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Joanna Golińska
THE SPECTRUM PROBLEM FOR THE LANGUAGE WITH HENKIN
QUANTIFIERS1
Probably under the influence of The Completeness Theorem for first order logic
[Gödel 1930] and Gödel’s work [Gödel 1931] proving essential incompleteness of logics for
richer languages, logicians focused on semantics for first order logic. This attitude got
intensified due to Trachtenbrot’s discovery [Trachtenbrot 1950] that the logic restricted to
finite models cannot be axiomatizable except in the case of very pure vocabularies. As a
result, until the questions and problems concerning researches on the foundations of computer
science arose, most of the mathematicians did not pay too much attention to semantics
restricted to finite models. The most extreme example of this is the monograph [Chang –
Keisler 1990] reassuming some important stage of the model – theoretic researches. However,
apart from the main direction of researches, problems concerning logics restricted to finite
models and attempts to solve them arose. The most important of them are the problems of
Scholz and Asser. Now we know that these problems are equivalent to crucial problems in
foundations of computer science.
In particular, the most important open mathematical problem „does P=NP?” is
equivalent to some questions concerning finite model theory. Assuming that the way of a
human brain’s work is based on the same rules as the way the machines made by a man work,
this problem is also equivalent to a question about the real human cognitive abilities.
Classes of finite models recognizable in real time, that is the ones belonging to the
class P, according to our programming experience determine the highest level of practical
computation. On the other hand, many practically difficult from the algorithmic point of view
problems belong to NP class. The importance of NP lies in the fact that many problems for
which efficient algorithms would be highly desirable, but are yet unknown, belong to NP. The
travelling salesman problem and the integer-programming problem are two such problems
among many others. Therefore the question concerning classification of concepts in finite
models can be treated as the crucial problems related to human cognitive abilities in the
technological sense and presumable in the absolute sense.
The proof of Trachtenbrot’s theorem stating that there is no algorithm, which would
decide for any formula whether it is satisfied in finite models, was presented on a symposium
organized by Institut für Mathematische Logik und Grundlagenforschung University in
Münster. Heinrich Scholz put there the question about sets of cardinalities of all finite models
of first order formulae (called the spectrum). The official formulation of the problem asks
about characterization of spectra for first order formulae (including „=” but without function
symbols) has been given in [Scholz 1952]. G. Asser in [Asser 1955] has stated the weaker
version of this problem „does complement of every spectrum is also a spectrum?”. Later,
Scholz problem has been considered for various logics. Thus, the spectrum problem applies to
finitely axiomatizable classes of finite models (spectra of these classes are called generalized
first order spectra) and to any n – order logics.
1
Many ideas and concepts used in the paper originate from the discussions with dr hab. M. Mostowski and
Konrad Zdanowski. I would like to thank for all this paper owes to them.
The most important and most interesting theorem related to the spectrum problem is
Fagin’s theorem. In [Fagin 1974] Fagin considered generalized spectra of existential second
order formulae. He proved that any class of finite models closed on isomorphism is a
generalized spectrum if and only if it is recognizable by nondeterministic Turing’s machine in
polynomial time, i.e., it is in NP. Of course we should restrict our attention to classes of
models closed on isomorphism. Blass and Gurevitch [Blass–Gurevitch 1986] showed that the
same holds for positive formulae with Henkin quantifiers. These above theorems allows to
prove that Asser problem is equivalent to well–known open problem of theory of
computational complexity „does NP=co–NP?”. Hence Scholz and Asser problems are still
open.
Therefore it seems natural to consider the spectrum problem for simpler case, e.g.
logics in empty vocabulary. It is known that in this case generalized spectra and spectra in the
sense of Scholz are equivalent. This so because in the case of empty vocabulary, models are
uniquely up to isomorphism determined by their cardinalities.
In this paper the spectrum problem for the logic with branched quantifiers in empty
vocabulary is considered. Particular attention is devoted to spectra of some syntactically
defined sublogics of the logic with all branched quantifiers. The characterization of spectra of
these logics would additionally solve the problem whether there is any essentially infinite
class of branched quantifiers not equivalent to all Henkin quantifiers. One of the simplest
examined essentially infinite classes is the class of quantifiers of the form:
x1 y1
...........
...........
xn yn
The problem whether these quantifiers are equivalent to all Henkin quantifiers is still open.
Therefore particular attention is devoted to the logic with these quantifiers. All considered
results can be also treated as results about expressive power of certain languages in finite
models
This paper does not contain final solutions to the title problem. Even though some of
general theorems giving partial characterization of spectra of considered sublogics of the logic
with all Henkin quantifiers are formulated, the examples of spectra of mentioned sublogics
are presented in the first place.
The paper consists of four sections. In the first section – Henkin Quantifiers – the
basic ideas and concepts related to the logic with branched quantifiers are presented. The
second section – The spectrum problem. General formulation – contains the various versions
of formulation of the spectrum problem with special attention concentrated on the logic with
branched quantifiers in the empty vocabulary. In the third section – The spectrum problem for
languages with Henkin quantifiers – the solution of the spectrum problem for the logic with
function quantifiers is presented. This logic is one of the few sublogics of the logic with all
Henkin quantifiers L* about which it is known that it is semantically weaker than L*. This
solution can be treated as paradigmatic solution to considered version of the spectrum
problem presented in the paper. In the last, fourth section – Examples of spectra of formulae
with branched quantifiers – examples of sets of natural numbers being the spectra of formulae
with specified branched quantifiers and the partial characterization of spectra for some special
class of formulae with branched quantifiers are presented. The paper ends with the summary
of received results.
The way of defining of classical logical concepts and formulation of the basic
theorems is based on the works [Chang–Keisler 1990], [Adamowicz–Zbierski 1991] and
[Krynicki–Mostowski 1995]. This work is based on a portion of the author’s master thesis in
Faculty of Philosophy and Sociology of Warsaw University. In the paper only main theorems
directly concerning the considered problem are being proved. The proofs of theorems
presented earlier in the literature within the limits of logical research on branched quantifiers
are substituted with references to literature.
HENKIN QUANTIFIERS
In this section we present basic ideas and concepts related to Henkin quantifiers. Our
presentation is based on the survey paper [Krynicki–Mostowski 1995]. Because we are
interested mainly in the logic with Henkin quantifiers restricted to finite models, so we skip
such semantics as relational or weak. In finite models these two semantics are equivalent to
the one given by Henkin in [Henkin 1961].
1.1 Definition (Henkin prefixes as dependency relations )
A Henkin prefix (a branched prefix) is a triple Q = (AQ, EQ, DQ), where AQ and EQ are disjoint
finite sets of variables called respectively universal and existential variables of Q, and DQ is a
relation between universal and existential variables of Q (DQ  AQ  EQ), called the
dependency relation of Q. If (x, y)  DQ then we say that the existential variable y depends on
the universal variable x in Q.
Logicians use the term quantifier quite ambiguously (see e.g., discussion in [Krynicki–
Mostowski 1995a]). Particularly in the case of Henkin quantifiers several concepts are equally
well applied to quantifier prefixes and to quantifiers. A quantifier prefix is a quantifier with
fixed bound variables. Having any class of quantifier prefixes we define an equivalence
relation for prefixes by an isomorphism determined by any permutation of all relevant
variables. Then we define a quantifier as an equivalence class of this equivalence relation.
Traditionally we define basic concepts parallel for prefixes and quantifiers, using slightly
ambiguous terminology (see e.g., [Krynicki–Mostowski 1995]). In this paper notation for
prefixes and related quantifiers will be the same.
By Hn we denote a Henkin prefix of the following form:
x1 y1
...........
...........
xn yn
Prefixes of this form we will call a simple Henkin prefix.
By A kn we denote a Henkin prefix of the following form:
x1 y1
..............
..............
xn yn
z 11 z 12 w1
..............
z 1k z 2k wk
1.2 Definition (Language with branched quantifiers )
Let H be a set of all branched prefixes. We define F(H) – the set of all formulae of
vocabulary  build up with connectives: ¬, , , , , elementary quantifiers: , , and
elements of H as additional quantifiers – in the same way as formulae of elementary logic
taking the following additional construction rule:
if Q  H and  is a formula then Q is a formula.
For formulae of the form Q we define: x is a free variable in the formula Q, if x is an
element of the set V(Q) = V()\{x1, ..., xn}, where x1, ..., xn are all variables occurring in Q
and V() is the set of free variables in .
1.3 Definition (Skolemization of branched prefixes )
Let Q be a Henkin prefix binding universal variables x1, ..., xn and existential variables y1, ...,
yk ; let xi be a sequence of the universal variables of Q on which yi depends in Q, for i = 1, ...,
k. Then we define a skolemization of Q relative to Q, sk(Q, ), as the results of substituting
in  of fi(xi) in place yi, for i = 1, ..., k. The function symbols f1, ..., fk introduced in this way
have to be new and distinct from one another. They are called the Skolem functions introduced
by skolemization of Q relative to Q.
1.4 Definition (Logic with branched quantifiers L*)
We define a logic L* as an assignment to every vocabulary  of a pair L * = (F(H),╞L  ),
where╞L  is an extension of the satisfaction relation for elementary logic by the following
condition:
M╞L  Q[p] if and only if there are operations F1, ..., Fk defined on the universe of M such
that (M, F1, ..., Fk)╞L   x sk(Q, )[p], where F1, ..., Fk interpret the respective Skolem
functions introduced by skolemization of Q relative to Q, ’ is the extension of the
vocabulary  by these Skolema functions, and x is a sequence of all universal variables of Q.
By L(Hn)n, L(A 1n )n, L( A kn )n,k we denote respectively logics which are the extensions of
the elementary logic by all formulae with quantifiers Hn, A 1n , A kn , for n, k = 2, 3, ...
By a simple positive formula of a given logic with branched quantifiers we will call the
formula Q, where  is a quantifier free formula and Q is a quantifier prefix.
1.5 Definition (Lformula)
By Lformula (for the fixed vocabulary) we will call any formula, which belongs to the set of
formulae of the logic L.
The logic with branched quantifiers is much stronger than the elementary logic. In this
logic we can express many of the concepts, which are not expressible in the elementary logic.
For example, in the logic L* the concept of infinity is expressible. Thus there is a sentence of
the logic L* (called the Ehrenfeucht sentence) true exactly in the infinite models. The
Ehrenfeucht sentence has the following form:
x1 y1
t
[(x1 = x2  y1 = y2)  t  y1]
x2 y2
1.1 Theorem ( [Enderton 1970] )
For every second order formula  which belongs to the class  11 ( 11 ) there is a L*–formula
 such that  is equivalent to .
1.2 Theorem ( [Enderton 1970] )
For every L*–formula  there is a second order formula  which belongs to the class  12 such
that  is equivalent to .
The aforementioned theorems allow to evaluate the semantic power of the logic L*.
These theorems show that the logic with branched quantifiers is weaker than the class
 12 ,
but at least so strong as the class of Boolean combinations of  11 . It is known that in infinite
models the above inequalities are sharp, while in the case of finite models we cannot even
decide whether the class  11 is equivalent to L*.
1.6 Theorem
In any models:
L(Hn)n  L(A 1n )n  L( A kn )n,k  L*
Proof
Let K and K’ be classes of quantifiers on the left and right member of any inequalities in our
theorem accordingly. Then, any prefix Q  K is subprefix of some Q’  K’. Therefore, the
formula of the form Q, where Q  K, is equivalent to some formula Q’, where Q’  K’,
and additional variables occurring in Q’ are not free variables in .
Q.E.D.
Still it is not known whether and which of the above inequalities are strict. In
particular, is interesting it whether mentioned logics (and which) are equivalent in finite
models. Considering the above hierarchy we can put another question: whether quantifiers
A kn determine increasing hierarchy relative to k, that is whether L( A kn )n  L(A kn 1 )n, for
fixed k ?
The weaker version of branched quantifiers is function quantifiers, also called
Krynicki quantifiers. It is known that the logic, which is the extension of the elementary logic
by all formulae with function quantifiers, is much weaker than the logic L*. For example, the
theory of equality in this logic is decidable, whereas it is not decidable in L*. Moreover, the
logic with function quantifiers in language with the empty vocabulary is equivalent to the
logic with divisibility quantifiers.
The logic with Krynicki quantifiers was defined as sublogic of L*. It is the only
sublogic of L* defined by infinite class of quantifiers, about which it is known that is
essentially semantically weaker than L* [see Krynicki–Mostowski 1995]. However this logic
is not determined by any class of Henkin quantifiers still we do not know any essentially
infinite class of Henkin quantifiers X such that L(X) is semantically weaker than L*.
1.6 Definition (Krynicki quantifier)
For any natural number n1 Krynicki quantifier (or function quantifier) Fn is a quantifier
binding 2n variables in one formula and it is defined by the following equivalence:
M╞ Fn x1 ... xn y1 ... yn  (x1, ..., xn, y1, ..., yn)[p] iff there is a function F : MM such
that (M, F)╞  (x1, ..., xn, f(x1), ..., f(xn))[p], where f is interpreted as F.
By L(Fn)n we denote the logic which is the extension of the elementary logic by all
formulae with quantifiers Fn, for n = 2, 3, ...
1.7 Definition. (Divisibilit y quantifier )
For any natural number n1 divisibility quantifier Dn is defined by the following equivalence:
Dn x (x) iff card({x : (x)}) is divisible by n or infinite
By L(Dn)n we denote the logic which id the extension of the elementary logic by all
formulae with quantifiers Dn, for n = 2, 3, ...
THE SPECTRUM PROBLEM – GENERAL FORMULATION
Actually, the spectrum problem is usually considered in the framework of feasible –
descriptive correspondence. In this section we will consider classical model theoretic
formulation of the spectrum problem, particularly for the case of empty vocabulary.
2.1 Definition (The spectrum of a formula )
Let  be any sentence for fixed vocabulary . The spectrum of a formula , Spec(), is the
set:
Spec() = df {n   : there is a model M such that card(M) = n and M╞  }.
2.2 Definition (The spectrum of a logic L)
Let FL be a set of sentences of the logic L for the fixed vocabulary . The spectrum of the
logic L, Spec(L), is the set:
Spec(L) =df {Spec() :   FL}
If there is a formula  of the logic L such that the spectrum of  is the set A  , we will say
that the set A is L–spectrum.
Original versions of the spectrum problem
I Generalized Scholz problem.
Let L be any logic for the fixed vocabulary . What is Spec(L) ?
II Asser problem.
Let L be any logic for the fixed vocabulary . Does for every set A  , which belongs to
Spec(L), complement of A also belongs to Spec(L)?
Of course, the solution of Scholz problem immediately gives the solution of Asser
problem.
Weak versions of the spectrum problem
I The spectrum problem for a language in empty vocabulary.
Let L be any logic in the empty vocabulary. What is Spec(L)?
II The spectrum problem for first order language in empty vocabulary.
Let L’ be elementary logic in the empty vocabulary. What is Spec(L’)?
2.1 Theorem
Let L’ be elementary logic in the empty vocabulary. Then:
Spec(L’) = {A   : card(A) < }  {A   : card( \ A) < }.
Proof
() It is obvious.
() Let us assume that there is a set A  Spec(L’) such that A is not finite and is not co–finite.
Since A  Spec(L’) there is a formula  of the logic L’ such that Spec() = A. Then  and
also ¬ have finite models of arbitrary large cardinality. It follows that  and ¬ have
countable models. Let M be a countable model for , and M¬ a countable model for ¬.
Since the logic L’ is in the empty vocabulary M  M¬.. Therefore M ╞  and M ╞ ¬., so
M ╞  and ¬M ╞ . Contradiction.
Q.E.D
The subject of our investigations is the spectrum problem formulated for the logics
with branched quantifiers in the empty vocabulary. Particularly we are interested in the
characterization of spectra of the logics L*, L(Hn)n, L(A 1n )n, L( A kn )n,k.
The weak spectrum problem for languages with Henkin quantifiers.
Let L*, L(Hn)n, L(A 1n )n, L( A kn )n,k be the logics in the empty vocabulary defined in
section I. What are these :
Spec(L*) = ?
Spec(L(Hn)n)) = ?
Spec(L(A 1n )n) = ?
Spec(L( A kn )n,k) = ?
Let us observe that we have the following obvious theorem:
2.2 Theorem
Let L and L’ be logics such that L  L’. Then Spec(L)  Spec(L’).
Further in the paper we restrict our interest to logics in the empty vocabulary.
THE SPECTRUM PROBLEM FOR LANGUAGES WITH HENKIN QUANTIFIERS
We will start with discussion of the solution for the spectrum problem for the logic
with Krynicki quantifiers. There are two reasons for this. Firstly, the solution can be treated as
paradigmatic for our case of logics with Henkin quantifiers. Secondly, because the spectrum
of the logic with Krynicki quantifiers is a subset of the spectrum of the logic with all Henkin
quantifiers.
3.1 Definition (The restriction of a formula to a formula )
Let  be a formula of the logic with Krynicki quantifiers and (z) any formula with
distinguished variable z. We define inductively (z) – the restriction of  to  relative to z:
1) (z) = , if  is a quantifier free formula;
2) ((z)) = (z);
3) (1  2)(z) = (1(z)  2(z));
4) (x )(z) = x ((x)  (z));
5) [Fn x1 ... xn y1 ... yn  (x1, ..., xn, y1, ..., yn)](z) = Fn x1 ... xn y1 ... yn [ (x1)   (xn)   (y1) 
....   (yn)  ( (x1, ..., xn, y1, ..., yn))(z)].
Let us note that in a case of conflict of bounded variables in  we should rename these
variables.
3.1 Theorem2
For each natural number n>1 the quantifier Dn is definable by the quantifier Fn in the
following way:
Dn x  (x)  Fn x1 ... xn y1 ... yn {[¬(x1 = y1)]  [
 n (x1, ..., xn, y1, ..., yn)} (x)
ij(yi = yj  xi = xj)]  n(x1, ..., xn, y1, ..., yn)
Where formulae n and n have the following form:
n (x1, ..., xn, y1, ..., yn) = [( in11 yi = xi+1)  yn = x1],
n’(x1, ..., xn, y1, ..., yn) = [( in11 yi = xi+1)  (yn = x1)],
n (x1, ..., xn, y1, ..., yn) = [(i1, ..., ik)  (1, ..., n) k’(xi1, ..., xik, yi1,..., yik)]
3.2 Theorem2
For any natural number n1 a function quantifier Fn is definable by a simple Henkin
quantifier Hn in the following way:
Fn x1 ... xn y1 ... yn  (x1, ..., xn, y1, ..., yn) 
Hn x1 ... xn y1 ... yn { (x1, ..., xn, y1, ..., yn)  [
ij (xi = xj  yi = yj)]}.
3.3 Theorem ([Krynicki – Mostowski 1992])
The theory of equality in L(Fn)n and the theory of equality in L(Dn)n are recursively
equivalent i.e., for every formula of the theory of equality in L(Fn)n there is effectively
found semantically equivalent formula of the theory of equality in L(Dn)n.
3.4 Theorem ([Mostowski 199?])
Spec(L(Dn)n) = Bool({A   : card(A) < }  {{x   : x  a (mod b), a < b, a, b  }})
3.1 Corollary (from theorem 3.3)
Spec(L(Dn)n) = Spec(L(Fn)n)
3.2 Corollary (from theorem 3.2)
Spec(L(Fn)n)  Spec(L(Hn)n)
It is known that the above inclusion is proper (see [Krynicki–Mostowski 1995]).
3.3 Corollary (from theorem 1.5)
Spec(L(Hn)n)  Spec(L(A 1n )n)  Spec(L( A kn )n,k)  Spec(L*)
It is not known which of the inclusions (and if any) are proper.
The characterization of the spectrum of the logic with divisibility quantifiers (hence
also with function quantifiers) and corollary 3.3 supply the large class of sets of natural
numbers belonging to the spectrum of the logic with Henkin quantifiers.
2
For the proof see [Golińska 1999]
Example 1
For any natural number n1 there is L(Hn)n– formula n such that:
Spec(n) = {m : m is divisible by n}
This formula has the following form:

Hn x1 ... xn y1 ... yn { ij (xi = xj  yi = yj)  ¬(x1 = y1) 
n (x1, ..., xn, y1,..., yn)  n (x1, ..., xn, y1, ..., yn)},
where n and n have the following form:
n (x1, ..., xn, y1, ..., yn) = [( in11 yi = xi+1)  yn = x1)
 yi = xi+1)  (yn = x1)],
n (x1, ..., xn, y1, ..., yn) = [(i1, ..., ik)  (1, ..., n) k’ (xi1, ..., xik, yi1, ..., yik)].
n’(x1, ..., xn, y1, ..., yn) = [(
n 1
i 1
Example 2
Formula of which the spectrum is a set of even positive numbers has the following form:
x1 y1
x2 y2
(x1 = x2  y1 = y2)  ¬(x1 = y1)  (y1 = x2  y2 = x1)
While the formula of which the spectrum is a set of natural numbers divisible by 4 has the
following form:
ij (xi = xj  yi = yj)]  ¬( x1 = y1 ) ·
x1 y1
{[
x2 y2
(y1 = x2  y2 = x3  y3 = x4  y4= x1) [
x3 y3
x4 y4
[
ij yi = xj  ¬(yj
= xi)] 
ijk, ik (yi = xj  yj = xk  ¬(yk = xi))] }
Theorem 3.4 and corollary 3.1 supply a nice characterization of spectra of the logics L(Dn)n
and L(Fn)n.
We will look for similar characterization of the spectrum for syntactically defined
sublogics of the logic L*.
Fact
The classes of spectra of the logic with branched quantifiers are closed on Boolean
combinations.
The question arises for which general operations the spectra of the logics of our interest
are closed.
3.2 Definition (Arithmetical operations )
Let A, B be sets of natural numbers. We define the arithmetical operations of addition,
multiplication, subtraction and exponentiation in the following way:
A ⊕ B = {n + m : n  A, m  B}
A ⊗ B = {n · m : n A, m  B, n, m > 1}
A ⊕ c = { n + c : n  A}, where c  
A ⊗ c = { n · c : n  A}, where c  
A ⊝ B = {n – m : n  A, m  B, n > m}
A ⊝ c = {n – c : n  A, n > c}, where c  
kA = {kn : n  A}, where k  , k > 1
3.3 Definition
Let T be a family of sets of natural number. We say that T is closed on a given arithmetical
operation ‘○’, if for each A, B  T also A ○ B  T.
The question arises whether the spectrum of a given logic with Henkin quantifiers is
closed on the above defined arithmetical operations. In a case of a positive answer we would
obtain a more general characterization of the spectrum for considered language. In section IV
we will show that some nontrivial classes are spectra of formulae of some mentioned
sublogics of L*.
EXAMPLES OF SPECTRA OF FORMULAE WITH BRANCHED QUANTIFIERS
An example of formula with quantifier H9 that describes a full binary tree (hence is
satisfied in models of cardinality 2m – 1, for m  ) is given in [Krynicki–Mostowski 1995].
The idea applied in this construction can be generalized for the case of a full k–ary tree, for
fixed k>1, therefore for constructing a formula which is satisfied in models of cardinality
(km – 1) / (k – 1), for some m  .
4.1 Lemma
Let us fix k  , then M has (km – 1) / (k – 1) elements, for some m    M is finite and
there are functions f, g  MM such that the following conditions are satisfied:
(a) f has exactly one fixpoint being also a fixpoint of g,
(b) for each a, b  M, (g(a) = g(b)  g(f(a)) = g(f(b))),
(c) for each a  M, if a is not a fixpoint of f, then f–1{a}) is empty or has exactly k
elements,
(d) if r is a fixpoint, then f –1({r}) \ {r} is empty or has exactly k elements,
(e) for each a, b  M, if g(a) = g(b) and f –1({a}) =  then f –1{b}) = .
Proof
() Let us assume thatM has (km –1) / (k –1) elements. Then elements of M can be
nodes in a full k–ary tree with m levels (Picture1). Let f(x) be a father of x in this tree and g(x)
the leftmost element at the same level as x. Functions f, g defined in this way satisfy
conditions (a) – (e):
a) f has exactly one fixpoint being also a fixpoint of g t. It is the root of a tree.
b) Any two nodes a, b are at the same level (g(a) = g(b)) if and only if the father of a is at the
same level as the father of b (g(f(a)) = g(f(b))).
c) Any node a which is not the root, has no child or has exactly k children (f –1({a}) is empty
or has exactly k elements).
d) The root of a tree has exactly k children different from it or has not children if M has
exactly one element.
e) If two nodes are at the same level and one of them has no child then the second one has no
child either.
Picture 1
r = f(r) = f(x1) = ... = f(xk)
x1 = g(x1) = ... = g(xk)
·
·
 r
P0
 x1 . . . . . . . . . .  xk
P1
·
·
 y = f(p1) = ... = f(pk) . . . . . . . . . . . z = f(s1) = ... = f(sk) 
Pm–2
 p1 . . . pk  . . . . . . . . . . . . . . . . . . . . . . . . . . . s1 . . . sk 
Pm–1
k – children of y
k – children of z
km–1 – children of elements from Pm–2.
() Let us assume that there are functions f, g defined on the universe of M satisfying
conditions (a) – (e) and M is finite. We will show that these functions describe a full k–ary
tree as at Picture 1.
Let r be a fixpoint of f.
We define
P0 = {x : f(x) = x}
Pi+1 = {x : f(x)  Pi  ¬(f(x) = x)}
L1) The function f has no cycles. Let us assume that there is an element x0  Mand
minimal j such that j  2 and f j(x0) = x0. Therefore x0  r, because if x0 = r, then j = 1 (from
condition (a)). From condition (c) follows that card(f –1({x0})) = k (the set f –1({x0}) is
nonempty because j  2 and f j(x0) = x0). Thus, there are x1, x2, ..., xk Mdifferent from r
and x0 such that f(x1) = f(x2) = ... = f(xk) = x0. From condition (b) we obtain: g(f(xi)) =
= g(f(xp))  g(xi) = g(xp) (for i, p  {1, ..., k}). Since xi  x0 and xp  x0, so from (e) and (c),
card(f –2({x0})) = k2. We can repeat this reasoning and then we obtain that for every m
card(f –m({x0})) = km and x0  f –m({x0}). Therefore M is infinite, but it contradicts to our
assumption.
L2) Let P = i   Pi. Every element of the universe belongs to some Pi that is P = M.
Let us assume that there is an element x0  M such that x0  P. Then there is x1  r such
that x1 = f(x0) and x1  P (because if x1  P then there is such i that x1  Pi, but then x0 
Pi+1). Let us take a sequence {xk}k such that xk = f k(x0). For every i, j such that i  j holds
xi  xj, so M is infinite, but it is inconsistent with our assumption. Thus there are i, j, i  j
such that xi = xj, which means that a function f has cycles, but it is impossible from L1.
Therefore xMi x  Pi. Moreover, because card(M) < , so there is such m that
Pm = .
L3) For every i, j such that i  j holds Pi  Pj = . Let us assume that there is such element
a M which belongs to Pi and Pj, for i  j. Let us note that i = 0  j = 0 (because if a  P0
then a = r, but r (from definition) belongs only to P0, thus there is no i  0 such that a  Pi).
Let i, j be such that there is a  Pi  Pj, i  j and min(i, j) is the smallest with this property.
However, in this case f(a) Pi-1  Pj-1, the contradiction.

m 1
We have shown that
i  0 Pi = Mand Pi  Pj = , for every i  j. From condition (a) we
obtain that P0 has exactly one element. From condition (d) and definition of P1 it follows that
P1 has exactly k elements. More generally: Pi+1 has k times more elements than Pi. Thus Pi
has ki elements (it is term of geometrical series which has the quotient k).
Let ai = card(Pi), for i = 0, ..., m – 1, and a = card(M).
Therefore a =  im01 ai =  im 0 ki = (km – 1) / (k –1).
Hence M has (km – 1) / (k – 1) elements and this ends the proof.
Q.E.D.
4.1 Theorem3
There is a formula k  L(Hn)n such that Spec(k) = {(km – 1)/ (k –1) : m  }.
This formula has the following form:
x0 y0
.............
xk+1 yk+1
z0 t0
k : = r
............
(0  ...  7)
z3 t3
u0 w0
............
uk–1 wk–1
v s
where
0 : = 0i<jk+1 (xi = xj  yi = yj)  0i<j3 (zi = zj  ti = tj)  0i<jk–1 (ui = uj  wi = wj );
1 : = (x0 = r  x0 = y0)  (z0 = r  z0 = t0);
3
For the proof of this theorem it is sufficient to justify the fact that the above formula holds in a model M iff one
can put up the elements of M in a tree in which every element has k or 0 successors and each two elements from
the same level have the same number of successors. Details of the proof – see [Golińska 1999].
2 : = ((x0 = z0  x1 = z1  y0 = z2  y1 = z3)  (t0 = t1  t2 = t3));
3 : = (( ik01 (yi = yi+1)   ik 0 (¬(yi = r)))  0i<jk (xi = xj));
4 : = ( ik01 ¬(yi = r)  x0 = u0 


5 : = ((
k 2
j o
j
i0

k 2
i 0
(xi+1 = wi) 

k 2
i 0
(wi = ui+1))  ((y0 = ... = yk–1) 
¬(wj = ui));
0i<jk+1 (xi = xj));
6 : = (xk = r  
¬(xi = r)  x0 = u0   (xi+1 = wi)  
... = yk)  
 ¬(wj = ui));
k
i0
(yi = yi+1)  x0 = r) 
k 1
i 0
k 1
j o
k 1
i 0
k 1
i 0
(wi =
= ui+1))  ((y0 =
j
i0
7 : = ((z0 = y0  t0 = t1  z1 = v  x1 = s)  (y1 = v)).
Q.E.D.
Let us observe that the idea above applied also enables construction of a formula which
is satisfied in models of cardinality {[s · (km – 1)/ (k – 1)] + 1}, for fixed s, k   and for
some m  . It is enough to change the conditions for functions f, g so that they describe s
full k–ary trees with one main root, which means a change only of the condition (d). What
follows is a suggestion that we can define the functions f, g in such way that these functions
describe ‘various combinations of the same trees’ (e.g., the tree whose root has exactly s–
children, every son of the root has exactly m–children and all the other nodes have k–children,
and so on). This method provides a quite large class of examples of spectra belonging to
Spec(L(Hn)n).
Below we present examples of formulae whose spectra belong to Spec(L(A 1n )n). It is
unknown whether this spectra also belong to Spec(L(Hn)n). A negative answer would mean
that the logic with simple Henkin quantifiers is weaker than the logics with quantifiers A 1n .
4.2 Lemma
The cardinality of a model M is a nonprime  M is finite and there are functions f, g defined
on the universe of a model M such that the following conditions are satisfied:
(a) Let’s take equivalence classes of relations Rf, Rg defined as follows:
(x, y)  Rf =df f(x) = f(y)
(x, y)  Rg =df g(x) = g(y)
These classes have at least two elements.
(b) Each equivalence class of the first relation has exactly one element in common with each
equivalence class of the second one.
Proof
() Let us assume that card(M) is a nonprime. Thus there are n, k  , n > 1 and k > 1 such
that n · k = card(M). We can enumerate elements of the universe of M in such way that:
M=

n
i 1
{xi1, ..., xik}.
Let
(xij) =df xnj
(xij) =df xik .
Those defined functions f, g satisfy the conditions (a) – (b) (Picture 2).
From the above definitions of f, g we obtain that equivalence classes of the relation Rf have
the following form Pj = {xij : i = 1, ..., n}, for j = 1, ..., k, and equivalence classes of the
relation Rg have the following form Si = {xij : j = 1, ..., k}, for i = 1, ..., n. Each equivalence
class of the relation Rf and Rg has at least two elements (of course, if card(M) is a nonprime).
Thus condition (a) is satisfied. Moreover Pj  Si = {xij}, for any i  {1, ..., n}, j  {1, ..., k}.
Therefore each equivalence class of the relation Rf has exactly one element in common with
each equivalence class of the relation Rg. Thus condition (b) is satisfied.
() Let us assume that M is finite and there are functions f, g defined on the universe of a
model M satisfying conditions (a) and (b). We will show that these functions describe a
diagram, which has n · k elements (Picture 2), for some n, k  .
Since any two equivalence classes of a given relation are identical or disjoint, thus we


n
k
can assume thatM=
i 1 Si =
j 1 Pj, where Si – the equivalence class of the relation
Rg, Pj – the equivalence class of the relation Rf. From condition (a) we have that each
equivalence class of the relation Rf has at least two elements. Let Pj = {x1j, ..., xpj), (p  2).
We will show that for each j  {1, ..., k}, p = n. From (b) it is known that for each j {1, ...,
k}, Pj has exactly one element in common with each equivalence class of the relation Rg.
Since all equivalence classes of the relation Rg are disjoint, there are n these classes and each
element of Pj also belongs to some equivalence class of the relation Rg, therefore Pj has
exactly n elements. Thus we obtain that each equivalence class of the relation Rf has exactly n
elements and n  2. In the same way we obtain also that k  2. Because there are k of
equivalence classes of the relation Rf and each of them has n elements (n, k  2), the universe
of the model M has n · k elements, what ends the proof.
Q.E.D.
Picture 2
P1
P2
P3
Pk
x11
x12
x13
. . . . . .
x1k
S1
x21
x22
x23
. . . . . .
x2k
S2
x31
.
x32
.
x33
.
. . . . . .
x3k
.
S3
.
xn1
.
xn2
.
xn3
f(xij) = xnj
. . . . . .
.
xnk
Sn
g(xij) = xik
4.3 Lemma
There are functions f, g defined on the universe of a model M satisfying conditions (a) and (b)
from lemma 4.2  M╞ f g x y z w p s (1  2  3), where
1 : = (f(x) = f(z)  ¬(x = z)),
2 : = (g(x) = g(w)  ¬(x = w)),
3 : = {f(x) = f(p)  g(y) = g(p)  [(f(x) = f(s)  g(y) = g(s))  p = s]}
Proof
Let us observe that the formula f g x y z w p s (1  2  3) is equivalent to the
formula f g {x z (f(x) = f(z)  ¬(x = z))  x w (g(x) = g(w)  ¬(x = w))  x y p
s{f(x) = f(p)  g(y) = g(p)  [(f(x) = f(s)  g(y) = g(s))  p = s]}}.
Therefore:
M╞ f g {x z (f(x) = f(z)  ¬(x = z))  x w (g(x) = g(w)  ¬(x = w)) 
xyps{f(x) = f(p)  g(y) = g(p)  [f(x) = f(s)  g(y) = g(s)  p = s]}}  there are f, g
defined on the universe of M such that:
(1) for each element x, there is different from it element y such that
[f(x) = f(y)];
(2) for each element x, there is different from it element y such that
[g(x) = g(y)];
(3) for each x, y, there is exactly one element z such that
[f(x) = f(z)  g(y) = g(z)];
 there are f, g defined on the universe of M such that the equivalence classes of Rf and Rg
defined in lemma 4.2 has at least two elements and each equivalence class of the relation R f
has exactly one element in common with each equivalence class of the relation Rg.
Q.E.D.
4.2 Theorem
There is a formula   L(A 1n )n such that Spec() = {k   : k is a nonprime}.
This formula has the following form:
x1 y1
x2 y2
x3 y3
z1 w1
z2 w2
z3 w3
s t
p r
a b c
(0  1  2  3), where
0 : = (0i<j3 (xi = xj  yi = yj)  0i<j3 (zi = zj  wi = wj);
1 : = (t = x2  s = x1)  (y1 = y2  ¬(x1 = x2));
2 : = (r = z2  p = z1)  (w1 = w2  ¬(z1 = z2));
3 : = (c = x2  c = z2  a = x1  b = z1  x3 = z3)  [y1 = y2  w1 = w2  ( y1 = y3  w1 = w3 
x3 = c)]
Proof
We have to show that for any finite model M:
 is true in M  card(M) is a nonprime.
() Let us assume that  is true in some finite model M. After skolemization of  we obtain
the following the formula  equivalent to  :
f1  f2  f3 g1 g2 g3 h j k x1x2 x3z1 z2 z3 s p a b (0  ...  3), where
0 : =(0i<j3 (xi = xj  fi(xi) = fj(xj) )  0i<j3 (zi = zj  gi(zi) = gj(zj));
1 : = (h(s) = x2  s = x1  f1(x1) = f2(x2)  ¬(x1 = x2));
2 : = (j(p) = z2  p = z1  g1(z1) = g2(z2)  ¬(z1 = z2));
3 : = {(k(a, b) = x2  k(a, b) = z2  a = x1  b = z1  x3 = z3)  [ f1(x1) = f2(x2)  g1(z1) =
= g2(z2)  ( f1(x1) = f3(x3)  g1(z1) = g3(z3))  x3 = k(a, b))]}.
From 0 follows that functions f1, f2, f3 and g1, g2, g3 are the same for arguments x1, x2, x3 and
z1, z2, z3, respectively. Therefore we can simplify the formula  obtaining the following
formula :
f g h j k x1 x2 x3 z1 z2 z3 s p a b (1  ...  3), where
1 : = (h(s) = x2  s = x1  f(x1) = f(x2)  ¬(x1 = x2));
2 : = (j(p) = z2  p = z1  g(z1) = g(z2)  ¬(z1 = z2));
3 : = {( k(a, b) = x2  k(a, b) = z2  a = x1  b = z1  x3 = z3)  [f(x1) = f(x2)  g(z1) = g(z2) 
((f(x1) = f(x3)  g(z1) = g(z3))  x3 = k(a, b))]}.
While the formula  is equivalent to the following formula:
f g h j k x y z (1  2  3), where
1 : = (f(x) = f(h(x))  ¬(x = h(x))),
2 : = (g(y) = g(j(y))  ¬(y = j(y))),
3 : ={f(x) = f(k(x, y))  g(y) = g(k(x, y))  [(f(x) = f(z)  g(y) = g(z)  z = k(x, y)]}.
Let us observe that:
M╞ f g h j k x y z (1  2  3) 
M╞ f g {x z (f(x) = f(z)  ¬(x = z))  x w (g(x) = g(w)  ¬(x = w))  x y p
s{f(x) = f(p)  g(y) = g(p)  [(f(x) = f(s)  g(y) = g(s))  p = s]}} 
M╞ f g x y z w p s {(f(x) = f(z)  ¬(x = z))  (g(x) = g(w)  ¬(x = w)) [f(x) = f(p)
 g(y) = g(p)  (f(x) = f(s)  g(y) = g(s)  p = s]}
The above formula (hence the formula ) is true in M iff there are functions f , g defined on
the universe of the model M satisfying conditions (a) and (b) from lemma 4.2 (it follows from
lemma 4.3). Since M is finite we have from lemma 4.2 that card(M) is a nonprime.
() Let us assume that the cardinality of the model M is a nonprime. Then from lemma 4.2
there are functions f and g satisfying conditions (a) and (b). Thus from lemma 4.3 the
following formula is true in M:
f g x y z w p s (1  2  3), where
1 : = (f(x) = f(z)  ¬(x = z)),
2 : = (g(x) = g(w)  ¬(x = w)),
3 : ={f(x) = f(p)  g(y) = g(p)  [(f(x) = f(s)  g(y) = g(s))  p = s]}.
From the proof of () of theorem 4.2 it is known that this formula is equivalent to the
formula . Therefore the formula  is true in M, what ends the proof.
Q.E.D.
4.3 Theorem [Golińska 1999]
There is a formula   L(A 1n )n such that Spec() = {k2 : k   and k > 1}.
This formula has the following form:
x1 y1
x2 y2
x3 y3
z1 w1
z2 w2
z3 w3
s t
p r
a b c
(0  ...  5), where
0 : = (0i<j3 (xi = xj  yi = yj)  0i<j3 (zi = zj  wi = wj);
1 : = (t = x2  s = x1)  (y1 = y2  ¬(x1 = x2));
2 : = (r = z2  p = z1)  (w1 = w2  ¬(z1 = z2));
3 : = (c = x2  c = z2  a = x1  b = z1  x3 = z3)  [y1 = y2  w1 = w2 ( y1 = y3  w1 = w3 )
x3 = c)];
4 : = [(y1 = x2  y1 = y2)  (w1 = z2  w1 = w2)];
5 : = [z1 = x1  (x1 = y1  z1 = w1)];
In section III we left a question unanswered whether the mentioned logics are closed on
defined operations of addition, multiplication, subtraction and exponentiation. In a case of a
positive answer we would obtain a more general characterization of the spectrum for
considered language. We have two theorems giving partial answers.
4.4 Theorem
Let A, B   be spectra of simple positive formulae of the following forms respectively:
Q (x1, ..., xn, y1, ..., ym)
Q’’(p1, ..., ps, q1, ..., qt) , where
Q and Q’ are Henkin quantifiers,
 and ’ are quantifier free formulae,
x1, ..., xn and y1, ..., ym – are universal and existential variables from Q respectively,
p1, ..., ps and q1, ..., qt – are universal and existential variables from Q’ respectively,
Then A ⊕ B = Spec() , where  has the following form:
a b
1 1
..........
..........
k k
Q
Q’
(0  ...  3) , where
k = n + m + s + t;
0 : =  ik 2 (1 = i  1 = i);
1 : = [¬(a = b)  (1 = 1  (1 = a  1 = b)  (1 = a  1 = b))];
2 : = {[ in1 (xi = i)   im1 (yi = n+i)   in1 (a = i)]  [   im1 (a = n+i)]};
3 : = {[ is1 (pi = n+m+i) 

t
i 1

t
i 1
(qi = n+m+s+i) 

s
i 1
(b = n+m+i)]  [’ 
(b = n+m+s+i)]};
Proof
We have to show that (M╞  iff card(M) is the number of the form n + m, where n A, m 
B)
After skolemization we obtain the following equivalent to  formula:
f1 ...fk g1 ...gmh1 ...hta bx1 ...xnp1 ...ps1 ...k (0  ...  3), where:
0 : =  ik 2 (1 = i  f1(1) = fi(i));
1 : = [¬(a = b)  (1 = f1(1)  (1 = a  1 = b)  (f1(1) = a  f1(1) = b))];
2 : = [ in1 (xi = i) 

m
i 1

m
i 1
(gi(xi) = n+i) 

n
i 1
(a = fi(i))]  [sk(Q, ) 
(a = fn+i(n+i))];
3 : = [ is1 (pi = n+m+i) 

t
i 1

(hi(pi) = n+m+s+i)
s
i 1
(b = fn+m+i(n+m+i))] 

 [skQ’’ it 1 (b = fn+m+s+i(n+m+s+i))];
where f1, ..., fk, g1, ..., gm, h1, ..., ht are Skolem functions obtained by skolemization of , and
xi and pi are sequences of universal variables on which existential variables yi and qi depend
respectively in Q and Q’.
We can simplify the above formula, because from 0 it follows that functions f1, ..., fk are the
same. Then we obtain the following formula equivalent to  :
f g1 ... gm h1 ... ht a b x1 ...xn p1 ... ps 1 ...k (’1  ...  ’3),
where:
’1 : = [¬(a = b)  (1 = f(1)  (1 = a  1 = b)  (f(1) = a  f(1) = b))];
’2 : = [ in1 (xi = i) 

m
i 1
t
i 1
(gi(xi) = n+i) 

(hi(pi) = n+m+s+i)

m
i 1
n
i 1
(a = f(i))]  [sk(Q, ) 
(a = f(n+i))];
’3 : = [ is1 (pi = n+m+i) 



t
i 1
s
i 1
(b = f(n+m+i))]  [skQ’’
(b = f(n+m+s+i))];
Therefore the formula  is true in M iff there is a function f defined on the universe of M,
which has two different fixpoints a and b, for each element of M the value of f is either a or b
(it is expressed by ’1), and M restricted to elements for which f takes the value a is closed on
functions g1, ..., gm and satisfies the skolemization of Q (it is expressed by ’2), and M
restricted to elements for which f takes the value b is closed on functions h1, ..., ht and satisfies
the skolemization of Q’’ (it is expressed by ’3) iff the universe of M can be split up into
disjoint subsets, in the first of them the formula Q is satisfied and in the second one Q’’ iff
the cardinality of M is of the form n + m, where n belongs to the spectrum of Q and m
belongs to the spectrum of Q’’ .
Q.E.D.
4.5 Theorem
Let A, B   be spectra satisfying assumptions of theorem 4.4.
Then A ⊗ B is the spectrum of , where  is of the following form:
s1 t1
..........
..........
sn+m tn+m
r1 w1
..........
(0  ...  6) , where
..........
rs+t ws+t
a b
c d
   
Q
Q’
0 : = (0i<jn+m (si = sj  ti = tj)  0i<js+t (ri = rj  wi = wj);
1 : = (s2 = t1  t1 = t2)  (r2 = w1  w1 = w2);
2 : = (b = s2  a = s1  t1 = t2  ¬(s1 = s2));
3 : = (d = r2  c = r1  w1 = w2  ¬(r1 = r2));
4 : = {( = s2   = r2   = s1   = r1  s3 = r3)  [t1 = t2  w1 = w2  ( t1 = t3  w1 = w3 
s3 = )]};
5 : ={[ in1 (xi = si)   in1 (si = ti)   im1 (yi = si+n)]  [   im1 (si+n = ti+n)]};
6 : ={[ is1 (pi = ri)   is1 (ri = wi)   it 1 (qi = ri+s)]  [’   it 1 (ri+s = wi+s)]};
Proof
We have to show that (M╞  iff card(M) is the number of the form n · m, where n A, m 
B, n, m > 1)
After skolemization we obtain the following formula equivalent to  :
f1 ... fn+m g1 ... gs+t h j k l1 ... lm v1 ...vt x1...xn p1 ...ps s1 ... sn+m r1
...rs+t a c   (0  ...  6), where:
0 : = (0i<jn+m (si = sj  fi(si)= fj(sj))  0i<js+t (ri = rj  gi(ri)= gj(rj));
1 : = (s2 = f1(s1)  f1(s1) = f2(s2))  (r2 = g1(r1)  g1(r1) = g2(r2));
2 : = (h(a) = s2  a = s1  f1(s1) = f2(s2)  ¬(s1 = s2));
3 : = (j(c) = r2  c = r1  g1(r1) = g2(r2)  ¬(r1 = r2));
4 : = {(k(, ) = s2  k(, ) = r2   = s1   = r1  s3 = r3)  [f1(s1) = f2(s2)  g1(r1) =
g2(r2)  (f1(s1) = f3(s3)  g1(r1) = g3(r3)  s3 = k(, ))]};
5 : = {[ in1 (xi = si) 

(si = fi(si)) 

(ri = gi(ri)) 

n
i 1
(li(xi) = si+n)]  [sk(Q, ) 

m
i 1
(si+n =
(vi(pi) = ri+s)]  [sk(Q’, ’) 

t
i 1
(ri+s =
m
i 1
fi+n(si+n))]};
6 : = {[ is1 (pi = ri) 

s
i 1
t
i 1
gi+s(ri+s))]};
where xi and pi are sequences of universal variables on which existential variables yi and qi
depend in Q and Q’ respectively.
We can simplify the above formula, because from 0 it follows that functions f1, ..., fn+m and
g1, ..., gs+t are the same for the arguments s1, ..., sn+m and r1, ..., rs+t respectively. Then we
obtain the following formula equivalent to  :
f g h j k l1 ... lm v1 ... vt x1...xn p1 ...ps s1 ... sn+m r1 ...rs+t a c  
(’1  ...  ’6), where:
’1 : = (s2 = f(s1)  f(s1) = f(s2))  (r2 = g(r1)  g(r1) = g(r2));
’2 : = (h(a) = s2  a = s1  f(s1) = f(s2)  ¬(s1 = s2));
’3 : = (j(c) = r2  c = r1  g(r1) = g(r2)  ¬(r1 = r2));
’4 : = {(k(, ) = s2  k(, ) = r2   = s1   = r1  s3 = r3)  [f(s1) = f(s2)  g(r1) = g(r2) 
 (f(s1) = f(s3)  g(r1) = g(r3)  s3 = k(, ))]};
’5 : = {[ in1 (xi = si) 

(si = f(si)) 

(ri = g(ri)) 

n
i 1
(li(xi) = si+n)]  [sk(Q, ) 

m
i 1
(si+n =
(vi(pi) = ri+s)]  [sk(Q’, ’) 

t
i 1
(ri+s =
m
i 1
= f(si+n))]};
’6 : = {[ is1 (pi = ri) 

s
i 1
t
i 1
= g(ri+s))]};
Thus we obtain that M╞  iff there are functions f, g defined on the universe of M such that:
(1) f, g satisfying conditions (a) and (b) of lemma 4.2 (it is described by formulae ’2, ’3 and
’4);
(2) for each element x  M, [f(f(x)) = f(x) and g(g(x)) = g(x)] (from ’1);
(3) each fixpoint of f satisfies the formula sk(Q, ) and every element li(xi) is also a fixpoint
of f, where xi is a sequence of fixpoints of f , for i  {1, ..., m};
(4) each fixpoint of g satisfies the formula sk(Q’, ’) and every element vi(pi) is also a
fixpoint of g, where pi is a sequence of fixpoints of g, for i  {1, ..., t}.
Since f, g satisfy conditions (a) and (b) of lemma 4.2, so card(M) is nonprime. It easy to see
that the spectrum of  contains numbers of the form n · m, where n and m are the number of
fixpoints of f and g respectively. The formula ’5 implies that the set of fixpoints of f is
closed on functions l1, ..., lm obtained during skolemization of Q; moreover Q holds in this
set, then n belongs to the spectrum of Q. Similarly the formula ’6 says that m belongs to the
spectrum of Q’’.
Q.E.D.
Remark
Theorems 4.4 and 4.5 will be also true for formulae of the form: x1 ... xn Q, where Q is a
simple positive formula.
Our theorems give as corollaries the following:
4.1 Corollary
Let L’(Hn)n be the class of formulae of the form z1 ... zz Q, where Q is simple positive
formula with a quantifier Hn. Then Spec(L’(Hn)n) is closed on addition.
4.2 Corollary
Let L’( A kn )n,k be the class of formulae of the form z1 ... zz Q, where Q is simple
positive formula with a quantifier A kn . Then Spec(L’( A kn )n,k) is closed on addition and
multiplication.
(Let us note that we cannot formulate similar corollaries with respect to the class of formulae
with quantifiers A 1n .)
4.3 Corollary
There is quantifier free formula  and prefix Q of type A kn 1 such that the spectrum of Q is
the set {m   : j1 ... jk (m =  ik1 ji) and is (ji > 1)}.
The proof of the last corollary follows from the form of quantifier defining the set of
nonprimes and from the construction of spectrum A ⊗ B.
Below, we present another one example of the spectrum belonging
Spec(L( A kn )n,k). This example is interesting considering the applied construction.
to
4.4 Lemma.
The cardinality of a model M is a number of the form k(k+1)/2, for some k    M is finite
and there is a function f defined on the universe of M satisfies conditions:
(a) for each element x  M, f(f(x)) = f(x);
(b) card(M\{x : f(x) = x}) = card({{x, y} : f(x) = x  f(y) = y}).
Proof
() Let us assume that the cardinality is the number of the form k(k+1)/2, k  . Then
elements of the universe of the model we can enumerate in the following way (Picture 4):
M=

k
j 1
{xij : i = 1, ..., k +1 – j}.
We define a function f in the following way:
f(xij) = x(k+1–j)j for i, j  {1, ..., k}
The obtained function satisfies conditions (a) and (b).
() Let us assume that there is a function f defined on the universe of M satisfying conditions
(a) – (b). Let x1, ..., xk be elements such that f(xi) = xi, for i = 1, ..., k. The set {x1, ..., xk} is
nonempty (condition (a)). Therefore there is k fixpoints of f. Thus:
card({{x, y} : f(x) = x  f(y) = y}) =

k
2
Therefore, we obtain that card(M) =
  + k = k(k+1) / 2 .
k
2
Q.E.D.
Picture 4
x11
x21
x31
·
·
·
·
·
·
x12
x22
x32
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
· ·
· ·
· ·
·
x1,k–1
x2,k–1 
x1k 
xk–1,2 
xk1
f(xij) = x(k+1–j)j, for i, j  {1, ..., k}
The symbol ‘’ at the picture denotes a fixpoint of f.
4.6 Theorem [Golińska 1999]
There is a formula  with a quantifier A kn such that:
Spec() = {m   : m(m+1) / 2}.
This formula has the following form:
x1 y1
x2 y2
x3 y3
z w
s t
a b c
p q r
(0  ...  4), where:
0 : = 1i<j3 (xi = xj  yi = yj)  [((a = p  b = q)  (a = q  b = p))  c = r];
1 : = (x2 = y1  y1 = y2);
2 : = (a = x1  b = x2  c = x3)  [(x1 = y1  x2 = y2  ¬(x1 = x2))  ¬(x3 = y3)];
3 : = {(z = x1  s = x1  w = x2  t = x3  a = w  b = t)  [(¬(x1 = y1))  (y2 = x2  y3 = x3
 ¬(z = w)  c = x1)]};
4 : = {c = r  [(a = p  b = q)  (a = q  b = p)]};
THE SUMMARY
With presented results many questions arise. First of all, whether the presented
examples of spectra of formulae of the logic L ( A kn )n,k are also the spectra of the logic
L( Hn )n. A negative answer would mean that the logic L( Hn )n is semantically weaker
than the logic with all Henkin quantifiers. Maybe, the set of nonprimes is a good candidate for
the spectrum, which is not spectrum of a formula with any quantifier Hn . But it is not obvious
how it can be shown. The related problem is the following: to find a formula with the minimal
quantifier A kn , which defines a given spectrum of L( A kn )n,k – formula. We have shown that
the set of nonprimes is definable by the formula with quantifier A 18 . This result is improved in
[Golińska 2000] with quantifier A14 .
The second problem is the following: whether the complement of the considered spectra
is definable by a simple positive formula of the same logic. In [Golińska 2000] it is shown
that the complement of the set of k–ary full tree (for fixed k) and the complement of the set
of primes are definable by simple positive formulae of the logics L( Hn )n and L( A kn )n,k
respectively.
The third problem concerns corollaries 4.1 and 4.2. We have shown that some sublogics
of L( Hn )n and L( A kn )n,k are closed respectively on addition and addition and
multiplication. Thus, the question arises: „whether these theorems can be generalized for all
formulae of the mentioned logics”.
The next open problem is “which of considered spectra are closed on others operations,
for example subtraction and exponentiation”. In [Golińska 2000] is shown that considered
logics are closed on exponentiation and that the sublogics of simple positive formulae of
L( A kn )n,k is closed on subtractions defined as follows:
A ⊝1 c = {n – c : n  A, n > c}, where c  .
A ⊝2 B = {n – m : n  A, m  B, n > m, (n – m)2  n}.
For the subtraction defined in the most general way i.e. A ⊝ B = {n – m : n  A, m  B, n >
m} it is known from [Mostowski 2000] that for A, B being NP spectra (or equivalent spectra
of simple positive formulae) A ⊝ B can be of arbitrary recursively enumerable degree.
Affiliation
Joanna Golińska
Institute of Philosophy, Warsaw University
e–mail: [email protected]
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