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Presburger Arithmetic

Yuxi Fu

BASICS

Yuxi Fu Presburger Arithmetic 1 / 36

Presburger Arithmetic is seen as a test for the limit of decidability.

It is the arithmetic with only the addition operator; in other wordsit is the Peano Arithmetic without the multiplication operator.


Mojzesz Presburger (A polish student, 1929).

Uber die Vollstandigkeit eines gewissen Systems der Arithmetikganzer Zahlen, in welchem die Addition als einzige Operationhervortritt. In Comptes Rendus du Premier Congres desMathematicienes des Pays Slaves, 395:92-101, WarsawMathematics Congress.

R. Stansifer (1984).

Presburger’s Article on Integer Arithmetic: Remarks andTranslation. Technical report, Computer Science Department,Cornell University.


Synopsis

1. Presburger Arithmetic

2. Decidability of PA

3. Complexity of Validity Checking


First Order Theory

1. First Order Logic: >,⊥,∧,∨,¬,⇒,∃,∀.

2. First Order Theory with Equality:

P{r1(t11 , . . . , t1k1

)/X1, . . . , rn(tn1 , . . . , tnkn

)/Xn} P a tautology

t = ts = t ⇒ t = s

r = s ∧ s = t ⇒ r = t∧1≤i≤k ti = t ′i ⇒ f(t1, . . . , tk) = f(t ′1, . . . , t

′k) f a k-ary function∧

1≤i≤k ti = t ′i ⇒ r(t1, . . . , tk)⇒ r(t ′1, . . . , t′k) r a k-ary relation

∀x .φ⇒ φ{t/x}φ⇒ ∀x .φ x not in φ

(∀x .(ϕ⇒ ψ))⇒ (∀x .ϕ⇒ ∀x .ψ)


Presburger Arithmetic, PA

1. Function: nullary 0, unary s.

2. Relation: binary <. Let ≤ be the union of < and =.

3. Axiom:

PA1 ∀x .(s(x) 6= 0)PA2 ∀xy .(s(x) = s(y)⇒ x = y)PA3 ∀x .(x = 0 ∨ ∃y .s(y) = x)PA4 ∀x .(x < s(x))PA5 ∀xy .(x < y ⇒ s(x) ≤ y)PA6 ∀xy .(¬(x < y)⇔ y ≤ x)PA7 ∀xy .((x < y) ∧ (y < z)⇒ x < z)

PA8 ∀x .(x + 0 = x)PA9 ∀xy .(x + s(y)) = s(x + y)



Fact. ∀x , y .x < y ⇔ s(x) < s(y).

Fact. ∀x , y .x < y ∨ x = y ∨ y < x .

Fact. ∀x , y .x 6< y ⇔ (x = y ∨ y < x).

Fact. ∀x , y .x = y ⇔ (x < s(y) ∧ y < s(x)).

Fact. ∀x , y .x 6= y ⇔ (x < y ∨ y < x).

So the relations 6<,=, 6= can be dispensed with in favour of ’<’.


Numeral

For a natural number i , the closed term

s(s(. . . s︸︷︷︸i times

(0) . . .))

is called a numeral and is denoted by i (by abusing notation).


Term

Multiplication by constant

ixdef= x + x + . . .+ x︸︷︷︸

i times

.


Term

The terms(s(. . . s︸︷︷︸i times

(t) . . .))

is equal to t + i .

A term containing variable x1, . . . , xk is equal to a term of the form

n0 + n1x1 + n2x2 + nkxk .

When we write kx + t, we always mean that x does not appear in t.


Formula

In this talk a term in Z is seen as an abbreviation for a term in N.

For example x ← 3 stands for x + 3 < 0 and −y + 3 < x for3 < x + y .


Expressiveness

Finite quantification

∃x ∈ S .ϕ(x)def=

∨i∈S

ϕ(i).

Unique existence

∃!x .ϕ(x)def= ∃x .ϕ(x) ∧ ∀y .ϕ(y)⇒ y = x .


Expressiveness

The predicate m = µx .ϕ(x) can be defined by

ϕ(m) ∧ ∀z .ϕ(z)⇒ m ≤ z .

Similarly m = max(S) and m = min(S) are definable.


Congruence

A useful predicate is the congruence predicate defined by

s ≡k tdef= ∃z .kz+s=t ∨ kz+t=s.

The divide predicate i |x can be defined by x ≡i 0.

Fact. The congruence ≡k is decidable.


Let gcd(S) be the greatest common divisor of S .

Let lcm(S) be the least common multiple of S .


Decidability of PA


Presburger Arithmetic is Decidable

Theorem. (Presburger, 1929)

The validity of a closed Presburger formula is decidable.

Proof. The basic idea is quantifier elimination.


Presburger Arithmetic is Decidable

All decidability proofs make use of quantifier elimination.

The purpose of quantifier elimination is to replace in a Presburgerformula all the explicit occurrences of the quantifiers by the implicitoccurrences in the form of the congruence relations ≡1,≡2,≡3, . . ..

Fact. The validity of a closed Presburger formula without anyexplicit occurrences of the quantifiers is decidable.


Quantifier Elimination

H. Enderton (First Edition, 1972; Second Edition, 2001).

A Mathematical Introduction to Logic. Harcourt/Academic Press.

J. Monk (1976).

Mathematical Logic. Springer-Verlag, New York.



Suppose (. . . (Qx .ϕ) . . .) is a Presburger formula in which Q is aquantifier and ϕ is quantifier free.

We will turn Qx .ϕ into an equivalent quantifier free formulacomposed of the atomic formulae (of the forms s < t, s ≡i t) andthe logical operators ∧,∨.



1. Universal Quantifier Elimination

I If Q = ∀, then apply the De Morgan law

∀x .ϕ(x)⇔ ¬∃.¬ϕ(x)

to get rid of ∀.

2. Negation Elimination

I A negation operator is pushed down all the way to the front ofatomic formulae.

I s 6< t is replaced by t < s ∨ s < t + 1 ∧ t < s + 1.

I s 6≡i t is replaced by (s ≡i t + 1) ∨ . . . ∨ (s ≡i t + i − 1).



3. Normalization

I The propositional subformula is converted to a dnf.

I The formula

∃x .((α0 ∧ . . . ∧ αi ) ∨ . . . ∨ (γ0 ∧ . . . ∧ γk))

is then converted to

∃x .(α0 ∧ . . . ∧ αi ) ∨ . . . ∨ ∃x .(γ0 ∧ . . . ∧ γk).



4. Coefficient Uniformization

I So we only have to consider formulae of the following form

∃x .

(∧i

nix < ti

)∧

∧j

tj < njx

∧(∧k

nkx ≡ck tk

).

I Let n = lcm({ni , nj , nk}i ,j ,k). We get the equivalent formula

∃x .

(∧i

x < si

)∧

∧j

sj < x

∧(∧k

x ≡dk sk

)∧ x ≡n 0.



I The formula∧

i x < si is equivalent to

∨i

x < si ∧∧i ′ 6=i

si ≤ si ′

.

Similarly the formula∧

j sj < x is equivalent to

∨j

sj < x ∧∧j ′ 6=j

sj ′ ≤ sj

.



5. Substitution

I So we have reduced the original problem to that of decidingthe validity of formulas of the form

∃x .t ′ < x < t ′′ ∧∧l

x ≡dl tl .

I Let ddef= lcm({dl}l). The above formula is equivalent to

∨0<m≤d

(t ′+m < t ′′ ∧

∧l

t ′+m ≡dl tl

).


Theorem (Presburger, 1929).

Presburger Arithmetic admits quantifier elimination.

Corollary.

Presburger Arithmetic is consistent, complete and decidable.


Cooper’s Algorithm

D. Cooper (1972).

Theorem Proving in Arithmetic without Multiplication. In B.Meltzer and D. Michie, eds., Machine Intelligence. EdinburghUniversity Press: 91-100.



Cooper’s algorithm is a decision procedure for integer PresburgerArithmetic Z.

It works for N by imposing the condition: all variables ≥ 0.



The basic idea of Cooper’s algorithm is to do as less formulaconversion as possible.

∀-elimination, ¬-elimination (but keep 6≡i ), coefficientuniformization are all that is necessary.

The trade-off is that it must test a lot of instantiation cases.



Let ϕ be a negation-free propositional formula whose atomicformulae are

{x < si}i ∪ {sj < x}j ∪ {x ≡dk 0}k ∪ {x 6≡dl 0}l .

Let ϕ−∞(x) be obtained from ϕ by replacing

I all x < si by >, and

I all sj < x by ⊥.

Lemma. ϕ(x)⇔ ϕ−∞(x) for sufficiently small x .



Theorem. Suppose d = lcm{dk , dl}k,l . Then

∃x .ϕ(x) ⇔d∨

m=1

ϕ−∞(m) ∨d∨

m=1

∨j

ϕ(sj + m).

Proof. Stare at

{x < si}i ∪ {sj < x}j ∪ {x ≡dk 0}k ∪ {x 6≡dl 0}l ;

and think . . .



Let ϕ+∞(x) be obtained from ϕ by replacing

I all x < si by ⊥, and

I all sj < x by >.

Lemma. ϕ(x)⇔ ϕ+∞(x) for sufficiently large x .

Theorem. Suppose d = lcm{dk , dl}k,l . Then

∃x .ϕ(x) ⇔d∨

m=1

ϕ+∞(−m) ∨d∨

m=1

∨i

ϕ(si −m).


FMVE, Fourier-Motzkin Variable Elimination

FMVE decides the validity of a Presburger formula in Q.

After ∀-elimination and ¬-elimination, ∃-elimination is achieved byapplying the following equivalence:

∃x .

(∧h

chah≤ x

)∧

(∧i

ciai< x

)∧

∧j

x ≤djbj

∧(∧k

x <dkbk

)

if and only if∧h,j

chah≤

djbj∧∧h,k

chah

<dkbk∧∧i ,j

ciai<

djbj∧∧i ,k

ciai<

dkbk.


Complexity of Validity Checking


M. Fischer and M. Rabin (1974).

Super-Exponential Complexity of Presburger Arithmetic. InComplexity of Computation edited by Richard M. Karp, AmericanMathematical Society, Providence, Rhode Island, 27-41.

D. Oppen (1978).

A 222pn

Upper Bound on the Complexity of Presburger Arithmetic.Journal of Computer and System Sciences, 16:323-332.


Mojzesz Presburger (1904-1943) was a Polish Jewishmathematician, logician, and philosopher. As a student of AlfredTarski, he invented Presburger Arithmetic in 1929. He died in aconcentration camp in 1943.

In 2010, the European Association for Theoretical ComputerScience began conferring the annual Presburger Award to youngtheoretical computer scientists for outstanding contributions.


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