Universität Ulm

Fakultät für Informatik

On Sets Turing Reducibleto P-Selective Sets

Hans-Jörg BurtschickTV-Berlin

Wolfgang LindnerUniversität Ulm

Nr. 95-09

Ulmer Informatik-Berichte

August 1995

On Sets Turing Reducible to P-Selective Sets

Hans-Jörg BurtschickTU - Berlin, Fachbereich Informatik

Franklinstr. 28/29D-10587 Berlin

Wolfgang LindnerUniversität Ulm, Abt. Theor. Informatik

Oberer Eselsberg 3D-89069 Ulm

Abstract

We consider sets Turing reducible to p-selective sets under various re-source bounds and restricted number of queries to the oracle. We showthat there is a hierarchy among the sets polynomial-time Turing reducibleto p-selective sets with respect to the degree of a polynomial boundingthe number of adaptive queries used by a reduction. We give a charac-terization of EXP/poly in terms of exponential-time Turing reducibilityto p-selective sets. Finally we show that EXP can not be reduced to thep-selective sets under 2hn time reductions with at most nk queries for anyfixed k € N.

1 Introduction

Selman [13] introduced p-selective sets as a polynomial time analogue to thesemirecursive sets as studied by Jockusch [8]. Roughly speaking, a set is p-selective if there is a polynomial-time procedure which decides for a pair ofstrings which of them is "more likely" to be in the set. Selman used p-selectivesets to show that polynomial-time Turing reducibility and many-one reducibilitydiffer on NP, unless E = NE.

Since then much attention has been paid to sets reducible to p-selectivesets under various polynomial-time reducibilities. This extends a long line ofresearch of sets reducible to sets of low density, such as tally and sparse sets (fora survey, see [16]). Toda [15] showed various collapses under the assumptionthat all sets in certain complexity classes are truth-table reducible to some p-selective set. In particular, he showed that if each set in UP is truth-tablereducible to some p-selective set then P = UP and if each set in A£ is truth-table reducible to some p-selective set then P = NP. Recently it has beenshown that if each set in NP is truth-table reducible to some p-selective set thenP = NP [1, 4, 12]. A rather detailed examination of the relationships betweensets reducible or equivalent to the p-selective sets under various reducibilities

has been given by Hemaspaandra et al.[6]. For a survey on results concerningp-selective sets we refer to [5]

Here we concentrate on sets Turing reducible to p-selective sets. We start byshowing that there is a proper hierarchy between the P«(P-sel) and Pr(P-sel)with respect to the degree of a polynomial bounding the number of adaptivequeries to some p-selective oracle. The proof is by diagonalization based on asimple fact concerning the number of sets selected by a Single selector-function.

Selman [13] showed that each tally set is polynomial-time Turing reducibleto some p-selective set. Ko [9] showed that the p-selective sets are containedin the nonuniform advice class P/poly which in turn is predsely the dass ofsets polynomial-time Turing redudble to some tally set. Hence P/poly can becharacterized by the sets polynomial-time Turing reducible to some p-selectiveset. In Section 4 we adress the question wether a similar characterization of thedasses EXP/poly and E/lin can be given. It should be mentioned that a characterization of EXP/poly via tally sets is impossible since every set is alreadyexponential-time many-one reducible to some tally set, namely its tally versionwhere each instance is encoded in unary. However, we can show that EXP /polyis predsely the dass of sets which are exponential-time Turing redudble tosome p-selective set using at most polynomially many adaptive queries. Fur-thermore we show that a set which is Turing redudble in time 0(2ltn) to somep-selective set using at most linearly many queries is contained in E/lin. Herethe converse falls. Concerning the non-uniform complexity of p-selective sets vialinear-length advices there is a recent result by Hemaspaandra and Torenvlietshowing P-sel C NP//m n coNP//m [7].

The proofs are based on an Observation which informally can be stated asfollows: Suppose that V is a iinite set and we know the number of strings inA fl V for some p-selective set A. Then we can dedde whether x is in A foreach string in F, by simply counting the strings in V that are "more likely"than x in A. We apply this Observation to sets Turing reducible to p-selectivesets considering various resource bounds. We thereby obtain a close relationbetween the number of oracle queries to some p-selective set and the length ofthe advice needed to decide a set reducible to some p-selective set nonuniformly.

In Section 5 we consider the relationship of (uniform) exponential-time complexity dasses and sets Turing redudble to some p-selective set. It is an openquestion whether EXP is included in P/poly. Regarding the characterization ofP/poly in terms of sets polynomial-time Turing reducible to some p-selective setit is natural to ask whether one can settle the relationship between subdasses ofP/poly and EXP, where this subclasses are obtained from restricting the accessto some p-selective oracle. In fact, Toda [15] showed that EXP is not includedin the class of sets polynomial-time truth-table reducible to some p-selectiveset. Here we extend this result to sets Turing reducible to some p-selectiveset where the reduction may use at most q{n) adaptive queries for every fixedpolynomial q{n).

2 Preliminaries

We write N to denote the set of nonnegative integers. A string is a finitesequence of characters over the two letter aiphabet E = {0,1}. We write E*for the set of all strings including the empty string and \x\ for the length of astring x € E*. We use ||j4|| to denote the cardinality of a finite set A. Let A-ndenote the set of strings in A of length at most n. A~n is the set of strings inA of length n. A tally set is a set A C {0n : n € N}. Let TALLY denote theclass of all tally sets.

We use deterministic-, nondeterministic- and oracle Turing machines andother notions of complexity theory, as can be found in [2]. We are espe-cially interested in the following deterministic time complexity classes P =(J*eN DTIME(n*), E= Uc€N DTIME(2cn) and EXP = UfceNDTIME(2nfc)-

A set A is Turing reducible to a set B if there exists an oracle Turing machinesuch that A = £(M, B). A query tree of an oracle Turing machine M on inputx is a binary tree in which the nodes are labeled with all possible queries M canask on input z, i.e. the root is labeled with the first query, and for each internalnode corresponding to a query g, the left (resp. right) successor corresponds tothe next query M asks with a positive (resp. negative) answer on q. For a timebound t(n) and a class of sets C, let DTIME(t(n))r(C) denote the class of allsets that are Turing reducible to some set in C via a t(n) time bounded oracleTuring machine. For a function q : N -+ N let DTIME(i(n))g(n).T(C) denotethe class of all sets in DTIME(<(7i))t(C) where the oracle Turing machine asksat most q(n) queries on every path of the query tree. We extend this notationto complexity classes and function classes, for example as Pg(n)-T(£)> ^Kn-T(C),Ypoiy-T(C) etc.

Throughout a computation an oracleTuring machine may ask queries whichdepend on the answers to previously asked queries. These kind of queries arecalled adaptive queries. In contrast, a set A is truth-table reducible to a set B ifthere exists an oracle Turing machine M such that A = L(M, B) and all queriesasked by M are nonadaptive, i.e. do not depend on the answers to previouslyasked queries. For a time bound t(n) and a class ofsets C,let DTIME(t(n))«(C)denote the class of all sets that are truth-table reducible to some set in C via at(n) time bounded oracle Turing machine asking only nonadaptive queries.

All time bounds and functions bounding the number of queries are assumedto be monotonic increasing and time constructible.

3 A hierarchy between polynomial-time truth-tableand Turing reducibility to p-selective sets

We show that there is a proper hierarchy between the class P«(P-sel) andPj(P-sel) with respect to the degree of a polynomial bounding the number ofqueries to some p-selective oracle. First we briefly review the definition and aStandard construction of p-selective sets from [13].

Definition 3.1 ([13]). Äset A C E* is p-selective if there is a polynomial-timecomputable function / : E* x E* —• E* such that for all strings x, y € Ev•>*

1- f(x,y) = x or f(x,y) = y, and

2. if x € j4 or y € A, then /(ar, y) € j4

A function / fulfilling Conditions (i) and (ii) is called a p-selectorfor A.

Let P-sel denote the class of all p-selective sets. It immediately follows fromthe definition that every set in P is p-selective. On the other hand, Selman [13]showed that for every tally set there exists a polynomial-time Turing equivalentp-selective set. Hence there are arbitrarily difiicult p-selective sets. The proofof this fact makes use of a of a subclass of the p-selective sets, namely the classof Standard leftcuts with respect to an infinite binary sequence (cf. [13, 9]).Recall that the dictionary ordering of binary strings over the aiphabet {0,1}can be defined as follows: 0 -< 1, and for x = x\. ..xmi y = 3/1.. .yn, x < yifF (i) m = n and (3i < m)(Vj < i)[xj —yj & X{ -< yt], or (ii) m < n and%•< y\... ym, or (iii) rn > n and x\...xn < y. The Standard leftcut La withrespect to an infinite binary sequence a is the set of strings x which are lessthan or equal to the initial segment of et of length \x\.

Proposition 3.2 ([13]). Every Standard leftcut is p-selective.

Proof. Every Standard leftcut is selected by the p-selector /(x, y) = x if x <y eise y. D

We go on to consider sets reducible to some p-selective set. It is knownthat 2k —1 nonadaptive queries to some p-selective set can be simulated by kadaptive queries of a polynomial-time bounded oracle Turing machine. A proofof this fact can be found in [6]. Though it is stated there only for a constantnumber of queries it can be easily generalized to the case where the nonadaptivequeries are bounded only by the running time of the reduetion.

Proposition 3.3 ([6]). P«(P-sel) = Po(logn)-T(P-sel).

We next show a hierarchy theorem among the sets polynomial-time Turingreducible to some p-selective set with respect to the number of adaptive queriesused by the reduetion. In the case of constant number of queries Hemaspaandraet al. [6] showed a tight hierarchy theorem: P*_T(P-sel) c Pfc+i-T(P-sel), k > 1.They use a construction of p-selective sets introduced by Naik et al. [11]. Ifthe number of queries depends on the length of the input we get a less tighthierarchy. The construeted set will be reducible to some p-selective leftcut. Weisolate the combinatorial part of the diagonalization in the following lemma (cf.

[6])-

Lemma 3.4. Let f be a P-selector and V a finite set of strings. Then

\\{W C V: fselects W}\\ < \\W\\.

Proof. Suppose that there are more than \\W\\ subsets of W* which are selectedby /. Hence among these sets there exists distinet sets W\, W2 Q W with thesame cardinality. That is, for some x\t x<i € W, x\ € W\—W2 and X2 € W2—W1.It follows that f(xi,X2) cannot select both of W\ and W25 D

Proposition 3.5. P * (P-sel) cPt+i (P-sel), k > 1.

Proof. Let Afo,Mi,... be an enumeration of all polynomial-time oracle ma-k

chines asking at most nä queries on an input of length n. Let /o,/i, • • • be anenumeration of all polynomial-time transducers such that /(s,y) 6 {x,y}.

Let fi(s) = 25+4, and let Fn denote the lexicographically first n? + (k + 1) •/o</(n) strings of length n.

We will construct a set A in stages. The set A will consist only of stringsof FM(S), s > 0. In stage s = (i,j), we will satisfy the following requirement:

(Rs) for any set B which is selected by fjy A is not accepted by Mi with oracleB.

The construction proceeds as follows. Let A! denote the set of strings put into Aprior to stage s. Let n = fi(s) and Qn be the set of all queries in the query-tree

of Mi on an all inputs xe Fn. Then ||Qn|| < nf+(k+iyiog(n)-2n*. By Lemma3.4, there exist at most ||Qn|| subsets of Qn which are selected by fj. Hence Mi

can agree on at most (n§ +(k +1) •log{n)) •2n* <n**1. 2n* =2n*+(fc+1)-M«)subsets of F„ with some oracle selected by fj. But there are 2n*X^-M")distinct subsets of Fn. Hence there exists some (smallest) set Dn C Fn whichcannot be accepted by Mt- with any oracle selected by fj. Setting A = A'UDnwe thus established (Rs).

It remains to show that A is Turing reducible to some p-selective set withat most n 2 adaptive queries. Let rfn, for n ~ fi>{s), the string of lengthnä + (k + 1) • log(n) denoting the (finite) chara; ristic function of Dn in theconstruction of A in stage s. Define a p-selectiv< t B to be the leftcut withrespect to the infinite sequence ^(0)^(1)^(2) — ^ ne membership ofsome xof length n = fj,(s) in A is fixed by dn. It thus suffices to reconstruct dn fromthe oracle B. By prefix search, this requires at most St=4 (2*) 2+ (k+1) •i <log(n) -n? + (k -f 1) •log2(n) < y/n-n* = n 2 adaptive queries. D

The announced hierarchy between P«(P-sel) and Pt(P-scI) now follows asa corollary.

Corollary 3.6. 1. P«(P-sel) C Pn.T(P-sel).

2. Pnfc.T(P-sel) C Pnfe+1.T(P-sel), k>\.

3. Pnfc.T(P-sel) C PT(P-sel), k > 1.

4 Exponential-time advice classes

We consider the nonuniform complexity of sets Turing reducible to p-selectivesets in terms of advice classes. As the main result we obtain a characterization

of EXP/po/y in terms of reducibility to p-selective sets.

Definition 4.1. An advice function is a function h : N -* E*. For a function q : N -* N, let ADV(q(n)) denote the set of all advice functions hwith \h(n)\ < q(n) for all n e N. For functions t,q : N -»• N, the adviceclass DT1ME(t(n))/ADV(q(n)) is the class of sets B for which there exists aset A € DTIME(*(n)) and an advice function h{n) e ADV(q(n)) such thatB = {x:(x,h(\x\))eA}.

Using Definition 4.1 we can redefine P/po/y as U*eN T>TIME(nk)/ADV(nk).Additionally we consider the advice classes E/lin and EXP/po/y which can bedefined similarly.

The following lemma is the key Observation which leads to all subsequentresults of this paper.

Lemma 4.2. Let A be a p-selective set with selector-function f. Let V a fi-nite set of strings. Then for each x € V, x € A if and only if \\{x' : x' €V and f(x,x') = x'}|| < ||A n V||.

Proof. Fix a string x € V. First assume that x is in A. By the definition of ap-selector, /(x, a;') = x' implies x' € A. Therefore, the number of strings x' inV for which f(x,x') = x' is at most \\A n V||. If x is not in A, then for all x'in A, f(x,x') = x'. Additionally, /(&,x) = x. Hence for more than ||A n V||strings x' in V it holds that /(x,x;) = x'. D

Consider a p-selective set A and a p-selector / for A. Let the advice bethe binary representation of the number of strings in A of length n. Thenthe length of the advice is at most n + 1. By Lemma 4.2, for each stringx of length n, the membership of x A can be decided with the help of theadvice by counting the strings x' of length n for which f(x,x') = x'. Since/ is polynomial-time computable, this can be done in time 0(22n). That is,P-sel C DTIME(22n)/i4I>y(n + 1). This argument can be generalized to setsTuring reducible to p-selective sets, whereby we obtain a relationship betweenthe number of oracle queries and the length of the advice.

Lemma 4.3.

DTIME(t(n))g(n).T(P-sel) C (J DTIME(*(n)* •22*W+2n)/ADV(q{n) + n+ 1)

Proof. Let A be a set Turing reducible to a p-selective set B via a 0(t(n)) timebounded oracle Turing machine which asks at most q(n) queries on every pathof the query tree on some input of length n. Let / be the p-selector for B andassume that / is computable in time 0(nk) for some constant A: € N.

Let Qn = \Jix\=nQ(x) where Q(x) denotes the set of all queries in thequery-tree of M on an input x. Define the advice function h : N —> E* to bethe binary representation of \\B DQn||- Thus the length of h(n) is less or equalto q(n) + n+ 1.

For a string x of length n, we decide x € A by the following algorithm. Firstgenerate a list of all queries q € Qn by traversing the query trees of M for allinputs of length n. In order to avoid counting a Single query more than once in

the following step, eliminate multiple occurrences of queries in this list. Nowsimulate M on x. Whenever M asks a query g, count the strings q' in the listsuch that /(g, q') = q'. If this number is less or equal to h(n), continue with theanswer "YES", otherwise continue with "NO". Accept if and only if M acceptsx.

By Lemma 4.2, we always continue with the correct answer. Therefore, weaccept x iff x € A. To compute the list of all queries in Qn, we have to generatesuccessively Q(x) for allx oflength n. This can bedone in time 0{2n-29^-t(n)).Eliminating multiple occurrences of queries in this list requires additionally time0((2n-2g(n))2-t(n)). To determine theanswer for a query q, we have to computethe p-selector / on at most 2n «29^ strings oflength less or equal to t(n). Since/ is computable in time 0(nfc), this requires time 0((t(n))k •2n •2^n)). Weconclude that the whole algorithm runs in time 0(t(n)k •22«<n)+2n). D

Applying Lemma 4.3 and the Standard leftcut construction we obtain thecharacterization of EXP /poly.

Theorem 4.4. EXPpo/y.T(P-sel) = EXP/po/y

Proof. By Lemma 4.3, EXPpo/y.7'(P-sel) C EXP/po/y. To see the inverse in-clusion let A be accepted by an exponential-time Turing machine M with theadvice function h. Define an infinite binary sequence a = /i(0)01h(l)01...,where x denotes the string x with each bit doubled. In order to decide x € Awith the p-selectiveleftcut Ia, first compute /i(0)01/i(l)01.. .Ol/iflxl) from LQby prefix search. Then simulate M with input x and the advice /i(|x|). Sincethe length of h(\x\) is polynomially bounded, both the number and the lengthof the queries are bounded by some polynomial. It follows that h(\x\) can beobtained from the oracle La in polynomial-time. Hence the whole algorithmruns in exponential-time. D

Remark 4-5. Note that in the above proof the queries used by the exponential-time reduetion are polynomially length bounded. Thus it follows from Theorem4.4 that every set in EXPp0/y.T(P-sel) can be exponential-time Turing reducedto some p-selective set where both the number and the length of the queries arepolynomially bounded.

A characterization similar to Theorem 4.4 falls for E/lin. Adapting a proofin [3], we show that E/,n.r(P-sel) is properly included in E/lin. In the proofwe use a Kolmogorov-random sequence. For the definition of Kolmogorov complexity and related facts we refer to [10].

Theorem 4.6. EKn.r(P-sel) C E/lin

Proof. The inclusion Enn.T(P-sel) C E/lin follows from Lemma 4.3. Moreover,replacing Qn = \J\x\=nQ(x) bY Q<n = \J\x\<nQ(x) in the Proof of Lemma4.3 we see that for a set in E/tn.r(P-sel) all strings up to length n can bedeeided in exponential-time with an advice of linear length. This implies thatthe Kolmogorov complexity of all initial segments of the characteristic sequenceup to strings of length n is at most linear in n.

Now consider a binary Kolmogorov random infinite sequence p. That is, asequence such that the Kolmogorov complexity of its prefixes is at least linearin the length of the prefixes infinitely often. We define a set A which containsat most the first n strings of length n. Let x be the lexicographically i-th string,i <n, of length n. Then x€Aifand only ifthe (w(w2~^ +t)-th bit of pis 1.That is, we divide p into consecutive subsequences of length 1,2,..., n,... andthe n-th subsequence of length n denotes the membership of the first n stringsof length n in A. It follows that A is in E/lin. We use a prefix of length n2 ofp to define A up to strings of length n. Hence the Kolmogorov complexity ofthe prefixes of the characteristic sequence of A up to strings of length n is atleast quadratic in n infinitely often. Thus A is not in Ejtn.T(P-sel). D

Remark 4-7. Throughout this paper we consider only boundedquery reductionsto p-selective sets. The reason is that if we do not restrict the number ofqueries, then every set is reducible in linear exponential time to p-selective sets.In order to see that, fix any set A. Then A can be many-one reduced in linearexponential-time to its tally version tally(A) = {0* : s« € A}, where S{ is thei-th string in the lexicographical ordering on S*. Furthermore, every tally setcan be Turing reduced to some p-selective set in polynomial time [13]. HenceA is in Em(PT(P-sel)). Since Em(PT(P-sel)) C Er(P-sel), we conclude that Ais in ET(P-sel).

5 Turing reducibility to p-selective sets and uniformexponential-time complexity

We locate sets Turing reducible to p-selectivesets in (uniform) exponential-timeclasses using Lemma 4.3 and the following proposition.

Proposition 5.1. Let f : N -* N be a function with n < f(n) < 2n. ThenDTIME(24-'<n)) g T>TIME(2ttn))/ADV(f(n)).

Proof. Let Mo, Mi,... be an enumeration of all Turing machines M« runningin time i2^n\ We will construct a set A in stages. Each stage determines themembership of all strings of length n. In stage n, we will satisfy the followingrequirement:

(Rn) for any advice function h € ADV(f(n)), A is not accepted by Mn withadvice h.

This clearly implies A $ BTIUE(2^n^)/ADV(f(n)). Let A! denote the set ofstrings put into A prior to stage n. There are at most2^n) < 22" setsof stringsof length n which can be accepted by Mn with some advice of length f(n). Sincethere are 22 distinct subsets of S=n there is a (smallest) set Dn C S=n whichis not accepted by Mn with some advice of length f(n). Setting A = A! U Dnwe thus established (Rn).

In order to decide x € A (uniformly) for some string x of length n we onlyhave to determine the set Dn in the above construction. But this can be donein time 0(2'<n) •2n •n •2'<n)), hence A is in DTIME(24>(n)). D

Theorem 5.2. Fix c,fc G N. Then

1- E g Pcn.T(P-sel)

2. EXP g Enk.T(P-sel)

Proo/- (1) By Lemma 4.3, for every c € N, Pcn.T(P-sel) C DTIME(2c'n)/,4W(c'n)for some constant c' € N. But E g DTIME(2c'n)/ADV(c'n) by Proposition5.1. The proof of (2) is similar. D

Remark 5.3. Theorem 5.2 (1) holds not only for polynomial-time reducibility,but also for super polynomial-time bounds. More precisely, for all t(n) suchthat, for all kGN, t(n)k €0(2*), Eg DTIMEWn^rCP-sel).

6 Conclusion

We showed that there is a hierarchy among the sets Turing reducible to p-selective sets with respect to the degree of the polynomial bounding the numberof adaptive queries used by a reduction. Furthermore, we gave a characteriza-tion of EXP/poly in terms of Turing reducibility to p-selective sets.

Furthermore, weextended Toda's result EXP g P«(P-sel) to EXP g En*_T(P-sel)for every fixed k € N. Wilson [14] constructed an oracle relative to whichEXPNP (and hence EXP) is included in T/poly = PT(P-sel). Thus our Separation seems to be the best possible without using nonrelativizing techniques.However, since EXP C F/poly if and only if EXP /poly C P/po/y, our charac-terization might shed some light on the EXP C P/poly question.

References

[1] M. Agrawal and V. Arvind. Polynomial time truth-table reductions to p-selective sets. In Proc. 9th Structure in Complexity Theory. 1994.

[2] J. L. Balcazar, J. Diaz, and J. Gabarro. Structural complexity I. EATCSMonographs on Theoretical Computer Science. Springer, 1988.

[3] J. L. Balcazar and U. Schöning. Logarithmic advice classes. TheoreticalComp. Sei., 99:279-290,1992.

[4] R. Beigel, M. Kummer, and F. Stefan. Approximable sets. In Proc. 9thStructure in Complexity Theory. 1994.

[5] D. Denny-Brown, Y. Han, L. A. Hemaspaandra, and L. Torenvliet. Semi-membership algorithms: Some recent results. SIGACT News, 25(3): 12-23,1994.

[6] L. Hemaspaandra, A. Hoene, and M. Ogihara. Reducibility classes of p-selective sets. to appear in Theoretical Computer Science.

[7] L. A. Hemaspaandra and L. Torenvliet. Optimal advice. to appear inTheoretical Computer Science.

8] C. G. Jockusch. Semirecursive sets and positive reducibility. Trans. Amer.Math. Soc., 131:420-436,1968.

9] Ker-I Ko. On self-reducibility and weak P-selectivity. J. Comput. SystemSei., 26:209-221,1983.

10] M. Li and P. Vitänyi. An Introduction to Kolomogorov Complexity and itsApplications. Springer-Verlag, 1993.

11] A. V. Naik, M. Ogiwara, and A. L. Selman. P-selective sets, reducingsearch to decision vs. self-reducibility. In Proc. 8th Structure in ComplexityTheory. 1993.

12] M. Ogihara. Polynomial-time membership comparable sets. In Proc. 9thStucture in Complexity Theory. 1994.

13] A. L. Selman. P-selective sets, tally languages and the behavior of polynomial time reducebilities on NP. Math. Systems Theory, 13:55-65,1979.

14] C. B. Wilson. Relativized cireuit complexity. J. Comput. System Sei.,31:169-181,1985.

15] S. Toda. On polynomial-time truth-table reducibility of intractable sets top-selective sets. Math. Systems Theory, 24:69-82,1991.

16] P. Young. How reduetions to sparse sets collapse the polynomial-timehierarchy: A primer. SIGACT News, 1992. Part I (#3 pp. 107-117), PartII (#4, pp. 83-94), and Corrigendum (#4 p. 94).

10

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92-09* Hermann von Hasseln, Laura MartignonConsistency in Stochastic Network

92-10 Michael Schmitt

A SlightlyImproved Upper Bound on the Size of Weights Sufficient to RepresentAny Linearly Separable Boolean Function

92-11 Johannes Köbler, Seinosuke TodaOn the Power of Generalized MOD-Classes

92-12 V. Arvind, J. Köbler, M. MundhenkReliable Reductions, High Sets and Low Sets

92-13 Alfons GeserOn a monotonic semantic path ordering

92-14* Joost Engelfriet, Heiko VoglerThe Translation Power of Top-Down Tree-To-Graph Transducers

93-01 Alfred Lupper, Konrad FroitzheimAppleTalk Link Access Protocol basierend auf dem Abstract Personal CommunicationsManager

93-02 M.H. Scholl, C. Laasch, C. Rieh, H.-J. Schek, M. TreschThe COCOON Object Model

93-03 Thomas Thierauf, Seinosuke Toda, Osamu WatanabeOn Sets Bounded Truth-Table Reducible to P-selective Sets

93-04 Jin-Yi Cai, Frederic Green, Thomas ThieraufOn the Correlation of Symmetrie Functions

93-05 K.Kuhn, M.Reichert, M. Nathe, T. Beuter, C. Heinlein, P. DadamA Conceptual Approach to an Open Hospital Information System

93-06 Klaus GaßnerRechnerunterstützung für die konzeptuelle Modellierung

93-07 Ullrich Keßler, Peter DadamTowards Customizable, Flexible Storage Structures for Complex Objects

94-01 Michael Schmitt

On the Complexity of Consistency Problems for Neurons with Binary Weights

94-02 Armin Kühnemann, Heiko VoglerA Pumping Lemma for Output Languages of Attributed Tree Transducers

94-03 Harry Buhrman, Jim Kadin, Thomas ThieraufOn Functions Computable with Nonadaptive Queries to NP

94-04 Heinz Faßbender, Heiko Vogler, Andrea WedelImplementation of a DeterministicPartial E-Unification Algorithm for Macro Tree Transducers

94-05 V. Arvind, J. Köbler, R. SchulerOn Helping and Interactive Proof Systems

94-06 Christian Kalus, Peter DadamIncorporating record subtyping into a relational data model

94-07 Markus Tresch, Marc H. SchollA Classification of Multi-Database Languages

94-08 Friedrich von Henke, Harald RueßArbeitstreffen Typtheorie: Zusammenfassung der Beiträge

94-09 F.W. von Henke, A. Dold, H. Rueß. D. Schwier, M. StreckerConstruction and Deduction Methoas for the Formal Development ofSoftware

94-10 Axel DoldFormalisierung schematischer Algorithmen

94-11 Johannes Köbler, Osamu WatanabeNew Collapse Consequences of NP Having Small Circuits

94-12 Rainer SchulerOn Average Polynomial Time

94-13 Rainer Schuler, Osamu WatanabeTowards Average-Case Complexity Analysis of NP Optimization Problems

94-14 Wolfram Schulte, Ton VullinghsLinkmg Reactive Software to the X-Window System

94-15 Alfred LupperNamensverwaltung und Adressierung in Distributed Shared Memory-Systemen

94-16 Robert RegnVerteilte Unix-Betriebssysteme

94-17 Helmuth PartschAgain on Recognition and Parsing of Context-Free Grammars: Two Exercisesin Transformational Programming

94-18 Helmuth PartschTransformational Development of Data-Parallel Algorithms: an Example

95-01 Oleg VerbitskyOn the Largest Common Subgraph Problem

95-02 Uwe SchöningComplexity of Presburger Arithmetic with Fixed Quantifier Dimension

95-03 Harry Buhrman, Thomas ThieraufThe Complexity of Generating and Checking Proofs of Membership

95-04 Rainer Schuler, Tomoyuki YamakamiStructural Average Case Complexity

95-05 Klaus Achatz, Wolfram SchulteArchitecture Indepentent Massive Parallelization of Divide-And-Conquer Algorithms

95-06 Christoph Karg, Rainer SchulerStructure in Average Case Complexity

95-07 P. Dadam, K. Kuhn, M. Reichert, T. Beuter, M. NatheADEPT: Ein integrierender Ansatz zur Entwicklung flexibler, zuverlässigerkooperierender Assistenzsysteme in klinischen Anwendungsumgebungen

95-08 Jürgen Kehrer, Peter SchulthessAufbereitung von gescannten Röntgenbildern zur filmlosen Diagnostik

95-09 Hans-Jörg Burtschick, Wolfgang LindnerOn Sets Turing Reducible to P-Selective Sets

Ulmer

Informatik-Berichte

ISSN 0939-5091

Herausgeber: Fakultät für Informatik

Universität Ulm, Oberer Eselsberg. D-89069 Ulm