Multi-Dimensional Interpretations of Presburger Arithmetic in Itself
aa r X i v : . [ m a t h . L O ] A p r Multi-Dimensional Interpretations of PresburgerArithmetic in Itself
Fedor Pakhomov and Alexander Zapryagaev Institute of Mathematics of the Czech Academy of Sciences, ˇZitn´a 25, 115 67, Praha1, Czech Republic Steklov Mathematical Institute of Russian Academy of Sciences, 8, Gubkina Str.,Moscow, 119991, Russian Federation National Research University Higher School of Economics, 6, Usacheva Str.,Moscow, 119048, Russian Federation
Abstract.
Presburger Arithmetic is the true theory of natural num-bers with addition. We study interpretations of Presburger Arithmeticin itself. The main result of this paper is that all self-interpretations aredefinably isomorphic to the trivial one. Here we consider interpretationsthat might be multi-dimensional. We note that this resolves a conjec-ture by A. Visser. In order to prove the result we show that all linearorderings that are interpretable in ( N , +) are scattered orderings withthe finite Hausdorff rank and that the ranks are bounded in the termsof the dimensions of the respective interpretations. Presburger Arithmetic
PrA is the true theory of natural numbers with addition.Unlike Peano Arithmetic PA , it is complete, decidable and admits quantifierelimination in an extension of its language[11].The method of interpretations is a standard tool in model theory and in thestudy of decidability of first-order theories [17,8]. An interpretation of a theory T in a theory U is essentially a uniform first-order definition of models of T in models of U (see details in Section 2). In the paper we study certain ques-tions about interpretability for Presburger Arithmetic that were well-studied inthe case of stronger theories like Peano Arithmetic PA . Although from techni-cal point of view the study of interpretability for Presburger Arithmetic usescompletely different methods than the study of interpretability for PA (see, forexample, [19]), we show that from interpretation-theoretic point of view, PrA has certain similarities to strong theories that prove all the instances of math-ematical induction in their own language, i.e. PA , Zermelo-Fraenkel set theory ZF , etc.A reflexive arithmetical theory ([19, p. 13]) is a theory that can prove theconsistency of all its finitely axiomatizable subtheories. Peano Arithmetic PA and Zermelo-Fraenkel set theory ZF are among well-known reflexive theories.In fact, all sequential theories (very general class of theories similar to PA , see[6, III.1(b)]) that prove all instances of the induction scheme in their languagere reflexive. For sequential theories reflexivity implies that the theory cannotbe interpreted in any of its finite subtheories. A. Visser has conjectured thatthis purely interpretational-theoretic property holds for PrA as well. Note that
PrA satisfies full induction scheme in its own language but cannot formalize thestatements about consistency of formal theories.Unlike sequential theories, Presburger Arithmetic cannot encode tuples ofnatural numbers by single natural numbers. And thus, for interpretations inPresburger Arithmetic it is important whether individual objects are interpretedby individual objects (one-dimensional interpretations) or by tuples of objectsof some fixed length m ( m -dimensional interpretations).J. Zoethout [21] considered the case of one-dimensional interpretations andproved that if any one-dimensional interpretation of PrA in ( N , +) gives amodel definably isomorphic to ( N , +), then Visser’s conjecture holds for one-dimensional interpretations, i.e. there are no one-dimensional interpretations of PrA in its finite subtheories. Moreover, he proved that any interpretation of
PrA in ( N , +) is isomorphic to ( N , +); however, he hadn’t proved that the iso-morphism is definable. We improve the latter result and establish the definabilityof the isomorphism. Theorem 1.1.
The following holds for any model A of PrA that is one-dimen-sionally interpreted in the model ( N , +) :(a) A is isomorphic to ( N , +) ,(b) the isomorphism is definable in ( N , +) . Then, by a more sophisticated technique, we establish Visser’s conjecture formulti-dimensional interpretations.
Theorem 1.2.
The following holds for any model A of PrA that is interpretedin ( N , +) :(a) A is isomorphic to ( N , +) ,(b) the isomorphism is definable in ( N , +) . In the present paper we obtain both Theorem 1.1 (a) and Theorem 1.2 (a) asa corollary of a single fact about linear orderings interpretable in ( N , +). Recallthat any non-standard model of Presburger arithmetic has the order type of theform N + Z · A , where A is a dense linear ordering. In particular, it means thatthe order types of non-standard models of PrA are never scattered (a linearordering is called scattered if it contains no dense suborderings). We show thatany linear ordering that is interpretable in ( N , +) is scattered.In fact, we establish an even sharper result and estimate the ranks of theinterpreted orderings. The standard notion of rank of a scattered linear orderingis the Cantor-Bendixson rank that goes back to Hausdorff [7]. However, in ourcase a more precise estimation is obtained using a slightly different notion of VD ∗ -rank from [10]. Theorem 1.3.
Suppose a linear ordering ( L, ≺ ) is m -dimensionally interpretablein ( N , +) . Then ( L, ≺ ) is scattered and has VD ∗ -rank at most m . n order to prove Theorem 1.1 (b), we show that the (unique) isomorphismof the interpreted model A and ( N , +) is in fact definable in ( N , +). This isomor-phism is trivially definable using counting quantifiers, while the theorem thatin Presburger Arithmetic first-order formulas with counting quantifiers have thesame expressive power as ordinary first-order formulas is due to H. Apelt [1] andN. Schweikardt [13].The proof of Theorem 1.2 relies on a theory of cardinality functions p
7→ | A p | for definable families of finite sets h A p ⊆ N m | p ∈ P ⊆ N n i .We note that the present work essentially is an expanded version of the paper[20]. Results of Theorem 1.1(a,b), Theorem 1.2(a), Theorem 1.3, Theorem 5.1,and Corollary 5.1 were already present in [20]. Theorem 1.2(b) is new.The work is organized in the following way. Section 2 introduces PresburgerArithmetic and interpretations. In Section 3, we define notion of dimensionfor Presburger-definable sets and prove Theorem 1.3. In Section 4 we proveTheorem 1.1. In Section 5 we prove Theorem 1.2. In this section we give some general results about Presburger Arithmetic anddefinable sets in ( N , +). In this paper the set of all natural numbers N includeszero. Definition 2.1.
Presburger Arithmetic (
PrA ) is the elementary theory of themodel ( N , +) of natural numbers with addition. It is easy to define the constants 0 ,
1, relation ≤ and modulo comparisonrelations ≡ n , for all n ≥
1, in the model ( N , +). In the language extended by theseconstants and predicates, Presburger arithmetic admits quantifier elimination[11]. Furthermore, PrA is decidable.
PrA has non-standard models. Unlike PA , however, where it is impossible toproduce an explicit non-standard model by defining some recursive addition andmultiplication (Tennenbaum’s Theorem [18]), examples of non-standard modelsof PrA can be given explicitly (see [14]). By a usual argument one can showthat any non-standard model of
PrA has the order type N + Z · L , where L is adense linear ordering without endpoints. In particular, any countable model of PrA has the order type of either N or N + Z · Q . Definition 2.2.
For vectors c, p , . . . , p n ∈ Z m we call the set { c + P k i p i | k i ∈ N } ⊆ Z m a lattice (or a linear set ) generated by { p i } from c . If { p i } are linearlyindependent, we call this set a fundamental lattice . According to [5], definable subsets of N m are exactly the unions of a finitenumber of (possibly intersecting, possibly non-fundamental) lattices (such unionsare also called semilinear sets in literature). Ito has shown in [9] that any set in m which is a union of a finite number of (possibly intersecting, possibly non-fundamental) lattices (a semilinear set) can be expressed as a union of a finitenumber of disjoint fundamental lattices. Hence, Theorem 2.1.
All subsets of N k definable in ( N , +) are exactly the subsets of N k that are disjoint unions of finitely many fundamental lattices. Definition 2.3.
For a fundamental lattice J generated by v , . . . , v n from c wecall a function f : J → N linear if it is of the form f ( c + x v + . . . + x n v n ) = a + a x + . . . + a n x n for some a , . . . , a n ∈ N .For an ( N , +) -definable set A we call a function f : A → N piecewise linear ifthere is a decomposition of A into disjoint fundamental lattices J , . . . , J n suchthat the restriction of f on any J i is linear . Theorem 2.2.
Functions f : N n → N definable in ( N , +) are exactly piecewiselinear functions.Proof. The definability of all piecewise linear functions in Presburger Arithmeticis obvious. A function f : N n → N is definable if and only if its graph G = { ( a , . . . , a n , f ( a , . . . , a n )) | ( a , . . . , a n ) ∈ N n } is definable. According to Theorem 2.1, G is a finite union of fundamental lattices J ⊔ . . . ⊔ J k . For 1 ≤ i ≤ k we denote by J ′ i the projections of J i along the lastcoordinate, J ′ i = { ( a , . . . , a n ) | ∃ a n +1 (( a , a , . . . , a n , a n +1 ) ∈ J i ) } . Clearly, all J ′ i are fundamental lattices. Furthermore, the restriction of the function f oneach of J ′ i is linear. We define the notion of a multi-dimensional first-order non-parametric interpre-tation, following [17].
Definition 2.4. An m -dimensional interpretation ι of some first-order language K in a model A consists of first-order formulas of language of A :1. D ι ( y ) defining the set D ι ⊆ A m (domain of interpreted model);2. P ι ( x , . . . , x n ) , for predicate symbols P ( x , . . . , x n ) of K including equality;3. f ι ( x , . . . , x n , y ) , for functional symbols f ( x , . . . , x n ) of K . Here all vectors of variables x are of length m , and f ι ’s should define graphsof some functions (modulo interpretation of equality).Naturally, ι and A give a model B of the language K on the domain D ι / ∼ ι ,where equivalence relation ∼ ι is given by = ι ( x , x ). We will call B the internalmodel . In our work, we use the word ‘piecewise’ only in the sense of the current definition. f B | = T , then ι is an interpretation of the theory T in A . If for a first-ordertheory U an interpretation ι is an interpretation of T , for any A | = U , then ι isan interpretation of T in U .Interpretations are a very natural concept, appearing in mathematics when,for example, Euclidean geometry is interpreted in the theory of real numbers R (two-dimensionally, by defining points as pairs of real numbers) in analyticgeometry, or the field C of complex numbers is two-dimensionally interpreted in R by defining a + bi ↔ ( a, b ) and addition and multiplication are declared bydefinition. We note that in ( N , +) itself, the field ( Z , +) can be interpreted. Thisis achieved by mapping the negative numbers to odd, positive to even and 0 to 0and defining the addition case-by-case (through non-negative subtraction, whichis definable).We will be interested in interpretations of theories in the standard model ofPresburger Arithmetic, that is, in ( N , +). Definition 2.5. An m -dimensional interpretation ι in a model A has absoluteequality if the symbol = ∈ K is interpreted as the coincidence of two m -tuples. Definition 2.6.
An interpretation ι, κ in a model A are definably isomorphic ,if there is a first-order formula F ( x, y ) . of the language of A defining an isomor-phism between the respective internal models. The following theorem is a version of [21, Lemma 3.2.2], extended to multi-dimensional interpretations. It shows that it suffices to consider only the inter-pretations with absolute equality.
Theorem 2.3.
Suppose ι is an interpretation of some theory U in ( N , +) . Thenthere is an interpretation κ of U in ( N , +) with absolute equality which is defin-ably isomorphic to ι .Proof. Indeed, there is a definable in ( N , +) well-ordering ≺ of N m :( a , . . . , a m − ) ≺ ( b , . . . , b m − ) def ⇐⇒ ∃ i < m ( ∀ j < i ( a j = b j ) ∧ a i < b i ) . Now we could define κ by taking the definition of + from ι , interpreting theequality trivially, and declaring the domain of κ to be the part of the domain of ι that contains exactly the ≺ -least elements of equivalence classes with respectto ι -interpretation of equality. It is easy to see that this κ is definably isomorphicto ι . Henceforth, we will talk only about definability in the model ( N , +). By a de-finable set we always mean a set A ⊆ N n definable in ( N , +), and a definablefunction f : A → B will always be understood as a function between definablesets A and B that is definable in ( N , +) itself. efinition 3.1. We say that a natural number k ≥ is the dimension dim( A ) of an infinite definable set A ⊆ N m if there is a definable bijection between A and N k . The following theorem shows that the definition above uniquely defines thedimension for each infinite definable set.
Theorem 3.1.
Suppose A ⊆ N n is an infinite definable set. Then there is aunique m ∈ N such that there is a Presburger-definable bijection between A and N m , ≤ m ≤ k. Proof.
First we show that there is some m possessing the property. Accordingto Theorem 2.1, all sets definable in ( N , +) are disjoint unions of fundamentallattices J , . . . , J n of the dimensions k , . . . , k n , respectively (the dimension of afundamental lattice is the number of generating vectors). It is easy to see thatfor each J i there is a linear bijection with N k i , which is obviously definable. Letus put m to be the maximum of k i ’s.Now we notice that for each sequence r , . . . , r m ∈ N and u = max( r , . . . , r m ), u ≥
1, we are able to split N u into a disjoint union of definable sets B , . . . , B m ,for which we have definable bijections with N r , . . . , N r m , respectively. This isproved by induction on m .Let us show that there is no other m with this property. Assume the contrary.Then, for some m > m , there is a definable bijection f : N m → N m . Let usconsider a sequence of expanding cubes I m s def = { ( x , . . . , x k ) | ≤ x , . . . , x n ≤ s } . We define function g : N → N to be the function which maps a natural number x to the least y such that f ( I m x ) ⊆ I l y . Clearly, g is a definable function. Thenthere should be some linear function h : N → N such that g ( x ) ≤ h ( x ), for all x ∈ N . But since for each x ∈ N and y < x m /m the cube I m x contains morepoints than the cube I m y , from the definition of g we see that g ( x ) ≥ x m /m .This contradicts the linearity of the function h .As far as we know, this definition of dimension for Presburger definable setswas first introduced in [4] and restated in [20]. It can be seen that the dimen-sion of a set A ⊆ N n is equal to the maximal m such that there exists an m -dimensional fundamental lattice which is a subset of A . Definition 3.2.
For a set A ⊆ N n + m and a ∈ N n we define the section A ↾ a = { b ∈ N m | a ⌢ b ∈ A } , where a ⌢ b is the concatenation of the tuples a and b . Definition 3.3.
For a definable set P ⊆ N n a family of sets h A p ⊆ N m | p ∈ P i is called definable if there is a definable set A ⊆ P × N m such that A p = A ↾ p ,for any p ∈ P . emma 3.1. Suppose h A p ⊆ N n | p ∈ P i is a definable family of sets, andthe set P ′ ⊂ P (possibly undefinable) is such that for p ∈ P ′ the sets A p are n -dimensional and pairwise disjoint. Then P ′ is finite.Proof. Let us consider the set A = { p ⌢ a | p ∈ P and a ∈ A p } . By Theorem 2.1,the set A is a disjoint union of finitely many fundamental lattices J i ⊆ N k + n . Itis easy to see that if some set A p is n -dimensional, then for some i the section J i ↾ p = { a | p ⌢ a ∈ J i } is an n -dimensional set. Thus it is enough to showthat for each J i there are only finitely many p ∈ P ′ for which the section J i ↾ p is an n -dimensional set.Let us now assume for a contradiction that for some J i there are infinitelymany p ∈ P ′ for which J i ↾ p are n -dimensional sets. Let us consider some p ∈ P ′ such that the section J i ↾ p is an n -dimensional set. Then there exists an n -dimensional fundamental lattice K ⊆ J i ↾ p . Suppose the generating vectorsof K are v , . . . , v n ∈ N n and initial vector of K is u ∈ N n . It is easy to see thateach vector v j is a non-negative linear combination of generating vectors of J i ,since otherwise for large enough h ∈ N we would have c + hv j J i . Now noticethat for any p ∈ P and a ∈ J i ↾ p the n -dimensional lattice with generatingvectors v , . . . , v n and initial vector a is a subset of J i ↾ p .Thus infinitely many of the sets A p , for p ∈ P ′ , contain some shifts of the same n -dimensional fundamental lattice K . It is easy to see that the latter contradictsthe assumption that all the sets A p , for p ∈ P ′ , are disjoint. A linear ordering ( L, ≺ ) is called scattered ([12, pp. 32–33]) ifit does not have an infinite dense subordering. Definition 3.5.
Let ( L, ≺ ) be a linear ordering. We define a family of equiva-lence relations ≃ α , for ordinals α ∈ Ord by transfinite recursion: – ≃ is just equality; – ≃ λ = S β<λ ≃ α , for limit ordinals λ ; – a ≃ α +1 b def ⇐⇒ |{ c ∈ L | ( a ≺ c ≺ b ) or ( b ≺ c ≺ a ) } / ≃ α | < ℵ .Now we define VD ∗ - rank rk ( L, ≺ ) ∈ Ord ∪ {∞} of the ordering ( L, ≺ ) . TheVD ∗ -rank rk ( L, ≺ ) is the least α such that L/ ≃ α is finite. If, furthermore, forall α ∈ Ord the factor-set L/ ≃ α is infinite, we put rk ( L, ≺ ) = ∞ .By definition we put α < ∞ , for all α ∈ Ord . The definition given above corresponds to the procedure of condensation thatglues the points at finite distance from each other. The VD ∗ -rank is now theminimal number of iterated condensations required to reach some finite ordering. VD stand for very discrete ; see [12, p. 84-89]. roposition 3.1. Linear orderings ( L, ≺ ) such that rk ( L, ≺ ) < ∞ are exactlythe scattered linear orderings.Proof. ( ⇒ ) Let ( L, ≺ ) be not scattered. This means there is a dense subordering( S, ≺ ) in L with the induced order relation. However, after a single condensationoperation, any two points of S remain separate as there is an infinite numberof points even from S between them. This means that the condensed orderingstill contains S as subordering. By transfinite induction, this holds now for allordinal-numbered iterations. Hence, L cannot have a VD ∗ -rank < ∞ .( ⇐ ) Let rk ( L, ≺ ) = ∞ . We have to prove that there is an embedded denseordering in L . Consider the equivalence relation on the points of L : x ∼ y ⇔ “ x and y have been identified on some step of condensations”. As the rank doesnot equal to any ordinal, the number of equivalence classes is infinite. Picking arepresentative from each, we obtain the required dense subordering: as no twopoints are identified, there is always an infinite number of points between them.Indeed, were any two groups at a finite distance from each other in the inducedordering, they would have been joined into a single group at some step.The orderings with the VD ∗ -rank equal to 0 are exactly finite orderings, andthe orderings with VD ∗ -rank ≤ N , − N and 1 (one-element linear ordering). Remark 3.1.
Each scattered linear ordering of VD ∗ -rank 1 is 1-dimensionallyinterpretable in ( N , +). There are scattered linear orderings of VD ∗ -rank 2 thatare not interpretable in ( N , +). Proof.
The interpretability of linear orderings with rank 0 and rank 1 followsfrom the description above.Since there are uncountably many non-isomorphic scattered linear orderingsof VD ∗ -rank 2 and only countably many linear orderings interpretable in ( N , +),there is some scattered linear ordering of VD ∗ -rank 2 that is not interpretablein ( N , +).Now we prove the rank condition. Theorem 1.3.
Suppose a linear ordering ( L, ≺ ) is m -dimensionally interpretablein ( N , +) . Then ( L, ≺ ) is scattered and has VD ∗ -rank at most m .Proof. Since any ordering with a finite VD ∗ rank is scattered, it is enough that rk ( L, ≺ ) ≤ m . We prove the theorem by induction on m ≥ m -dimensionally interpretableordering ( L, ≺ ) with rk ( L, ≺ ) > m . By the definition of VD ∗ -rank, there areinfinitely many distinct ≃ m -equivalence classes in L . Hence, either there is aninfinite ascending a ≺ a ≺ . . . or descending a ≻ a ≻ . . . chain of elementsof L such that a i m a i +1 , for each i . Let L i be the intervals ( a i , a i +1 ) in theorder ≺ , if we had an ascending chain, or the intervals ( a i +1 , a i ) in the order ≺ ,if we had a descending chain. Since a i m a i +1 , the set L i / ≃ m − is infinite and rk ( L i , ≺ ) > m − L i are definable. Let us show that dim( L i ) ≥ m , foreach i . If m = 1 then it follows from the fact that L i is infinite. If m > L i ) < m . Also notice that in this case( L i , ≺ ) would be ( m − N, +), which contradictsthe induction hypothesis and the fact that rk ( L i , ≺ ) > m −
1. Since L i ⊆ N m ,we conclude that dim( L i ) = m , for all i .Now consider the definable family of sets { ( a, b ) | a, b ∈ L } . We see that all L i are in this family. Thus we have infinitely many disjoint sets of the dimension m in the family and hence there is a contradiction with Lemma 3.1.We have proved that if ( L, ≺ ) is m -dimensionally interpretable in ( N , +),then its VD ∗ -rank is at most m . Hence, by Proposition 3.1, L is scattered. Let us now consider the extension of the first-order predicate language with anadditional quantifier ∃ = y x, called a counting quantifier (notion introduced in[2]). The syntax is as follows: if f ( x, z ) is an L -formula with the free variables x, z, then F = ∃ = y z G ( x, z ) is also a formula with the free variables x, y. We extend the standard assignment of truth values to first-order formulas inthe model ( N , +) to formulas with counting quantifiers. For a formula F ( x, y ) ofthe form ∃ = y z G ( x, z ), a vector of natural numbers a , and a natural number n wesay that F ( a, n ) is true if and only if there are exactly n distinct natural numbers b such that G ( a, b ) is true. H. Apelt [1] and N. Schweikardt [13] have establishedthat such an extension does not change the expressive power of PrA : Theorem 4.1. ([13, Corollary 5.10]) Every formula F ( x ) in the language ofPresburger arithmetic with counting quantifiers is equivalent in ( N , +) to a quan-tifier-free formula. Theorem 1.1.
The following holds for any model A of PrA that is one-dimen-sionally interpreted in the model ( N , +) :(a) A is isomorphic to ( N , +) ,(b) the isomorphism is definable in ( N , +) .Proof. From Theorem 2.3 it follows that it is enough to consider the case whenthe interpretation that gives us A has absolute equality.Let us denote the relation given by the PrA definition of < within A by < A .Clearly, < A is definable in ( N , +). Hence, by Theorem 1.3, the order type of A isscattered. But since any non-standard model of PrA is not scattered, the model A is isomorphic to ( N , +).It is easy to see that the isomorphism f from A to ( N , +) is the function f : x
7→ |{ y ∈ N | y < A x }| . Now we use a counting quantifier to express thefunction: f ( a ) = b ⇐⇒ ( N , +) | = ∃ = b z ( z < A a ) . ow apply Theorem 4.1 and see that f is definable in ( N , +).This implies Visser’s Conjecture for one-dimensional interpretations. Theorem 4.3.
Theory
PrA is not one-dimensionally interpretable in any ofits finitely axiomatizable subtheories.Proof.
Assume ι is an one-dimensional interpretation of PrA in some finitelyaxiomatizable subtheory T of PrA . In the standard model ( N , +) the interpre-tation ι gives us a model A for which there is a definable isomorphism f with( N , +). Now let us consider theory T ′ that consists of T and the statement thatthe definition of f establishes an isomorphism between (internal) natural num-bers and the structure given by ι . Clearly, T ′ is finitely axiomatizable and truein ( N , +), and hence it is a subtheory of PrA . But now note that T ′ proves thatif something was true in the internal structure given by ι , it is true. And since T ′ proved any axiom of PrA in the internal structure given by ι , the theory T ′ proves every axiom of PrA . Thus T ′ coincides with PrA . But it is known that
PrA is not finitely axiomatizable, contradiction.
Our goal is to prove Theorem 1.2. In order to prove that all multi-dimensionalinterpretations of
PrA in ( N , +) are isomorphic to ( N , +), we use the sameargument as in one-dimensional case: an interpretation of a non-standard modelwould entail an interpretation of a non-scattered order, which is impossible byTheorem 1.3.However, in order to show that the isomorphism is definable, we first needto develop theory of cardinality functions for the definable families of finite sets. Definition 5.1.
Let J ⊆ Z n be a fundamental lattice generated by vectors p , . . . , p m from c . We say that f : J → N is polynomial if there is a polynomialwith rational coefficients P f ( x , . . . , x m ) such that f ( c + p x + . . . + p m x m ) = P f ( x , . . . , x m ) , for all x , . . . , x m ∈ N . We note that if f is a polynomial function on J , then the polynomial P f isuniquely determined. Definition 5.2.
Let A ⊆ Z n be a definable set. We call a function f : A → N piecewise polynomial if there is a decomposition of A into finitely many funda-mental lattices J , . . . , J k such that the restriction of f on each J i is a polynomial.The degree deg( f ) is the maximum of the degrees of the restrictions f ↾ J i . We note that our definition of the degree is independent of the choice of thedecomposition J , . . . , J k . Indeed, for a piecewise polynomial function f : A → N consider the function h f : N → N that maps x ∈ N to max { f ( a ) | a ∈ A and | a | ∞ ≤ x } . Here as usual | ( a , . . . , a n ) | = max( | a | , . . . , | a n | ). Observethat if f has degree m (according to a particular decomposition) then h f hashe asymptotic growth rate of m -th degree polynomial. Thus the degree is inde-pendent of the choice of decomposition.By the same argument as above we get the following Lemma 5.1.
Suppose piecewise polynomial functions f, g : A → N are such that g ( x ) ≤ f ( x ) , for any x . Then deg( g ) ≤ deg( f )The following theorem is a slight modification of the theorem by G.R. Blak-ley [3]. Theorem 5.1.
Suppose M is a m × n matrix of integer numbers. Let the func-tion ϕ M : Z m → N ∪ {ℵ } be defined as follows: ϕ M ( u ) def = |{ a ∈ N n | M a = u }| . Additionally suppose that the values of ϕ M are always finite. Then ϕ M is apiecewise polynomial function of the degree ≤ n − rk ( M ) .Proof. In [3] it had been proved that ϕ M is a piecewise polynomial function.Further we prove that deg( ϕ M ) ≤ n − rk ( M ). Our goal will be to find a polyno-mial P ( x ) of the degree ≤ n − rk ( M ) such that ϕ M ( u ) ≤ P ( | u | ∞ ). After this wecould derive that deg( ϕ M ) ≤ n − rk ( M ) by Lemma 5.1.Note that each value ϕ M ( u ) is the number of natural points (we call a =( a , . . . , a m ) natural if a , . . . , a m ∈ N ) in the hyperplane H u = { a ∈ R | M a = u } . We are going to find a linear in | u | ∞ bound on | a | ∞ , for natural points a ∈ H u .Since ϕ M ( u ) is always finite, there could be no non-zero a ∈ N n such that M a = 0. Hence there are no non-zero a ∈ ( Q + ) n such that M a = 0. Furthermore,since M was a matrix with integer coefficients, there are no non-zero a ∈ ( R + ) n such that M a = 0. Thus there exists a rational ε > a ∈ ( R + ) n with | a | ∞ = 1 we have | M a | ∞ ≥ ε . Thus for any point a ∈ H u ∩ ( R + ) n we have a ≤ | u | ∞ ε .Henceforth all natural points of H u are contained in the hypercube [0 , | u | ∞ ε ] n .It is easy to see that the intersection of a k -dimensional plane with a cube [0 , b ] n always contains at most (( b + 1) n ) k natural points. Given that the planes H u are n − rk ( M )-dimensional, we see that ϕ M ( u ) ≤ (( | u | ∞ ε + 1) n ) n − rk ( M ) . We put P ( x ) = (( xε + 1) n ) n − rk ( M ) and finish the proof. Corollary 5.1.
For any definable family of finite sets h A p ⊆ N n | p ∈ P i , thefunction p
7→ | A p | is piecewise polynomial of the degree ≤ n .Proof. Let A = S p ∈ P { p ⌢ a | a ∈ A p } ⊆ N m + n . We have a decomposition of A into a disjoint union of fundamental lattices J , . . . , J n . A sum of piecewisepolynomial functions of degree ≤ n is piecewise polynomial of the degree ≤ n .Hence, it is enough to show that for all J i the function f i : p
7→ | J i ↾ p | is apiecewise polynomial function on P .uppose J i is generated by vectors v , . . . , v k from c . Let v ′ , . . . , v ′ k , c ′ bethe vectors consisting of first m coordinates of v , . . . , v k , c , respectively. Let M be the m × k -dimensional matrix corresponding to the function that maps( x , . . . , x k ) to v ′ x + . . . v ′ k x k . It is clear that rk ( M ) ≥ k − n . Now we see that | J i ↾ p | = ϕ M ( p − c ′ ) and thus f i is piecewise polynomial of the degree ≤ n . Lemma 5.2.
Each monotone piecewise polynomial function f : N → N of thedegree n + 1 is of the form Cx n +1 + g ( x ) , where C > is rational and g : N → N ,is piecewise polynomial of the degree n .Proof. Since f is piecewise polynomial, there is a splitting of N into infinitearithmetical progressions and one-element sets A , . . . , A n such that on eachof them f is given by a polynomial P , . . . , P n . From monotonicity of f , it iseasy to see that for all infinite A i the corresponding P i should have the samehighest degree term Cx n +1 . This determines g . On infinite A i , we see that g ( x ) = P i ( x ) − Cx n +1 (which is n -th degree polynomial). Thus, g is piecewise polynomialof the degree n . Corollary 5.2.
Suppose f : N → N is a monotone piecewise polynomial functionof the degree n + 1 . Then f ( x + 1) − f ( x ) is piecewise polynomial of the degree ≤ n . Theorem 1.2.
The following holds for any model A of PrA that is interpretedin ( N , +) :(a) A is isomorphic to ( N , +) ,(b) the isomorphism is definable in ( N , +) .Proof. As in the proof of Theorem 1.1 we may assume that the interpretationof A has absolute equality. And we show that A ≃ ( N , +) by the same method.So further we just prove that the isomorphism is definable.For i ∈ N , let S i be the maximal initial fragment of A such that | a | ∞ ≤ i , forall a ∈ S i . Clearly, h S i | i ∈ N i is a definable family of finite sets. Let h : N → N be the function x
7→ | S x | . From Corollary 5.1, it follows that the function h ispiecewise polynomial.Clearly, the degree of h is non-zero. First assume that h has the degree 1. Inthis case, since h is monotone, from Corollary 5.2 it follows that h ( x + 1) − h ( x )is piecewise polynomial of the degree 0 and hence bounded by some constant C .Thus, or any i we have | S i +1 \ S i | ≤ C . As we will see below this allows us tocreate a first-order definition of the required isomorphism f : A → ( N , +).If x ∈ S we define f ( x ) by separately considering the cases x = a , for allindividual a ∈ S . Further we define f ( x ) for x ∈ A \ S . We find the unique z such that x ∈ S z +1 \ S z . Let U x,z = { w ∈ S z +1 \ S z | w < A x } . Externally we know that f ( x ) = h ( z ) + | U x,z | . Since h is piecewise linear, byTheorem 2.2 it is definable. We know that 0 ≤ | U x,z | < C , which allows us toefine the value f ( x ) by separately considering the cases for all possible valuesof | U x,z | . More formally this description corresponds to the following definitionof the predicate f ( x ) = y : ^ a ∈ S (cid:0) x = a → y = f ( a ) (cid:1) ∧ ^ ≤ s 2. Our goal will be to show thatthis is in fact impossible. For this we consider the following definable function g : N → N : g ( x ) = min { y | ( ∀ z ∈ S x )( z + A z + A +1 A ∈ S y ) } In other words g ( x ) is the least y such that the initial fragment S y is at leasttwo times larger than S x . Thus we have h ( g ( x ) − < h ( x ) ≤ h ( g ( x )) . Since both h and g are monotone, by Lemma 5.2 we have rational C , C > h ( x ) = C x k (1 + o (1)) and g ( x ) = C x (1 + o (1)). Therefore h ( g ( x ) − 1) = C C k x k (1 + o (1)) and h ( g ( x )) = C C k x k (1 + o (1)). Hence 2 h ( x ) = C C k x k (1 + o (1)). At the same time 2 h ( x ) = 2 C x k (1 + o (1)). Thus 2 = C k and C = k √ 2. Contradiction with the fact that C is rational.In the same manner as Theorem 4.3 (but using Theorem 1.2 instead of Theorem 1.1)we prove Theorem 5.3. Theory PrA is not interpretable in any of its finitely axiomati-zable subtheories. Acknowledgments The authors thank Lev Beklemishev for suggesting to study Visser’s conjecture,a number of fruitful discussions of the subject, and his useful comments.Work of Fedor Pakhomov is supported by grant 19-05497S of GA ˇCR.Work of Alexander Zapryagaev was prepared within the framework of theAcademic Fund Program at the National Research University Higher School ofEconomics (HSE) in 2020 (grant No. 19-04-050) and by the Russian AcademicExcellence Project “5-100”. eferences 1. Apelt, H.: Axiomatische Untersuchungen ¨uber einige mit der PresburgerschenArithmetik verwandte Systeme. MLQ Math. Log. Q. 12.1, 131-168 (1966)2. Barrington, D., Immerman, N., Straubing, H.: On uniformity within NC1. §§