Collapse Of Hierarchy Research Articles

Transfer theorems via sign conditions

We show that P=PSPACE implies the collapse of the Boolean polynomial hierarchy over any structure which admits “efficient enumeration of sign conditions”. This fairly rich class of structures contains in particular R and C .

Parallel turing machines with one-head control units and cellular automata

Parallel Turing machines ( Ptm) can be viewed as a generalization of cellular automata ( Ca) where an additional measure called processor complexity can be defined which indicates the “amount of parallelism” used. In this paper Ptm are investigated with respect to their power as recognizers of formal languages. A combinatorial approach as well as diagonalization are used to obtain hierarchies of complexity classes for Ptm and Ca. In some cases it is possible to keep the space complexity of Ptm fixed. Thus for the first time it is possible to find hierarchies of complexity classes (though not Ca classes) which are completely contained in the class of languages recognizable by Ca with space complexity n and in polynomial time. A possible collapse of the time hierarchy for these Ca would therefore also imply some unexpected properties of Ptm.

What's up with downward collapse

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what before-the-fact seemed to be) lower complexity levels. For example, for each k ≥ 2 it. is now known that if one query to σ p k is as powerful as two queries to σ p k (i.e., P Σ[1] = P Σ[2] ), then PH = σ p k . This article surveys the history, the results, and the method---the so-called easy-hard technique---of the just-mentioned nine-paper progression:1. J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing , 17(6):1263--1282, 1988. Erratum appears in the same journal. 20(2):404.2. K. Wagner. Number-of-query hierarchies. Technical Report 158, Institut für Mathematik. Universität Augsburg, Augsburg, Germany, October 1987.3. K. Wagner. Number-of-query hierarchies. Technical Report 4. Institut für Informatik. Universität Würzburg, Würzburg, Germany, February 1989.4. R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing . 25(2):340--354. 1996.5. R. Beigel. R. Chang. and M. Ogiwara. A relationship between difference hierarchies and relativized polynomial hierarchies. Mathematical Systems Theory . 26(3):293--310, 1993.6. E. Hemaspaandra, L. Hemaspaandra. and H. Hempel. An upward separation in the polynomial hierarchy. Technical Report Math/Inf/96/15, Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany, June 1996.7. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. A downward collapse within the polynomial hierarchy. SIAM Journal on Computing , 28(2):383--393, 1999.8. H. Buhrman and L. Fortnow. Two queries. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity . 13--19. IEEE Computer Society Press, June 1998.9. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Translating equality downwards. Technical Report TR-657. Department of Computer Science, University of Rochester, Rochester, NY. April 1997.

Capturing Relativized Complexity Classes without Order

AbstractWe consider the problem of obtaining logical characterisations of oracle complexity classes. In particular, we consider the complexity classes LOGSPACENP and PTIMENP. For these classes, characterisations are known in terms of NP computable Lindström quantifiers which hold on ordered structures. We show that these characterisations are unlikely to extend to arbitrary (unordered) structures, since this would imply the collapse of certain exponential complexity hierarchies. We also observe, however, that PTIMENP can be characterised in terms of Lindström quantifers (not necessarily NP computable), though it remains open whether this can be done for LOGSPACENP.

On parallel hierarchies and Rki

This paper defines natural hierarchies of function and relation classes □ i, k c and Δ i, k c , constructed from parallel complexity classes in a manner analogous to the polynomial-time hierarchy. It is easily shown that □ i−1, k p ⊆ □ c, k c ⊆ □ i, k p and similarly for the Δ classes. The class □ i,3 c coincides with the single-valued functions in Buss et al.'s class F℘ 3 Σ i−1,3 p [wit, log O(1)] (see [11]), and analogously for other growth rates. Furthermore, the class □ i, k c comprises exactly the functions Σ i, k b -definable in S k i−1 , and if T k i−1 is ∀∃ Σ i, k b -conservative over S k i−1 , then □ i, k p is completely parallelizable. All functions in □ i, k c are Σ i, k b -definable in R k i ; this suffices to show that if the known ∀∃ Σ i, k b conservativity between R 3 i and S 3 i−1 extends to R 2 i and S 2 i−1 , then NC = NC 1 relative to an oracle in PH. We prove a KPT-style witnessing theorem for S k i using constantly many rounds of □ i, k c interactive computation, and thus show that if S k i ≡ R k i+1 then the bounded arithmetic hierarchy collapses, provably in S k i .

Disjunctive datalog

We consider disjunctive Datalog, a powerful database query language based on disjunctive logic programming. Briefly, disjunctive Datalog is a variant of Datalog where disjunctions may appear in the rule heads; advanced versions also allow for negation in the bodies which can be handled according to a semantics for negation in disjunctive logic programming. In particular, we investigate three different semantics for disjunctive Datalog: the minimal model semantics the perfect model semantics, and the stable model semantics. For each of these semantics, the expressive power and complexity are studied. We show that the possibility variants of these semantics express the same set of queries. In fact, they precisely capture the complexity class Σ P 2 . Thus, unless the Polynomial Hierarchy collapses, disjunctive Datalog is more expressive that normal logic programming with negation. These results are not only of theoretical interest; we demonstrate that problems relevant in practice such as computing the optimal tour value in the Traveling Salesman Problem and eigenvector computations can be handled in disjunctive Datalog, but not Datalog with negation (unless the Polynomial Hierarchy collapses). In addition, we study modularity properties of disjunctive Datalog and investigate syntactic restrictions of the formalisms.

Oracles That Compute Values

This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomial-time reducibilities, it is shown that a multivalued partial function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable via 2k-1 nonadaptive queries to NPMV; a characteristic function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable with k adaptive queries to NP; unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV are different than k+1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness, and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses.

On compact representations of propositional circumscription

Circumscription is a popular common-sense reasoning technique, used in the fields of Artificial Intelligence, Databases and Logic Programming. In this paper we investigate the size of representations (formulae, data structures) equivalent to the circumscription of a propositional formula T, taking into account three different definitions of equivalence. We find necessary and sufficient conditions for the existence of polynomial-size representations (formulae, data structures) equivalent to the circumscription of T in the three cases. All such conditions imply the collapse of the polynomial hierarchy. In particular, we prove that — unless the polynomial hierarchy collapses at the second level — the size of the shortest propositional formula T′ logically equivalent to the circumscription of T grows faster than any polynomial as the size of T increases. The significance of this result in the related field of closed-world reasoning is then analyzed.

Counting Quantifiers, Successor Relations, and Logarithmic Space

Given a successor relationS(i.e., a directed line graph), and given two distinguished pointssandt, the problem ORD is to determine whethersprecedestin the unique ordering defined byS. We show that ORD is L-complete (via quantifier-free projections). We then show that first-order logic with counting quantifiers, a logic that captures TC0over structures with a built-in total-ordering, cannot express ORD. Our original proof of this in the conference version of this paper employed an Ehrenfeucht–Fraı̈ssé Game for first-order logic with counting. Here we show how the result follows from a more general one obtained independently by Nurmonen. We then show that an appropriately modified version of the EF game is “complete” for the logic with counting in the sense that it provides a necessary and sufficient condition for expressibility in the logic. We observe that the L-complete problem ORD is essentially sparse if we ignore reorderings of vertices, a property which we term “pseudo-sparseness.” We then prove that there are no pseudo-sparse NP-complete properties (under P-time reductions) unless the polynomial time hierarchy collapses (toΣ3), revealing a structural property on which L and NP apparently differ.

The Boolean Hierarchy and the Polynomial Hierarchy: A Closer Connection

We show that if the Boolean hierarchy collapses to level $k$, then the polynomial hierarchy collapses to $BH_{3}(k)$, where $BH_{3}(k)$ is the $k$th level of the Boolean hierarchy over $\Sigma^{P}_{2}$. This is an improvement over the known results, which show that the polynomial hierarchy would collapse to $P^{NP^{NP}[O(\log n)]}$. This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. Also, this result points to some previously unexplored connections between the Boolean and query hierarchies of $Delta^{P}_{2}$ and $Delta^{P}_{3}$. Namely, \begin{eqnarray*} & BH(k) = {\rm co-}BH(k) \implies BH_{3}(k) = {\rm co-}BH_{3}(k), \\[\jot] & P^{{\rm NP}\|[k]} = P^{{\rm NP}\|[k+1]} \implies P^{{\rm NP}^{\rm NP}\|[k+1]} = P^{{\rm NP}^{\rm NP}\|[k+2]}. \end{eqnarray*}

On the power of generalized Mod-classes

The computational power of the counting class ModP, which generalizes the classes Mod/sub p/P, p prime, is investigated. It is shown that ModP is truth-table equivalent in power to MP, and that ModP is contained in the class AmpMP. As a consequence, the lowness of AmpMP or of ModP for MP would imply the collapse of the counting hierarchy (CH) to MP. Further, every set in C=P is shown to be reducible to a ModP set via a random many-one reduction that uses only logarithmically many random bits. Therefore, ModP and AmpMP are not closed under such reductions unless CH collapses. Finally, ModP is generalized to the class Mod/sub */P, which turns out to be a superclass of C=P. >

The Breakdown of Hierarchies in the Soviet Union and China: A Neoinstitutional Perspective

Why did modest attempts to decentralize the centrally administered Soviet system lead to its collapse, while more far-reaching decentralization in China left central political and administrative hierarchies intact? This article analyzes the disintegration of centrally planned organizations in the context of a neoinstitutional model of the breakdown of authority within hierarchies. An agency model of hierarchy is presented that incorporates the ambiguous property rights, authority relations, and risk-sharing conditions that prevailed under central planning and then persist during postcommunist transitions. This model suggests that decentralizing reforms could trigger an organizational “bank run,” prompting local agents to seize organizational assets under their control. The article also considers reputation-preserving strategies that central authorities might use to avert disintegration.As an application of this model, the collapse of Soviet political, industrial, and state fiscal hierarchies are considered and compared with the experience of analogous sectors in China. Reforms in both states transferred significant autonomy from the center to local field agents. In the Soviet case these agents appropriated organizational assets with little interference from the center. In China, by contrast, the center preserved both its capacity for monitoring and its reputation for disciplining transgressions; and the rise of hybrid ownership forms made expropriation of state and Party assets far less attractive.

PP Is Closed under Intersection

In this seminal paper on probabilistic Turing machines, Gill asked whether the class PP is closed under intersection and union. We give a positive answer to this question. We also show that PP is closed under a variety of polynomial-time truth-table reductions. Consequences in complexity theory include the definite collapse and (assuming P ≠ PP) separation of certain query hierarchies over PP. Similar techniques allow us to combine several threshold gates into a single threshold gate. Consequences in the study of circuits include the simulation of circuits with a small number of threshold gates by circuits having only a single threshold gate at the root (perceptrons) and a lower bound on the number of threshold gates that are needed to compute the parity function.

If NP has polynomial-size circuits, then MA = AM

It is shown that the assumption of NP having polynomial-size circuits implies (apart from a collapse of the polynomial-time hierarchy as shown by Karp and Lipton) that the classes AM and MA of Babai's Arthur-Merlin hierarchy coincide. This means that also a certain inner collapse of the remaining classes of the polynomial-time hierarchy occurs.

Carnival Culture: The Trashing of Taste in America

Untended gates - the triumph of vulgarity in an age of show business paperbacked culture - how candles became shoes peepshow America - Hollywood and popular taste programming television - reflections on the electronic midway vulgarity victorious - the collapse of cultural hierarchy in postmodern America.

Analog computation via neural networks

We pursue a particular approach to analog computation, based on dynamical systems of the type used in neural networks research. Our systems have a fixed structure, invariant in time, corresponding to an unchanging number of “neurons”. If allowed exponential time for computation, they turn out to have unbounded power. However, under polynomial-time constraints there are limits on their capabilities, though being more powerful than Turing machines. (A similar but more restricted model was shown to be polynomial-time equivalent to classical digital computation in the previous work (Siegelmann and Sontag, 1992).) Moreover, there is a precise correspondence between nets and standard nonuniform circuits with equivalent resources, and as a consequence one has lower bound constraints on what they can compute. This relationship is perhaps surprising since our analog devices do not change in any manner with input size. We note that these networks are not likely to solve polynomially NP-hard problems, as the equality “ p= np” in our model implies the almost complete collapse of the standard polynomial hierarchy. In contrast to classical computational models, the models studied here exhibit at least some robustness with respect to noise and implementation errors.

Metric extensions and the L1 hierarchy

A finite semimetric is L 1-embeddable if it can be expressed as a non-negative combination of Hamming semimetrics. A finite semimetric is called hypermetric if it satisfies the (2 k+1)- gonal inequalities which naturally generalize the triangle inequality. It is known that all L 1-embeddable semimetrics are hypermetric and the metric induced by K 7− P 3 is hypermetric but not L 1-embeddable. In the first part of the paper we show that there are infinite metrics that are hypermetric but are not L 1-embeddable, answering a question of Deza. We introduce the r-extension of a semimetric: this is the addition of new points from which all of the distances are r. We show that all 2-extensions of K 7− P 3 are hypermetric. Unfortunately, 2- extensions of hypermetrics in which all of the distances are either one or two may not be hypermetric in general. We illustrate this by showing that the metric induced by the star K 1,5 (which is hypermetric) has a nonhypermetric 2-extension involving three additional points. In the second part of the paper, we show that 1-extensions of the metric induced by K 4− P 3 form a family that span a hierarchy of metrics that contain L 1. A finite semimetric is of negative type if it is the square of a semimetric that can be isometrically embedded in Euclidean space. It is known that a semimetric that is hypermetric is of negative type; and the distance matrix of a semimetric of negative type has one positive eigenvalue. All of the inclusions in the above hierarchy are strict. We demonstrate this latter fact by the 1-extensions of K 4− P 3. Indeed, K 6− P 3 is L 1-embeddable and K 7− P 3 is hypermetric but not L 1-embeddable. Here we show that K 8− P 3 is hypermetric but K 9− P 3 is not; K 10− P 3 is of negative type but K 11− P 3 is not; and K 11− P 3 has one positive eigenvalue but K 12− P 3 does not. This situation can be contrasted with the collapse of the metric hierarchy for bipartite graphs.

Strangers and Liberals

CHARLES TAYLOR'S Multiculturalism and Politics of Recognition* is a meditation on the rise of the modern demand for recognition, which he suggests is at the root of contemporary calls for the protection and elevation of specific minority cultures in majority societies. He connects the rise of this demand to the collapse of social hierarchies and the persistence of the desire for honor that such hierarchies once fulfilled. In a democratic era, we seek equal recognition and equal dignity. Both of these quests are fundamentally associated with the intensification of a desire to achieve individual identity and the ideal of authenticity to which it has given rise. For Taylor, the difficulty with authenticity as an ideal is that it undermines the realization that a crucial feature of human life is its fundamentally dialogical (p. 32). Relying on the insights of Bakhtin, he contrasts the dialogical character of life with what he calls the monological ideal. The monological ideal seeks to limit the role of dialogue to the genesis of identity in language and self-understanding and not give dialogue a role in the rest of social life. In his reading, the monological ideal is expressed, not as an overt denial of the dialogical character of identity formation and its continued expression throughout life, but, paradoxically, as a response to the lack of recognition accorded individuals in modern life, a consequence of the persistence of the demand for dignity in a democratic age. Taylor writes, politics of difference grows organically out of the politics of universal dignity through one of those shifts with which we are long familiar, where a new understanding of the human social condition imparts a radically new meaning to an old principle (p. 39). The issue of

A second order version of S2i and U21

In [1] S. Buss introduced systems of bounded arithmetic , , , (i = 1, 2, 3, …). and are first order systems and and are second order systems. and are closely related to and respectively in the polynomial hierarchy, and and are closely related to PSPACE and EXPTIME respectively. One of the most important problems in bounded arithmetic is whether the hierarchy of bounded arithmetic collapses, i.e. whether = or = for some i, or whether = , or whether is a conservative extension of S2 = ⋃i. These problems are relevant to the problems whether the polynomial hierarchy PH collapses or whether PSPACE = PH or whether PSPACE = EXPTIME. It was shown in [4] that = implies and consequently the collapse of the polynomial hierarchy. We believe that the separation problems of bounded arithmetic and the separation problems of computational complexities are essentially the same problem, and the solution of one of them will lead to the solution of the other.

The Polynomial Time Hierarchy Collapses If the Boolean Hierarchy Collapses

It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that ${\text{co-NP}} \subseteq {\text{ NP}}^S $, and therefore the polynomial time hierarchy (PH) collapses to ${\text{P}}^{{\text{NP}}^{{\text{NP}}} [O(\log n)]} $, a subclass of $\Delta _3^{\text{P}} $. Since the BH is contained in ${\text{P}}^{{\text{NP}}} $, these results relate the internal structure of ${\text{P}}^{{\text{NP}}} $ to the structure of the PH as a whole. Other conditions that imply the collapse of the BH (and the collapse of the PH in turn include ${\text{D}}^{\text{P}} = {\text{co}}{\text{-}}{\text{D}}^{\text{P}} $, ${\text{P}}^{{\text{NP}}[k]} = {\text{P}}^{{\text{NP}}[k + 1]} $ for any k, and ${\text{P}}^{{\text{NP}}\|[k]} = {\text{P}}^{{\text{NP}}\|[k + 1]} $ for any k. ${\text{P}}^{{\text{NP}}[i]} $ is the class of languages recognizable 1n polynomial time with at most i queries to an oracle from NP, and ${\text{P}}^{{\text{NP}}\|[i]} $ is the class of languages recognizable with at most i parallel queries to an oracle from NP.