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In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P (polynomial time) and PSPACE (polynomial space). It was shown that the separability of two complexity classes can be reduced to a combinatorial property of the corresponding defining leaf languages. In the present paper, it is shown that every separation obtained in this way holds for every generic oracle in the sense of Blum and Impagliazzo. We obtain several consequences of this result, regarding, e.g., simultaneous separations and universal oracles, resource-bounded genericity, and type-2 complexity.

This paper introduces modal independence logic MIL, a modal logic that can explicitly talk about independence among propositional variables. Formulas of MIL are not evaluated in worlds but in sets of worlds, so called teams. In this vein, MIL can be seen as a variant of V\"a\"an\"anen's modal dependence logic MDL. We show that MIL embeds MDL and is strictly more expressive. However, on singleton teams, MIL is shown to be not more expressive than usual modal logic, but MIL is exponentially more succinct. Making use of a new form of bisimulation, we extend these expressivity results to modal logics extended by various generalized dependence atoms. We demonstrate the expressive power of MIL by giving a specification of the anonymity requirement of the dining cryptographers protocol in MIL. We also study complexity issues of MIL and show that, though it is more expressive, its satisfiability and model checking problem have the same complexity as for MDL.

In this paper we study the expressive power of Horn-formulae in dependence logic and show that they can express NP-complete problems. Therefore we define an even smaller fragment D-Horn* and show that over finite successor structures it captures the complexity class P of all sets decidable in polynomial time. Furthermore we study the question which of our results can ge generalized to the case of open formulae of D-Horn* and so-called downwards monotone polynomial time properties of teams.

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers., Comment: 10 pages, 1 figure

We introduce a new method to separate counting classes of a special type by oracles. Among the classes, for which this method is applicable, are NP, coNP, US (also called 1-NP), ? P, all other MOD-classes, PP, and C = P, classes of Boolean Hierarchies over the named classes, classes of finite acceptance type, and many more. As an important special case, we completely characterize all relativizable inclusions between classes NP(k) from the Boolean Hierarchy over NP and other classes defined by what we call bounded counting.

This paper shows that classical results about complexity classes involving “delayed diagonalization” and “gap languages”, such as Ladner's Theorem and Schoning's Theorem and independence results of a kind noted by Schoming and Hartmanis, apply at very low levels of complexity, indeed all the way down in Sipser's log-time hierarchy. This paper also investigates refinements of Sipser's classes and notions of log-time reductions, following on from recent work by Cai, Chen, and others.

The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First we define formally different notions of efficient enumeration in the context of parameterized complexity. Second we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems., Accepted for MFCS 2013; long version of the paper

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC 0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC 0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC 0 proof systems for a variety of languages ranging from regular to NP complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC 0 proof systems. We also show that Majority does not admit NC 0 proof systems. Finally, we present a general construction of NC 0 proof systems for regular languages with strongly connected NFA's.

There has been a great effort in giving machine-independent, algebraic characterizations of complexity classes, especially of functions. Astonishingly, no satisfactory characterization of the prominent class #P is known up to now. Here, we characterize #P as the closure of a set of simple arithmetical functions under summation and weak product. Based on that result, the hierarchy of counting functions, which is the closure of #P under substitution, is characterized, remarkably without using the operator of substitution, since we can show that in the context of this hierarchy the operation of modified subtraction is as powerful as substitution. This leads us to a number of consequences concerning closure of #P under certain arithmetical operations. Analogous results are achieved for the class Gap-P which is the closure of #P under subtraction.

We explore the potentially "off-by-one" nature of the definitions of counting (#P versus #NP), difference (DP versus DNP), and unambiguous (UP versus UNP; FewP versus FewNP) classes, and make suggestions as to logical approaches in each case. We discuss the strangely differing representations that oracle and predicate models give for counting classes, and we survey the properties of counting classes beyond #P. We ask whether subtracting a #P function from a P function it is no greater than necessarily yields a #P function.

We study Boolean circuits as a representation of Boolean functions and consider different equivalence, audit, and enumeration problems. For a number of restricted sets of gate types (bases) we obtain efficient algorithms, while for all other gate types we show these problems are at least NP-hard., 25 pages, 1 figure

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as $\SigmaPtwo$-complete, and the complexity of the credulous and skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete. Additionally, he investigated restrictions on the default rules, i.e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases., Corrected version

Autoepistemic logic extends propositional logic by the modal operator L . A formula φ that is preceded by an L is said to be “believed.” The logic was introduced by Moore in 1985 for modeling an ideally rational agent’s behavior and reasoning about his own beliefs. In this article we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of checking that a given set of formulae characterizes a stable expansion and that of counting the number of stable expansions of a given knowledge base. We improve the best known Δ 2 p -upper bound on the former problem to completeness for the second level of the Boolean hierarchy. To the best of our knowledge, this is the first paper analyzing counting problem for autoepistemic logic.

We investigate the application of Courcelle's Theorem and the logspace version of Elberfeld et al. in the context of the implication problem for propositional sets of formulae, the extension existence problem for default logic, as well as the expansion existence problem for autoepistemic logic and obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu), unless P=NP.

The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations---restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004). For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that, in cases where the combined complexity is of relevance, approaching CTL model checking by parallelism cannot be expected to result in any significant speedup. We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL+, and ECTL+.

Modal dependence logic was introduced recently by V\"a\"an\"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is determined by those of p_1,...,p_(n-1). Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man's dependence logic, the language consisting of formulas built from literals and dependence atoms using conjunction, necessity and possibility (i.e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend V\"a\"an\"anen's language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by V\"a\"an\"anen and Sevenster., Comment: 22 pages, full version of CSL 2010 paper

We study the extension of dependence logic $$\mathcal {D}$$D by a majority quantifier $$\mathsf{M}$$M over finite structures. We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities. Our results imply that, from the point of view of descriptive complexity theory, $$\mathcal {D}(\mathsf{M})$$D(M) captures the complexity class counting hierarchy. We also obtain characterizations of the individual levels of the counting hierarchy by fragments of $$\mathcal {D}(\mathsf{M})$$D(M).

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC0 proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC0 proof systems. We also present a general construction of NC0 proof systems for regular languages with strongly connected NFA's.

Seinosuke Toda introduced the class Mid P of functions that yield the middle element in the set of output values over all paths of nondeterministic polynomial time Turing machines. We define two related classes: Med P consists of those functions that yield the middle element in the ordered sequence of output values of nondeterministic polynomial time Turing machines (i.e. we take into account that elements may occur with multiplicities greater than one). [Formula: see text] P consists of those functions that yield the middle element of all accepting paths (in some resonable encoding) of nondeterministic polynomial time Turing machines. We exhibit similarities and differences between these classes and completely determine the inclusion structure between these classes and some other well-known classes of functions like Valiant’s # P and Köbler, Schöning, and Torán’s span-P, that hold under general accepted complexity theoretic assumptions such as the counting hierarchy does not collapse. Our results help in clarifying the status of Toda’s very important class Mid P in showing that it is closely related to the class PPNP .

The class NC1 of problems solvable by bounded fan-in circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC1 and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC1 and C=NC1, defined by a test for positivity and a test for zero, respectively, of arithmetic circuit families of logarithmic depth, sit in this complexity interval. We study the landscape of Boolean hierarchies, constant-depth oracle hierarchies, and logarithmic-depth oracle hierarchies over PNC1 and C=NC1. We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses--in particular, the constant-depth oracle hierarchy over PNC1 collapses to its first level PNC1, and the constant-depth oracle hierarchy over C=NC1 collapses to its second level.

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as $\Sigma^{\rm P}_2$-complete, and the complexity of the credulous and skeptical reasoning problem as $\Sigma^{\rm P}_2$-complete, resp. $\Pi^{\rm P}_2$-complete. Additionally, he investigated restrictions on the default rules, i. e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a trichotomy ($\Sigma^{\rm P}_2$-, NP-complete, trivial) for the extension existence problem, whereas for the credulous and sceptical reasoning problem we get a finer classification down to NL-complete cases.

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.

In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in P) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators., 27 pages, 7 figures

Imposing an extensional uniformity condition on a non-uniform circuit complexity class $\mathcal{C}$ means simply intersecting $\mathcal{C}$ with a uniform class $\mathcal{L}$. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining $\mathcal{C}$. We say that $(\mathcal{C},\mathcal{L})$ has the Uniformity Duality Propertyif the extensionally uniform class $\mathcal{C}\cap\mathcal{L}$ can be captured intensionally by means of adding so-called $\mathcal{L}$ -numerical predicatesto the first-order descriptive complexity apparatus describing the connection language of the circuit family defining $\mathcal{C}$. This paper exhibits positive instances and negative instances of the Uniformity Duality Property.

The propositional satisfiability problem SAT, i.e., the problemto decide, given a propositional formula φ (without loss ofgenerality in conjunctive normal form CNF), if there is anassignment to the variables in φ that satisfies φ, is thehistorically first and standard NP-complete problem [Coo71].However, there are well-known syntactic restrictions for whichsatisfiability is efficiently decidable, for example if everyclause in the CNF formula has at most two literals (2CNF formulas)or if every clause has at most one positive literal (Horn formulas)or at most one negative literal (dual Horn formulas), see [KL99].To study this phenomenon more generally, we study formulas with“clauses” of arbitrary shapes, i.e., consisting ofapplying arbitrary relations R ⊆ {0,1} k to (notnecessarily distinct) variables x 1,...,x k . A constraint languageΓ is a finite set of such relations. In the rest of thischapter, Γ and Γ′ will always denote Booleanconstraint languages. A Γ-formula is a conjunction of clausesR(x 1,...,x k ) as above using only relations R from Γ. Thefor us central family of algorithmic problems, parameterized by aconstraint language Γ, now is the problem to determinesatisfiability of a given Γ-formula, denoted byCsp(Γ).

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NP-complete, and identified all tractable cases. Schaefer's dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ≠ NP). We show that if one considers AC0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct).

Conjunctive-query containment is considered as a fundamental problem in database query evaluation and optimization. Kolaitis and Vardi pointed out that constraint satisfaction and conjunctive query containment are essentially the same problem. We study the Boolean conjunctive queries under a more detailed scope, where we investigate their counting problem by means of the algebraic approach through Galois theory, taking advantage of Post’s lattice. We prove a trichotomy theorem for the generalized conjunctive query counting problem, showing this way that, contrary to the corresponding decision problems, constraint satisfaction and conjunctive-query containment differ for other computational goals. We also study the audit problem for conjunctive queries asking whether there exists a frozen variable in a given query. This problem is important in databases supporting statistical queries. We derive a dichotomy theorem for this audit problem that sheds more light on audit applicability within database systems.

We consider the Boolean constraint isomorphism problem, that is, the problem of determining whether two sets of Boolean constraint applications can be made equivalent by renaming the variables. We show that depending on the set of allowed constraints, the problem is either coNP-hard and GI-hard, equivalent to graph isomorphism, or polynomial-time solvable. This establishes a complete classification of the complexity of the problem, and moreover, it identifies exactly all those cases in which Boolean constraint isomorphism is polynomial-time many-one equivalent to graph isomorphism, the best-known and best-examined isomorphism problem in theoretical computer science.

We define the counting classes NC/sup 1/, GapNC/sup 1/ PNC/sup 1/ and C/sub =/NC/sup 1/. We prove that Boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that NC/sup 1//spl sube/L and that C/sub =/NC/sup 1//spl sube/L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC/sup 0/ from MOD-PH as well as that of TC/sup 0/ from the counting hierarchy. Moreover we obtain that dlogtime-uniformity and logspace-uniformity for AC/sup 0/ coincide if and only if the polynomial time hierarchy equals PSPACE.

We discuss the use of operators of higher types in complexity theory. These are operators ranging over sets of words, i.e. over oracles. Depending on different oracle access mechanisms we consider two types of operators. In particular we examine existential, universal, and bounded-error probabilistic operators. We identify some of the emerging classes and we interpret recent results about interactive protocols in terms of these operators.

A Boolean constraint satisfaction instance is a set of constraint applications where the allowed constraints are drawn from a fixed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent and prove a dichotomy theorem by showing that for all finite sets C of constraints, this problem is either polynomial-time solvable or coNP-complete, and we give a simple criterion to determine which case holds. A more general problem addressed in this paper is the isomorphism problem, the problem of determining whether there exists a renaming of the variables that makes two given constraint satisfaction instances equivalent in the above sense. We prove that this problem is coNP-hard if the corresponding equivalence problem is coNP-hard, and polynomial-time many-one reducible to the graph isomorphism problem in all other cases.

We consider the following optimization problem for UMTS networks: For a specified teletraffic demand and possible base station locations, choose positions for base stations such that the construction costs are below a given limit, as much teletraffic as possible is supplied, the ongoing costs are minimal, and the intra-cell interference in the range of each base station is low. We prove that for a particular specification of teletraffic (the so called demand node concept), this problem has a polynomial-time approximation scheme, but cannot have a fully polynomial-time approximation scheme unless P = NP.

We examine the power of nondeterministic finite automata with acceptance of an input word defined by a leaf language, i.e., a condition on the sequence of leaves in the automaton's computation tree. We study leaf languages either taken from one of the classes of the Chomsky hierarchy, or taken from a time- or space-bounded complexity class. We contrast the obtained results with those known for leaf languages for Turing machines and Boolean circuits.

In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P (polynomial time) and PSPACE (polynomial space). It was shown that the separability of two complexity classes can be reduced to a combinatorial property of the corresponding defining leaf languages. In the present paper, it is shown that every separation obtained in this way holds for every generic oracle in the sense of Blum and Impagliazzo. We obtain several consequences of this result, regarding, e.g., simultaneous separations and universal oracles, resource-bounded genericity, and type-2 complexity.

This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straight-line programs which we call polynomial replacement systems. We contribute a classification of these systems and we investigate their complexity. Diverse problems falling in the scope of this study include, for example, counting proof circuits, and evaluating {∪, +}-circuits over the natural numbers. The former is shown #P-complete, the latter to be equivalent to a particular problem for replacement systems.

A language A ⊆ {0,1}* is called i.o. autoreducible if A is Turing-reducible to itself via a machine M such that, for infinitely many input words w, M does not query its oracle A about w. We examine the question if algorithmically random languages in the sense of Martin-Lof are i.o. autoreducible. We obtain the somewhat counterintuitive result that every algorithmically random language is polynomial-time i.o. autoreducible where the autoreducing machine poses its queries in a "quasi-nonadaptive" way; however, if in the above definition the "infinitely many" is replaced by "almost all," then every algorithmically random language is not autoreducible in this stronger sense. Further results obtained give upper and lower bounds on the number of queries of the autoreducing machine M and the number of inputs w for which M does not query the oracle about w.

First-order translations have recently been characterized as the maps computed by aperiodic single-valued nondeterministic finite transducers (NFTs). It is shown here that this characterization lifts to "V-translations" and "V-single-valued-NFTs", where V is an arbitrary monoid pseudovariety. More strikingly, 2-way V-machines are introduced, and the following three models are shown exactly equivalent to Eilenberg's classical notion of a bimachine when V is a group variety or when V is the variety of aperiodic monoids: V-translations, V-single-valued-NFTs and 2-way V-transducers.

In this chapter we want to examine a class which has turned out to be of immense importance in the theory of efficient parallel algorithms. What should be considered to be a “good” parallel algorithm? First the number of processors should certainly not be too high, i. e., still polynomial in the input length. Second, the algorithm should achieve a considerable speed-up compared to efficient sequential algorithms, which typically run in polynomial time for a polynomial of small degree. These requirements led researchers to consider the class NC of all problems for which there are circuit families that are simultaneously of polynomial size and polylogarithmic (i.e., (log n) o (1)) depth. By the results presented in Sect. 2.7 this corresponds to polynomial processor number and polylogarithmic time on parallel random access machines. NC is the class of problems solvable with very fast parallel algorithms using a reasonable amount of hardware.

A notion of generalized quantifier in computational complexity theory is explored and used to give a unified treatment of leaf language definability, oracle separations, type 2 operators, and circuits with monoidal gates. Relations to Lindstrom quantifiers are pointed out.

Suppose we are given two binary strings, each consisting of n bits, a = a n−1 a n−2 ... a 0, and b = b n−1 b n−2 ... b 0. We want to solve the following problem: Interpret a and b as binary representations of two natural numbers and compute their sum (again in binary). We refer to this problem as ADD.

The results of the previous chapter show that there is a circuit complexity class which contains every length-respecting function f : {0,1}* → {0,1}*, computable or not

In the previous chapters we studied mostly subclasses of the class NC. As we have stressed already, NC is meant to stand for problems with feasible highly parallel solutions (meaning very efficient runtime using a reasonable number of processors). In the theory of sequential algorithms, the class P = DTIME(n O (1)) is meant to capture the notion of problems with feasible sequential algorithms (meaning reasonable runtime on a sequential machine). Every feasible highly parallel solution can be used to obtain a feasible sequential solution (NC ⊆ P). The question whether every problem with a feasible sequential algorithm also has a feasible parallel algorithm was discussed in detail in Sect. 4.6.