …

+ Linear programming,

+ Integer Linear programming.

Reducing separation predicates to propositional logic (1/6)

A. Normalize (example):

: f(x) > f(y+1)

1. Uninterpreted functions equality logic

: (x=y+1f1=f2)(f1>f2)

2. Normal form

xy+1

f1=f2

: (x>y+1 y>x-1(f1 f2 f2 f1)) (f1>f2)

Now has no negations and only the ‘>’ and ‘’ predicate symbols.

3

1

z

y

-1

x

-3

-1

z

y

1

Reducing separation predicates to propositional logic (3/6)B. Encode + construct graph (example):

: (x > z +3 (z>y –1 yx+1))

Transitivity

constraints

))

(

’:

(

and its

dual:

Separation

graph:

3

1

z

y

-1

x

-3

-1

z

y

1

Reducing separation predicates to propositional logic (5/6)C. Add transitivity constraints for each simple cycle (example):

Transitivity

constraints

))

(

(

’:

((

))

))

(

(

’:

.....

Compact representation of constraints (1/4)n diamonds 2nsimple cycles.

Can we do better than that ?

In most cases - yes.

e.g. If the diamonds are ‘balanced’ (c1 +c2 =c3 +c4) O(n) constraints

c2

c1

c1+ c2

c4

c3

Compact representation of constraints (2/4)

Chordal graphs: each cycle of size greater than 3, has a ‘chord’.

G:

In the equality predicates case:

Let C be a cycle in G

Let be an assignment that violates C’s transitivity ( |C)

Theorem: there exists a cycle c of size 3 in G s.t. | c

Conclusion: add transitivity constraints only for triangles.

Now only a polynomial no. of constraints is required.

c1

c1+ c2

c3

c4

c5

Compact representation of constraints (3/4)- Our case is more complicated:
- G is directed
- G is a multi-graph
- Edges have weights
- There are two types of edges

G is chordal iff:

Every directed cycle of size greater than 3 has a chord which ‘accumulates’ the weight of the path between its ends.

2. If there are uniform weights c1 and c2, c1c2 on top and bottom

paths O(n2) constraints

c1

c1

c1

c1

c2

c2

c2

c2

Compact representation of constraints (4/4)Complexity of making the graph chordal:

1. If the diamonds are ‘balanced’ O(n) constraints

3. Worst case O(2n)

Extension to integer variables (1/2)

(c is an integer)

Given with integer separation predicates, derive R:

- Declare all variables as real.

- For each predicate x>y + c, add a constraint
- x > y + c x y + c + 1

Theorem: is satisfiable iff R is satisfiable

Experimental results (1/3)

d=2

.....

n diamonds

Each diamond has 2d edges

Top and bottom paths in each diamond are disjuncted.

There are 2nconjuncted cycles.

By adjusting the weights, we ensured that there is a single

satisfying assignment.

Experimental results (2/3)

To be continued...

Experimental results (3/3)

To be continued...

The procedure has recently been integrated into SyMP and Euclid.

We currently experiment with real software verification problems.

c3

c2

Next: Linear Arithmetic (1/2)Separation predicates:

c

x > y + c

y

x

Adding constraints according to accumulated cycle weight:

The testc1 + c2 + c3 > 0 results in a yes/no answer

2z + c

3

y

2

Next: Linear Arithmetic (2/2)Linear Arithmetic:

2z + c

x > y + 2z + c

y

x

The test1 + 2 + 3 > 0 results in a new predicate!

Shostak[81]: ‘Deciding linear inequalities by computing loop residues’

- Determine a fixed variable order

- Represent each predicate by its two ‘highest’ variables

This procedure guarantees termination.

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