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Decision Procedures for Presburger Arithmetic

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### Decision Procedures forPresburger Arithmetic

Presented by Constantinos Bartzis

Presburger formulas

- numeral ::= 0 | 1 | 2…
- var ::= x | y | z …
- relop ::= < | ≤ | = | | >
- term ::= numeral | term + term | -term | numeral term | var
- formula ::= term relop term | formula formula | formula formula | formula | var. formula | var. formula
numeral term isn't really multiplication; it's short-hand for term + term + … + term

Decision Procedures

- The aim is to produce an algorithm for determining whether or not a Presburger formula is valid with respect to the standard interpretation in arithmetic.
- Such an algorithm is a decision procedure if it is guaranteed to correctly return “True” or “False” for all closedformulas.
- Will discuss algorithms for determining truth of formulas of Presburger arithmetic:
- Fourier-Motzkin variable elimination (FMVE)
- Omega Test
- Cooper's algorithm
- Automata based

Quantifier Elimination

- All the methods we'll look at are quantifier eliminationprocedures.
- If a formula with no free variables has no quantifiers, then it is easy to determine its truth value,
e.g., 10 > 11 3+4 < 5 3 - 6

- Quantifier elimination works by taking input Pwith nquantifiers and turning it into equivalent formula P’ with mquantifiers, and where m < n.
- So, eventually P P’ … Qand Qhas no quantifiers.
- Qwill be trivially true or false, and that's the decision

Normalization

- Methods require input formulas to be normalized
- e.g., collect coefficients, use only < and ≤

- Methods eliminate innermost existentialquantifiers. Universal quantifiers are normalized with
(x. P(x)) (x. P(x))

- In FMVE, the sub-formula under the innermost existential quantifier must be a conjunction of relations.
- This means the inner formula must be converted to disjunctive normal form(DNF):
(c11c12 … c1n1) ... (cm1cm2 … cmnm)

Normalization (cont.)

- The formula under is in DNF. Next, the must be moved inwards
- First over disjuncts, using
(x. P Q) (x. P) (x. Q)

- Must then ensure every conjunct under the quantifier mentions the bound variable. Use
(x. P(x) Q) (x. P(x)) Q

- For example:
(x. 3 < x x +2y ≤ 6 y < 0)

(x. 3 < x x +2y ≤ 6) y < 0

Fourier-Motzkin theorems

- The following simple facts are the basis for a very simple quantifier elimination procedure.
- Over R (or Q), with a,b > 0:
(x. c ≤ax bx ≤d) bc ≤ad

(x. c < ax bx ≤d) bc < ad

(x. c ≤ax bx < d) bc < ad

(x. c < ax bx < d) bc < ad

- In all four, the right hand side is implied by the left because of transitivity
- e.g., (x < y y ≤z) x < z

Fourier-Motzkin theorems (cont.)

- For the other direction:
(bc < ad) (x.c < ax bx ≤d)

take xto be d/b: c < a( d/b ), and b( d/b ) ≤d.

- For (bc < ad) (x.c < ax bx < d)
take xto be (bc+ad)/2ab:

c < a(bc+ad)/2ab 2bc < bc+ad bc < ad

- Similarly for the other bound

Extending to a full procedure

- So far: a quantifier elimination procedure for formulas where the scope of each quantifier is 1 upper bound and 1 lower bound.
- The method needs to extend to cover cases with multiple constraints.
- No lower bound, many upper bounds:
(x: b1x < d1b2x < d2 … bnx < dn)

True!(take min(di/bi) as witness for x)

- No upper bound, many lower bounds: obviously analogous.

Combining many constraints

- Example:
(x.c ≤ax b1x ≤d1b2x ≤d2) b1c ≤ad1b2c ≤ad2

- From left to right, result just depends on transitivity.
- From right to left, take xto be min(d1/b1, d2/b2).
- In general, with many constraints, combine all possible lower-upper bound pairs.
- Proof that this is possible is by induction on number of constraints.

Combining many constraints

- The core elimination formula is
- With nconstraints initially, evenly divided between upper and lower bounds, this formula generates n2/4 new constraints.

FMVE example

x. 20+x ≤ 0 y. 3y +x ≤ 10 20 ≤y - x

(re-arrange)

x. 20+x ≤ 0 y. 20+x ≤y 3y ≤ 10 - x

(eliminate y)

x. 20+x ≤ 0 60+3x ≤ 10 - x

(re-arrange)

x. 20+x ≤ 0 4x +50 ≤ 0

(normalize universal)

x. 20+x ≤ 0 0 < 4x +50

(re-arrange)

x. -50 < 4x x ≤ -20

(eliminate x)

(-50 < -80) T

Complexity

- As before, when eliminating an existential over nconstraints we may introduce n2/4 new constraints.
- With kquantiers to eliminate, we might end with n2k/4kconstraints.
- If dealing with alternating quantifiers, repeated conversions to DNF may really hurt.

Expressivity over Integers

- Can do divisibility by specific numerals:
2|e x. 2x = e

for example:

x. 0 < x < 30 (2|x 3|x 5|x)

- Can do integer division and modulus, as long as divisor is constant. Use one of the following results (similar for division)
P(x mod d) q,r. (x = qd +r ) (0 ≤r < d d < r ≤ 0) P(r )

P(x mod d) q,r. (x = qd +r ) (0 ≤r < d d < r ≤ 0) P(r )

- Any formula involving modulus or integer division by a constant can be translated to one without.

Expressivity over Integers

- Any procedure for Z trivially extends to be one for N (or any mixture of N and Z) too: add extra constraints stating that variables are 0
- Relations < and ≤ can be converted into one another:
x ≤y x < y +1

x < y x +1 ≤y

- Decision procedures normalize to one of these relations.

Fourier-Motzkin for Integers?

- Central theorem is false. E.g.,
(xZ. 3 ≤ 2x 2x ≤ 3) 6 ≤ 6

- But one direction still works (thanks to transitivity):
(x. c ≤ax bx ≤d) bc ≤ad

- We can compute consequences of existentially quantified formulas

/

Fourier-Motzkin for Integers?

- We know (x. c ≤ax bx ≤d) bc ≤ad
- Thus an incomplete procedure for universal formulas over Z:
- Compute negation: (x. P(x)) (x. P(x))
- Compute consequences:
if (x. P(x)) then (x. P(x)) and (x. P(x)) T

- Repeat for all quantified variables.
- This is Phase 1 of the Omega Test

Omega Phase 1 - Example

x,yZ. 0 < x y < x y +1 < 2x

(normalize)

x,y. 1 ≤x y +1 ≤x 2x ≤y +1

x,y. 1 ≤x y +1 ≤x 2x ≤y +1

(eliminate y)

x. 1 ≤x 2x ≤x

(normalize)

x. 1 ≤x x ≤ 0

(eliminate x)

1 ≤ 0

Omega Phase 1 and the Interactive Theorem Provers

- The Omega Test's Phase 1 is used by systems like Coq, HOL4, HOL Light and Isabelle to decide arithmetic problems.
- Against:
- it's incomplete
- conversion to DNF
- quadratic increase in numbers of constraints

- For:
- it's easy to implement
- it's easy to adapt the procedures to create proofs that can be checked by other tools

Some Shadows

- Given x. (i ci≤aix) (j bjx ≤dj)
- The formula
i,j bjci≤aidj

is known as the real shadow.

- If all of the aior all of the bjare equal to 1, then the real shadow is exact.
- If the shadow is exact, then the formula can be used as an equivalence.

Exact Shadows

- When a = 1 or b = 1, the core theorem
(x. c ≤ax bx ≤d) bc ≤ad is valid because

- transitivity still holds
- take x = dif b = 1 or x = cif a = 1

- Omega Test's inventor, Bill Pugh, claims many problems in his domain (compiler optimization) have exact shadows.
- Experience suggests the same is true in other domains too, such as interactive theorem-proving.
- When shadows are exact, can pretend problem is over R rather than Z and life is easy.

Dark Shadows

- The formula i,j (ai-1)(bj-1) ≤ aidj - bjci
is known as the dark shadow.

- If all aior all bjare one, then this is the same as the real shadow (or exact).
- The real shadow provides a test for unsatisfiability.
- The dark shadow tests for satisfiability, because
(a-1)(b-1) ≤ad - bc (x. c ≤ax bx ≤d)

- This is the Phase 2 of the Omega Test

Omega Test Phases 1 & 2

- Problem is x. P(x)
- If input is exact for one elements of x, then eliminate them
x. P(x) x’. P’(x’)

- Otherwise, calculate real shadow R:
x. P(x) R

so, if R , then input formula is .

- Otherwise, calculate dark shadow D:
D x. P(x)

so, if D = T, then input formula is T.

Omega Phase 2 - Example

(a-1)(b-1) ≤ad - bc (x. c ≤ax bx ≤d)

x,y. 3x +2y ≤ 18 3y ≤ 4x 3x ≤ 2y +1

3y ≤ 4x 3x ≤ 2y +1 3y ≤ 4x 3x ≤ 18 - 2y

6 ≤ 8y + 4 - 9y 6 ≤ 72 - 8y - 9y

y ≤ -2 17y ≤ 66

y ≤ 3

- This gives a suitable value for y, and by back-substitution, finds
x = -1, y = -2 as a possible solution.

Splinters

- Purely existential formulas are often proved false by their real shadow; or proved true by their dark shadow
- But in “rare” cases, the main theorem is needed. Let mbe the maximum of all the djs. Then

splinter

dark shadow

Splinters

- A splinter doesrepresent a smaller problem than the original because the extra equality allows xto be eliminated immediately.
- When quantifiers alternate, and there is no exact shadow, the main theorem is used as an equivalence, and splinters can't be avoided.
- Splinters must also be checked if neither real nor dark shadows decide an input formula.

Eliminating Equalities

- In an expression
x. … cx = e …

the existential can be eliminated.

- First, multiply all terms involving xso that they have a common coefficient.
- Formula becomes
x. …c’x … c’x = e’ …c’x…

- This is equivalent to
…e’… c’|e’ …e’…

Eliminating Divisibilities

x. … c | dx + e …

- Note: d < c(otherwise, replace d with d mod c).
- Introduce temporary new existential variable:
x,y. … cy = dx + e …

- Rearrange:
x,y. … dx = cy -e …

- Use equality elimination to derive
y. … d | cy -e …

- Because d < c, this process must terminate with elimination of divisibility term.

Implementation - Normalization

- Omega Test's bigdisadvantage is that it requires the formula under quantifier to be in DNF
- Consider
x. (x 10 x 11 9 < x ≤ 12) x = 12

- Negate, remove , <:
x. (x ≤ 9 11 ≤x) (x ≤ 10 12 ≤x) 10 ≤x x ≤ 12 (x ≤ 11 13 ≤x)

- Evaluate 8 (= 23) DNF terms.
- Clever preparation of input formulas can make orders of magnitude difference

Implementation - Normalization

- The propositional tautology
(p (q q’)) (p q p q’)

justifies the following procedure:

- If Pis an atomic formula, then when processing P Q, assume Pis true while processing Q:
- If a sub-formula Q0 of Qis such that P Q0, then replace Q0 in Qby T.
- If a sub-formula Q0 of Qis such that P Q0, then replace Q0 in
Qby .

- Similarly, (p (q q’)) (p q p q’) for disjunctions.

Contextual Rewriting example

- Over :
0 ≤x + y + 4 (0 ≤x + y + 6 0 ≤ 2x + 3y + 6)

is equivalent to 0 ≤x + y + 4

- Whereas
0 ≤x + y + 4 0 ≤ -x -y -6 0 ≤ 2x + 3y + 6

is equivalent to

- Over :
0 ≤x + y + 4 0 ≤x + y + 1 0 ≤ 2x + 3y + 6

is equivalent to

0 ≤x + y + 4 0 ≤ 2x + 3y + 6

Cooper's Algorithm

- Cooper's algorithm is a decision procedure for Presburger arithmetic.
- A non-Fourier-Motzkin alternative
- It is also a quantifier elimination procedure, which also works from the inside out, eliminating existentials.
- Its bigadvantage is that it doesn't need to normalize input formulas to DNF.
- Description is of simplest possible implementation; many tweaks are possible.

Preprocessing

- To eliminate the quantifier in x. P(x):
- Normalize so that only operators are <, and divisibility (c|e), and negations only occur around divisibility leaves.
- Compute least common multiplec of all coefficients of x, and multiply all terms by appropriate numbers so that in every term the coefficient of x is c.
- Now apply
(x. P(cx)) (x. P(x) c|x).

Preprocessing Example

x,y Z. 0 < y x < y x +1 < 2y

(normalize)

x,y. 0 < y x < y 2y < x +2

(transform y to 2y everywhere)

x,y. 0 < 2y 2x < 2y 2y < x +2

(give y unit coefficient)

x,y. 0 < y 2x < y y < x +2 2|y

Two cases

- How might x. P(x) be true?
- Either:
- there is a least xmaking Ptrue; or
- there is no least x: however small you go, there will be a smaller xthat still makes Ptrue

- Construct two formulas corresponding to both cases.

Case 1:Infinitely many small solutions

- Look at the atomic formulas in P, and think about their values when xhas been made arbitrarily small:
- x < e: if xgoes as small as we like, this will be T
- e < x: if xgoes small, this will be
- c | x+e: unchanged

- This constructs P-, a formula where xonly occurs in divisibility terms.
- Say is the l.c.m.of the constants involved in divisibility terms. Need just test P- on 1,…, .

P- example

- For y. 0 < y 2x < y y < x +2 2|y
- 0 < ywill become as ygets small
- 2x < yalso becomes as y gets small
- y < x +2 will be T as y gets small
- 2|ydoesn't change (it tests if yis even or not)

- So in this case,
P- (y) ( T 2|y)

Case 2: Least solution

- The case when there is a least xsatisfying P.
- For there to be a least xsatisfying P, it must be the case that one of the terms e < xis T, and that if xwas any smaller the formula would become .
- Let B = {b | b < x is a term of P}
- Need just consider P(b+j), where b Band 1 ≤j ≤.
- Final elimination formula is:

Example continued

- For
y. 0 < y 2x < y y < x +2 2|y

- least solutions, if they exist, will be at
y = 1, y = 2, y = 2x +1, or y = 2x +2

- The divisibility constraint eliminates two of these.
- Original formula is equivalent to:
(2x < 2 0 < x) (0 < 2x +2 x < 0)

Which is unsatisfiable.

0,0,1

0 0 1

0,1,1

0

1

1

0

Symbolic RepresentationWe use finite automata to represent the integer solutions (in binary) of atomic linear constraints.

Example: The constraint x1x20 has solutions:

(0,0), (1,0), (1,1), (2,0), (2,1), (2,2), (3,0), …

The corresponding automaton

1

2

FA Construction- Based on the basic state machine (BSM)
- A finite state transducer for computing linear integer expressions

0 1

0 0

/ /

0 1

0

1

/

1

1

0

/

0

0 1

1 1

/ /

0 1

Example BSM

for x + 2y

0

1

/

0

1

1

/

1

1

1

/

0

010

+ 2 001

0

0

/

1

0 1

0 0

/ /

0 1

1

0

0

Equality With 0

- Construct the BSM
- All transitions writing 1 go to a sink state
- State labeled 0 is the only accepting state
- For disequations, state labeled 0 is the only rejecting state
- Size

Inequality (<0)

- Construct the BSM
- States with negative carries are accepting
- No sink state
- All other types of inequalities can be derived from this
- Size

Non-zero Constant Term c

- Same as before but now -c is the initial state
- If there is no such state, create one (and possibly some intermediate states)
- Size
- Alternatively, we can stack log2c BSMs that compare with 0 or 1, depending on the corresponding bit of c

Boolean Connectives

- For compute the intersection
- For compute the union
- For compute the complement

0,1,1

0 1

0,1

1

0

Automaton for x-y<1

1

0

1

0

-1

0

1

0

1

0 0 1

0,1,1

0 0

0,1

0 0

0,1

0,-1

0

0

1

1

1

0

1

1 1

0,1

0

1

Automaton for x-y<1 2x-y>0

1

0

0 1

0,1

Automaton

for 2x-y>0

0 1

0,1

0 1

0,1

-1,-1

-1,0

-1

0

0

1

0

0

0

0

0

1

0

1

0

1

0

0 1

1,1

0 1

1,1

1

0

0 1 1

1,0,1

-2,-1

-2,0

-2,1

-2

0

1

1

0

1

1

Conjunction exampleExistential Quantification

- Project the quantified variables away
- The resulting FA is non-deterministic
- Determinization results in exponential blowup of the FA size
- Rarely occurs in practice
- For universal quantification use negation

Constraints on All Integers

- Use 2’s complement arithmetic
- The BSM remains the same
- Create two clones for each state of the BSM one accepting and one rejecting
- For =0, accepting states contain loops that write 0
- For <0, accepting states contain loops that write 1
- Size

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