Skolem arithmetic

For other uses, seeSkolem arithmetic (disambiguation).

Inmathematical logic,Skolem arithmeticis thefirst-order theoryof thenatural numberswithmultiplication,named in honor ofThoralf Skolem.Thesignatureof Skolem arithmetic contains only the multiplication operation and equality, omitting the addition operation entirely.

Skolem arithmetic is weaker thanPeano arithmetic,which includes both addition and multiplication operations.Unlike Peano arithmetic, Skolem arithmetic is adecidable theory.This means it is possible to effectively determine, for any sentence in the language of Skolem arithmetic, whether that sentence is provable from the axioms of Skolem arithmetic. The asymptotic running-timecomputational complexityof thisdecision problemis triply exponential.

Axioms

We define the following abbreviations.

• a|b:= ∃n.(an=b)
• One(e):= ∀n.(ne=n)
• Prime(p):= ¬One(p) ∧ ∀a.(a|p→ (One(a) ∨a=p))
• PrimePower(p,P):= Prime(p) ∧p|P∧ ∀q.(Prime(q) ∧ ¬(q=p) → ¬(q|P))
• InvAdicAbs(p,n,P):= PrimePower(p,P) ∧P|n∧ ∀Q.((PrimePower(p,Q) ∧Q|n) →Q|P) [Pis the largest power ofpdividingn]
• AdicAbsDiff_n(p,a,b):= Prime(p) ∧p|ab∧ ∃P.∃Q.(InvAdicAbs(p,a,P) ∧ InvAdicAbs(p,b,Q) ∧Q=pⁿP) for each integern> 0. [The largest power ofpdividingbispⁿtimes the largest power ofpdividinga]

The axioms of Skolem arithmetic are:

1. ∀a.∀b.(ab=ba)
2. ∀a.∀b.∀c.((ab)c=a(bc))
3. ∃e.One(e)
4. ∀a.∀b.(One(ab) → One(a) ∨ One(b))
5. ∀a.∀b.∀c.(ac=bc→a=b)
6. ∀a.∀b.(aⁿ=bⁿ→a=b) for each integern> 0
7. ∀x.∃a.∃r.(x=arⁿ∧ ∀b.∀s.(x=bsⁿ→a|b)) for each integern> 0
8. ∀a.∃p.(Prime(p) ∧ ¬(p|a)) [Infinitude of primes]
9. ∀p.∀P.∀Q.((PrimePower(p,P) ∧ PrimePower(p,Q)) → (P|Q∨Q|P))
10. ∀p.∀n.(Prime(p) → ∃P.InvAdicAbs(p,n,P))
11. ∀n.∀m.(n=m↔ ∀p.(Prime(p) → ∃P.(InvAdicAbs(p,n,P) ∧ InvAdicAbs(p,m,P)))) [Unique factorization]
12. ∀p.∀n.∀m.(Prime(p) → ∃P.∃Q.(InvAdicAbs(p,n,P) ∧ InvAdicAbs(p,m,Q) ∧ InvAdicAbs(p,nm,PQ))) [p-adic absolute value is multiplicative]
13. ∀a.∀b.(∀p.(Prime(p) → ∃P.∃Q.(InvAdicAbs(p,a,P) ∧ InvAdicAbs(p,b,Q) ∧P|Q)) →a|b) [If thep-adic valuation ofais less than that ofbfor every primep,thena|b]
14. ∀a.∀b.∃c.∀p(Prime(p) → (((p|a→ ∃P.(InvAdicAbs(p,b,P) ∧ InvAdicAbs(p,c,P))) ∧ ((p|b) → (p|a)))) [Deleting from the prime factorization ofball primes not dividinga]
15. ∀a.∃b.∀p.(Prime(p) → (∃P.(InvAdicAbs(p,a,P) ∧ InvAdicAbs(p,b,pP))) ∧ (p|b→p|a))) [Increasing each exponent in the prime factorization ofaby 1]
16. ∀a.∀b.∃c.∀p.(Prime(p) → ((AdicAbsDiff_n(p,a,b) → InvAdicAbs(p,c,p)) ∧ (p|c→ AdicAbsDiff_n(p,a,b))) for each integern> 0 [Product of those primespsuch that the largest power ofpdividingbispⁿtimes the largest power ofpdividinga]

Expressive power

First-order logic with equality and multiplication of positive integers can express the relation $c=a\cdot b$.Using this relation and equality, we can define the following relations on positive integers:

• Divisibility:$b|c\ \Leftrightarrow \ \exists a.c=a\cdot b$
• Greatest common divisor:$d=\gcd(a,b)\ \Leftrightarrow \ d|a\land d|b\land \forall d'.(d'|a\land d'|b)\Rightarrow d'|d$
• Least common multiple:$m=\mathrm {lcm} (a,b)\ \Leftrightarrow \ a|m\land b|m\land \forall m'.(a|m'\land b|m')\Rightarrow m|m'$
• the constant$1$:$\forall a.1|a$
• Prime number:$\mathrm {prime} (p)\ \Leftrightarrow \ p\neq 1\land \forall a.a|p\Rightarrow (a=1\lor a=p)$
• Number$b$is a product of$k$primes (for a fixed$k$):$\exists a_{1},...a_{k}.\ \mathrm {prime} (a_{1})\land \ldots \land \mathrm {prime} (a_{k})\land b=a_{1}\cdot \ldots \cdot a_{k}$
• Number$b$is a power of some prime:$\mathrm {ppower} (b)\ \Leftrightarrow \ \exists p.\mathrm {prime} (p)\land \forall a.(a\neq 1\land a|b)\Rightarrow p|a$
• Number$b$is a product of exactly$k$prime powers:$\exists a_{1},...a_{k}.\ \mathrm {ppower} (a_{1})\land \ldots \land \mathrm {ppower} (a_{k})\land b=a_{1}\cdot \ldots \cdot a_{k}$

Idea of decidability

The truth value of formulas of Skolem arithmetic can be reduced to the truth value of sequences of non-negative integers constituting their prime factor decomposition, with multiplication becoming point-wise addition of sequences. The decidability then follows from theFeferman–Vaught theoremthat can be shown usingquantifier elimination.Another way of stating this is that first-order theory of positive integers is isomorphic to the first-order theory of finitemultisetsof non-negative integers with the multiset sum operation, whose decidability reduces to the decidability of the theory of elements.

In more detail, according to thefundamental theorem of arithmetic,a positive integer$a>1$can be represented as a product of prime powers:

$a=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots$

If a prime number$p_{k}$does not appear as a factor, we define its exponent$a_{k}$to be zero. Thus, only finitely many exponents are non-zero in the infinite sequence$a_{1},a_{2},\ldots$.Denote such sequences of non-negative integers by$N^{*}$.

Now consider the decomposition of another positive number,

$b=p_{1}^{b_{1}}p_{2}^{b_{2}}\cdots$

The multiplication$ab$corresponds point-wise addition of the exponents:

$ab=p_{1}^{a_{1}+b_{1}}p_{2}^{a_{2}+b_{2}}\cdots$

Define the corresponding point-wise addition on sequences by:

$(a_{1},a_{2},\ldots ){\bar {+}}(b_{1},b_{2},\ldots )=(a_{1}+b_{1},a_{2}+b_{2},\ldots )$

Thus we have an isomorphism between the structure of positive integers with multiplication,$(N,\cdot )$and of point-wise addition of the sequences of non-negative integers in which only finitely many elements are non-zero,$(N^{*},{\bar {+}})$.

FromFeferman–Vaught theoremforfirst-order logic,the truth value of a first-order logic formula over sequences and pointwise addition on them reduces, in an algorithmic way, to the truth value of formulas in the theory of elements of the sequence with addition, which, in this case, isPresburger arithmetic.Because Presburger arithmetic is decidable, Skolem arithmetic is also decidable.

Complexity

Ferrante & Rackoff (1979,Chapter 5) establish, usingEhrenfeucht–Fraïssé games,a method to prove upper bounds on decision problem complexity of weak direct powers of theories. They apply this method to obtain triply exponential space complexity for$(N^{*},{\bar {+}})$,and thus of Skolem arithmetic.

Grädel (1989,Section 5) proves that thesatisfiabilityproblem for thequantifier-freefragment of Skolem arithmetic belongs to theNP complexity class.

Decidable extensions

Thanks to the above reduction using Feferman–Vaught theorem, we can obtain first-order theories whose open formulas define a larger set of relations if we strengthen the theory of multisets of prime factors. For example, consider the relation$a\sim b$that is true if and only if$a$and$b$have the equal number of distinct prime factors:

$|\{p\mid \mathrm {prime} (p)\land (p|a)\}|\ =\ |\{p\mid \mathrm {prime} (p)\land (p|b)\}|$

For example,$2^{10}\cdot 3^{100}\sim 5^{8}\cdot 19^{9}$because both sides denote a number that has two distinct prime factors.

If we add the relation$\sim$to Skolem arithmetic, it remains decidable. This is because the theory of sets of indices remains decidable in the presence of theequinumerosityoperator on sets, as shown by theFeferman–Vaught theorem.

Undecidable extensions

An extension of Skolem arithmetic with the successor predicate,$succ(n)=n+1$can define the addition relation using Tarski's identity:

$(c=0\lor c=a+b)\Leftrightarrow (ac+1)(bc+1)=c^{2}(ab+1)+1$

and defining the relation$c=a+b$on positive integers by

$\mathrm {succ} (ac)\,\mathrm {succ} (bc)=\mathrm {succ} (c^{2}\mathrm {succ} (ab))$

Because it can express both multiplication and addition, the resulting theory is undecidable.

If we have an ordering predicate on natural numbers (less than,$<$), we can express$\mathrm {succ}$by

$\mathrm {succ} (a)=b\ \ \Leftrightarrow \ \ a<b\land \forall c.{\big (}a<c\Rightarrow (b=c\lor b<c){\big )}$

so the extension with$<$is also undecidable.

See also

• Presburger arithmetic
• Robinson arithmetic
• Quantifier elimination

Bibliography

• Bès, Alexis (2001)."A Survey of Arithmetical Definability"(PDF).In Crabbé, Marcel; Point, Françoise; Michaux, Christian (eds.).A Tribute to Maurice Boffa.Brussels: Societé mathématique de Belgique. pp. 1–54.

• Bès, Alexis; Richard, Denis (1998). "Undecidable Extensions of Skolem Arithmetic".Journal of Symbolic Logic.63(2): 379–401.CiteSeerX10.1.1.2.1139.doi:10.2307/2586837.JSTOR2586837.S2CID14566619.

• Cegielski, Patrick (1981), "Théorie élémentaire de la multiplication des entiers naturels", in Berline, Chantal; McAloon, Kenneth; Ressayre, Jean-Pierre (eds.),Model Theory and Arithmetic,Berlin: Springer, pp. 44–89.

• Ferrante, Jeanne; Rackoff, Charles W. (1979).The Computational Complexity of Logical Theories.Berlin Heidelberg New York: Springer-Verlag.doi:10.1007/BFb0062837.ISBN3-540-09501-2.

• Grädel, Erich (June 1989)."Dominoes and the complexity of subclasses of logical theories".Annals of Pure and Applied Logic.43(1): 1–30.doi:10.1016/0168-0072(89)90023-7.

• Nadel, Mark E. (1981)."The completeness of Peano multiplication".Israel Journal of Mathematics.39(3): 225–233.doi:10.1007/bf02760851.Retrieved2022-09-08.

• Robinson, Julia Hall Bowman(1949)."Definability and Decision Problems in Arithmetic"(PDF).Journal of Symbolic Logic.14(2): 98–114.doi:10.2307/2266510.JSTOR2266510.S2CID40861592.Retrieved2022-09-05.