Unsolvability Cores in Classification Problems

Classification problems have been introduced by M. Ziegler as a generalization of promise problems. In this paper we are concerned with solvability and unsolvability questions with respect to a given set or language family, especially with cores of unsolvability. We generalize the results about unsolvability cores in promise problems to classification problems. Our main results are a characterization of unsolvability cores via cohesiveness and existence theorems for such cores in unsolvable classification problems. In contrast to promise problems we have to strengthen the conditions to assert the existence of such cores. In general unsolvable classification problems with more than two components exist, which possess no cores, even if the set family under consideration satisfies the assumptions which are necessary to prove the existence of cores in unsolvable promise problems. But, if one of the components is fixed we can use the results on unsolvability cores in promise problems, to assert the existence of such cores in general. In this case we speak of conditional classification problems and conditional cores. The existence of conditional cores can be related to complexity cores. Using this connection we can prove for language families, that conditional cores with recursive components exist, provided that this family admits an uniform solution for the word problem.


Introduction
The concept of classsification problems was introduced by M. Ziegler ( [1]) as a generalization of promise problems due to S.Even ([5]).Promise problems are a generalization of decision problems.A classification problem is a vector A = (A 1 , . . ., A k ) where the A i are pairwise disjoint infinite subsets of a given basic set S. For a set family F ⊆ 2 S such a classification problem is F-solvable, if a vector Q = (Q 1 , . . ., Q k ) exists with If k = 2 we are faced with promise problems.In applications S = X * where X is a finite nonempty alphabet and F = L a language family and/or a complexity class.From an algorithmic point of view solutions of classification problems can be used to obtain constant size advices.In this case advices indicate the inputs to belong to certain subsets (c.f.[1] for further details).We extend the results about unsolvability cores in promise problems ( [4]) to unsolvability cores in classification problems.Again cohesiveness is the characterizing indicator.For unsolvable promise problems we can find in general unsolvability cores, if the set family is closed under union, intersection and finite variation.But for unsolvable classification problems with k > 2 the existence of unsolvability cores needs further conditions.We show, that we can assert the existence of unsolvability cores for k > 2 under the same assumption as needed for promise problems, if we fix one of the components.In this approach the fixed component is called the condition for the classification problem.The results are proven under assumptions which involve closure properties of F against some or all boolean operations union, intersection and complementation.Moreover, we can relate unsolvability cores for conditional classification problems to so called proper hard cores introduced by R. Book and D.-Z.Du in a general form ( [3]) and first defined by N. Lynch ([6]) for complexity classes.Using results and proof techniques from [3] we can apply our results to language families and complexity classes.Especially, we are able to construct unsolvability cores where the components are recursive.To do this, the language family or complexity class under consideration must allow an enumeration where the word problem has a uniform solution.We assume the reader to be familiar with the theory of recursive functions, languages and complexity (cf.[2], [7]).

Set and Language Families, Basic Notations
In the following an infinite basic set S is given.We assume that the elements of set families F are subsets of S.Moreover, sets are always subsets of S and singletons {s} are identified with s.We mainly deal with denumerable set families F; i.e. a function e F : N 0 → 2 S with e F (N 0 ) = F exists (enumeration of F).Consider the boolean operations A ∪ B union, A ∩ B intersection and A c = S\A complementation in connection with set families F. These operations can be lifted to binary operations between set families F 1 and F 2 and unary operations for F. Define and the closure operations We will frequently use Let fin(S) = {A ⊆ S|A finite}.Then F is closed under finite variation if F ⊕ fin(S) ⊆ F and F ⊙ fin(S) co ⊆ F. We call F nontrivial if ∅, S ∈ F and F is closed under finite variation.In this case fin(S) ⊆ F. Note, that fin(S) = fin(S) b .Moreover, F cc , F u , F s and F b are nontrivial, if F is nontrivial.Consider the case S = X * , where X * is the free monoid over X (a nonempty, finite alphabet) with concatenation of words as monoid operation and 1 as identity.As usual L ⊆ X * is called a language and L ⊆ 2 X * a language family.For a word w = x 1 . . .x n (x i ∈ X for 1 ≤ i ≤ n) |w| = n is the length of w and |1| = 0.For languages L 1 and L 2 the complex product is defined by There are various kinds of quotients available, for example the left quotient defined by L −1 1 L 2 = {w| ∃w 1 ∈ L 1 : w 1 w ∈ L 1 }.In this context we are mainly interested in handling leftmarkers, i.e. we consider the products wL and the quotients w −1 L where w ∈ X * and L is a language.With respect to language families L we get the closure operations L ltr = {wL|w ∈ X * , L ∈ L} and L -ltr = {w −1 L|w ∈ X * , L ∈ L}.In handling the leftmarkers (for example complementation of a leftmarked language) we use variation by L reg (X), the family of regular languages (for details see [4]).A language family L is closed under regular variation if L ⊕ L reg (X) ⊆ L and L ⊙ L reg (X) ⊆ L.
Consider the language families L r.e.(X) (recursively enumerable languages) and L rec (X) = L r.e.(X) dc (recursive languages).Let rec n (n ≥ 0) be the set of n-ary recursive functions.Using 0, 1 ∈ N 0 as truth values define for a language L the function λi.δ L (i) = "lex (i) ∈ L".Then a language L is recursive if and only if δ L ∈ rec 1 .Alternatively, a nonempty language L is recursive if and only if a function f : N 0 → X * exists such that λi.ord (f (i)) is nondecreasing and recursive.Classical language families and complexity classes are always denumerable.Of special interest are families with enumerations which are in a certain sense "effective".For our purpose it is important to assert that these enumerations allow a uniform solution for the word problem.More formular, we define for an enumeration e of a language family L the function λi, j.word e (i, j) = "lex (j) ∈ e(i)".If word e ∈ rec 2 then e is called WP-recursive.L is called WP-recursive, if a WP-recursive enumeration e of L exists.Note, that any WP-recursive L is a (proper) subfamily of L rec (X) and every complexity class with reasonable ressource bounds (time-and space-constructability [2]) is WP-recursive.

Solvability of Classification Problems
Let k > 0. We consider vectors A If S = N 0 then F-solvability of promise problems corresponds to the separation principle defined in [7] (exercise 5-33).Our definition of F-solvability for classification problems is stronger than the definition of F-separability given in [1], where a classification problem A is F-separable, if there exists a Q, which satisfies the conditions of Definition 2.1.except the condition "set (Q) = S", which may not necessarily be valid.Note that for such a Q, we always obtain Hence, the class of F-solvable classification problems with more than one components is identical with the class of F-separable classification problems, if F is a boolean algebra.That F-solvability is stronger than F-separability, follows from results in [7].Consider L r.e.(X) where X is a one-letter alphabet.Then a promise problem (A, B) consisting of recursively enumerable sets exists, which is not L r.e.(X)-solvable ( [7] exercise 5-34).But (A, B) is clearly L r.e.(X)-separable.We also find the interesting result that any promise problem (A, B) with A, B ∈ L r.e.(X) co is L r.e.(X) cosolvable ( [7] exercise 5-33).Hence all promise problems, which are L r.e.(X) co -separable are L r.e.(X) co -solvable.But L r.e.(X) co is not closed under complementation.
For k = 1 we identify A 1 with (A 1 ).If F is nontrivial then every A 1 is F-solvable.If k > 2 and F satisfies appropriate closure properties, then we can reduce the question of solvability of classification problems to solvability of promise problems.Directly from the definition we get Proposition 2.2.If F = F u then for all classification problems A and Then we can assume without loss of generality Proof.The "if part" follows by Proposition 2.2.Suppose that (A i , A j ) ∈ class 2 (F) for 1 ≤ i = j ≤ k.Now we proceed by induction over |A| = k.If k = 2 nothing is to prove.Let A = (A 1 , . . ., A k+1 ) and suppose (A 1 , . . ., A k ) ∈ class k (F).Then an F-partition As indicated in the introduction we generalize the notion of a classification problem to conditional classification problems by fixing one component as condition.Consider C ⊆ S and a classification problem A.
The following facts follow directly from the definition Proposition 2.5.Let F and k > 0 be given.

Unsolvability Cores in Classification Problems
As in the case of promise problems unsolvability of classification problems is closely related to cohesiveness.
Remark 3.2.It is interesting to compare our definition of cohesiveness with related classical definitions, as they are presented in [7].Consider the families L r.e.(X) cc , L r.e.(X) and L rec (X).Then L ∈ cohesive(L r.e.(X) cc ) if and only if L is cohesive in the classical sense.Moreover, cohesive(L rec (X)) = cohesive(L r.e.(X)), since a language Q is recursive if and only if Q and Q c are recursively enumerable.Furthermore the definition of recursively indecomposability coincides with the definition of L rec (X)-cohesiveness.In [7] we also find the notion of indecomposability.L is indecomposable if there exist no infinite sets Then we find the following results in [7].If L ∈ cohesive(L r.e.(X) cc ) then it is indecomposable and any indecomposable L is L rec (X)-cohesive.None of the converse implications hold.
In [4] (Theorem 5.1.)it is proven, that for a promise problem (A, B) and a nontrivial set family F A ∪ B ∈ cohesive(F) if and only if A, B ∈ cohesive(F) and (A, B) / ∈ class 2 (F).This result leads to a much stronger one.In the theory of complexity we find the notion of hard cores inside those sets which can be computed with bounded ressources (time, space, e.t.c.[3]).Similarily, we can consider unsolvability cores of classification problems which are not solvable.

and only if for all classification problems A
Clearly, any subproblem of a core is itself a core.This is especially true for subproblems, which are promise problems.This enables us to use the results about unsolvability cores for promise problems from [4].
Proof.Suppose A ∈ core k (F), then by definition (A i , A j ) ≤ A and therefore (A i , A j ) ∈ core 2 (F).Conversely, suppose that A / ∈ core k (F), i. e.
Now we can characterize cores by cohesiveness.Using Theorem 5.1.and Theorem 6.7. of [4] we can prove By Theorem 6.7. in [4] we know Again by Theorem 6.7. of [4] (A i , A j ) ∈ core 2 (F) and therefore by Lemma 3.4.A ∈ core k (F).

We can find to any classification problem
).But this is not true for classification problems A with |A| > 2. To see this we prove the following theorem, where we use S = X * with X = {a, b, c}.Define for A ⊆ X * the classification problem C(A) = (A ab , A bc , A ca ), where A xy = xA ∪ yA c for x, y ∈ X. Assertion : B ′ (x, y) ∈ fin(X * ) for all x, y ∈ X with x = y.Suppose to the contrary (without loss of generality) Remark 3.7.The basic idea behind the proof of Theorem 3.6.is due to M. Ziegler ( [1]).Note, that complexity classes and most of the known language families satisfy the conditions of Theorem 3.6.
Using conditional unsolvability, we can derive an existence theorem for cores.
∈ class 2 (F), we can find C i ⊆ C and B i ⊆ A i with (C i , B i ) ∈ core 2 (F) (Theorem 6.14. in [4]).By Theorem 3.5.
and we obtain B = (B 1 , . . ., B k ) ≤ A and by Theorem 3.5.B ∈ core k (F).Remark 3.9.Consider the situation of Theorem 3.6.Then set (C(A)) = XX * and there is no room for an infinite condition C to make the conditional classification problem (C, C(A)) L-solvable.

Cores in Conditional Classification Problems
Unsolvability of conditional classification problems can be related to cohesiveness, too.In contrast to the definition of core(F) subproblems The following lemma characterizes A ∈ ccore 1 (C, F) by conditional cohesiveness.
(ii) ⇒ (i): Suppose that A / ∈ cclass 1 (C, F) and A ∈ ccohesive(C c , F). Assume to the contrary that an infinite set we get the result.
(ii) ⇒ (i): Let the A i be given according to the assumption.Assume to the contrary that . This is a contradiction.Now, we are able to assert the existence of conditional cores in the case that both C and C c are infinite.Observe that under this assumption A ∈ cclass 1 (C, F) if and only if (C, A) considered as a promise problem is solvable for F, i.e. (C, A) ∈ class 1 (F).Lemma 4.5.Let F be denumerable and nontrivial with . By cor.5.16. in [4] we can find B ⊆ A such that for all infinite Using this lemma in connection with Theorem 4.4.we get

Conditional Cores and Hard Cores
For WP-recursive language families we can prove a much stronger result.This depends on the relation between A ∈ ccore 1 (C, F) and proper hard cores introduced by N. Lynch [6] for complexity classes and in a very general form by R. Book-D.-Z.Du [3].Note, that for A ′ ⊆ A with A ′ infinite every F-hardcore of A is a F-hardcore of A ′ .Rephrasing Lemma 7.2. of [4] we get the following Lemma 5.2.If F is nontrivial with F = F co and (C, A) a conditional classification problem then A is a proper F-hardcore of C c if and only if A ∈ ccore 1 (C, F).Now we can use a construction for proper hard cores from [3] in a modified form.Remark 5.5.The B i 's constructed in Theorem 5.4.are all infinite.By the Dekker-Myhill theorem ( §12.3 Theorem VI in [7]), we can find in every B i a L-cohesive B ′ i , but we cannot show, that B ′ i is recursive under the conditions of Theorem 5.4.The best result to our knowledge is the result of Friedberg ( §12.4 Theorem XI in [7]).The construction (due to Yates) in the proof given in [7] can be easily modified in such a way, that to any infinite, recursive A a L r.e.(X)-cohesive subset B with B c ∈ L r.e.(X) can be found.Since any WP-recursive language family L is a subfamily of L r.e.(X) this B is L-cohesive, too.

Concluding Remarks
This paper continues our research about unsolvability cores in promise problems ( [4]) generalizing the results to classification problems.Our approach is very general, though the applications in this paper deal mainly with language families and complexity classes.The main open problem in our approach is to construct cohesive sets with "nice" properties.
Especially, we get ccohesive(S, F) = cohesive(F) and therefore cohesive(F) ⊆ ccohesive(C, F) for all C ⊆ S. Rewriting the definition, we also find ccohesive(C, F)) = cohesive(F(C) cc ) where F(C) = {Q| Q ⊆ C and Q ∈ F}.Analogously, we define conditional cores by Definition 4.2.Let C ⊆ S and A a classification problem.Then A is a C-conditional core of F (A ∈ ccore |A| (C, F)) if and only if for all A ′ ≤ A with |A ′ | > 0 : A ′ / ∈ cclass |A ′ | (C, F).

Lemma 4 . 3 .
Let F be nontrivial and C, A ⊆ S with A infinite and A ∩ C = ∅.Then the following statements are equivalent

Theorem 4 . 4 .
Let F be nontrivial with F = F u and (C, A) a conditional k-classification problem.If A = (A 1 , . . ., A k ) then the following statements are equivalent

Lemma 4 . 6 .
Let F be denumerable and nontrivial with F = F u = F s and (C, A) a conditional classification problem where C and C c are infinite.If A = (A 1 , . . ., A k ) with A i / ∈ cclass 1 (C, F) for 1 ≤ i ≤ k then a B ≤ A exists with |B| = k and B ∈ ccore k (C, F).Proof.By Lemma 4.5.we find for each 1 ≤ i ≤ k B i ∈ ccore 1 (C, F) and B i ⊆ A i .Let B = (B 1 , . . ., B k ).Then B ≤ A and |B| = k.By Theorem 4.4.B ∈ ccore k (C, F).

Definition 5 . 1 .
B is a F-hardcore of A if and only if B is infinite and for all C ∈ F(A):B ∩ C ∈ fin(S).If additionally B ⊆ A then B is a proper F-hardcore of A. (Remind F(A) = {Q ⊆ A | Q ∈ F} for F and A.)

Theorem 5 . 3 .
If L is a nontrivial and WP-recursive language family with L = L b and (C, A) a conditional classification problem with A / ∈ cclass 1 (C, L) and C, A are recursive then a recursive B ⊆ A exists with B ∈ ccore 1 (C, L).