The efficient certification of knottedness and Thurston norm
Introduction
How difficult is it to determine whether a given knot is the unknot? The answer is not known. There might be a polynomial-time algorithm, but so far, this has remained elusive. The complexity of the unknot recognition problem was shown to be in NP by Hass, Lagarias and Pippenger [10]. The main aim of this article is to establish that it is in co-NP. This can be stated equivalently in terms of the Knottedness decision problem, which asks whether a given knot diagram represents a non-trivial knot.
Theorem 1.1 Knottedness is in NP.
In some sense, this result is not new. It was first announced by Agol [1] in 2002, but he has not provided a full published proof. In 2011, Kuperberg gave an alternative proof of Theorem 1.1, but under the extra assumption that the Generalised Riemann Hypothesis is true [19]. In this paper, we provide the first full proof of the unconditional result.
Combined with the theorem of Hass, Lagarias and Pippenger [10], Theorem 1.1 gives the following corollary.
Corollary 1.2 If either of the decision problems Unknot recognition or Knottedness is NP-complete, then NP = co-NP.
This is because if any decision problem in co-NP is NP-complete, then the complexity classes NP and co-NP must be equal. Since this is widely viewed not to be the case (see Section 2.4.3 in [7] for example), then it seems very unlikely that either of these decision problems is NP-complete.
Decision problems that lie in both NP and co-NP are viewed as good potential candidates for being solvable in polynomial time. However, it is very probable that some decision problems in NP ∩ co-NP do not lie in P. A notable example is the problem of factorising an integer which, when suitably rephrased as a decision problem, is in NP ∩ co-NP. It is a very interesting consequence of Theorem 1.1 that Unknot recognition now lies in that list of problems lying in NP ∩ co-NP but that is not known to lie in P.
Our proof of Theorem 1.1 follows the outline given by Agol and, as in his argument, we establish more information about the genus of the knot. Recall that the genus of a knot K in a 3-manifold is the minimal possible genus of a Seifert surface for K, which is a compact orientable embedded surface, with no closed components and with boundary equal to K. When no such surface exists, the genus of K is defined to be infinite. We provide an algorithm to determine the genus of a knot in the 3-sphere. More specifically, we consider the decision problem Classical knot genus. This takes, as its input a knot diagram D and a positive integer g (given in binary), and it asks whether the knot specified by D has genus g. We establish the following result.
Theorem 1.3 Classical knot genus is in NP.
Theorem 1.1 is a consequence of Theorem 1.3 as follows. Given a diagram D of a non-trivial knot K, we need a certificate for its knottedness. Specifically, this certificate needs to be verifiable in polynomial time, as a function of the crossing number of D. Now, the genus g of K lies between 1 and . The certificate provided by Theorem 1.3 for this diagram D and this genus g is the required certificate for knottedness.
The term ‘classical’ is meant to refer to knots in the 3-sphere. In fact, Theorem 1.3 is unlikely to generalise to knots in arbitrary closed 3-manifolds. Indeed, there is good reason to believe that the problem of determining whether a knot in a closed 3-manifold has genus g is not in NP (when both the knot and the 3-manifold are permitted to vary), because of the following theorem, which is a well known consequence of work of Agol, Hass and Thurston [2].
Theorem 1.4 If Knot genus in compact orientable 3-manifolds is in NP, then NP = co-NP.
Here, the decision problem Knot genus in compact orientable 3-manifolds takes as its input a triangulation of a compact orientable 3-manifold X, a knot K in X given as a subcomplex of the 1-skeleton and a positive integer g (in binary), and it asks whether the minimal genus of a Seifert surface for K is g. We will give a short proof of this result in Section 16. As a result of Theorem 1.4, it is highly unlikely that Knot genus in compact orientable 3-manifolds is in NP.
The reason why knots in arbitrary compact orientable 3-manifolds appear to be significantly more complicated than those in the 3-sphere is that when M is the exterior of a knot in the 3-sphere, is cyclic, and so there is only one possible homology class (up to sign) that can be represented by a Seifert surface for K. However, when M is the exterior of a knot in a general compact 3-manifold, may have rank bigger than 1, and so one must consider many different homology classes that could support a genus-minimising Seifert surface. However, we will see shortly that if we restrict attention to a single homology class, then an NP algorithm is available.
Although Theorem 1.3 is phrased in terms of the genus of a knot, it is not really the genus of a surface S that is its main measure of complexity in this paper. Instead, Thurston complexity plays this role. Recall that, for a compact connected orientable surface S, equals . When S is a compact orientable surface with components , then . So, when S is a Seifert surface for a knot K and , then has minimal Thurston complexity in its class in if and only if S has minimal possible genus. However, the same is not true when K is a link with more than one component. In this case, K may have disconnected Seifert surfaces as well as connected ones.
When M is a compact orientable 3-manifold, and z is a class in , then the Thurston norm of z is the minimal Thurston complexity of a compact oriented surface representing z. Theorem 1.1, Theorem 1.3 are special cases of a more general result which allows one to efficiently determine the Thurston norm of a homology class. We define the Thurston norm of a homology class decision problem as follows. The input is a triangulation of a compact orientable 3-manifold M, a simplicial 1-cocycle ϕ representing an element in and a non-negative integer n. The problem asks whether the Thurston norm of the Poincaré dual to is n. The measure of complexity is the size of the input, which is, up to a bounded linear factor, equal to the sum of the number of the tetrahedra in the triangulation of M, the number of digits of the integer n in binary and the sum of the number of digits of , as e ranges over all edges of the triangulation.
The following is the main theorem of this paper.
Theorem 1.5 Thurston norm of a homology class is in NP.
Note that Theorem 1.3, and hence Theorem 1.1, follow quickly from Theorem 1.5. Given a diagram D for a knot K, one can easily build a triangulation for its exterior M in polynomial time, and where the number of tetrahedra is at most a linear function of the crossing number of D. One can also readily build a simplicial 1-cocycle ϕ representing a generator of , where the maximal value of for each edge e is at most a linear function of the crossing number. Setting , we can apply Theorem 1.5 and thereby obtain an NP algorithm to determine whether the genus of K is g.
Our methods can also be used to certify the irreducibility of a 3-manifold. The decision problem Irreducibility of a compact orientable 3-manifold with toroidal boundary and takes, as its input, a triangulation of such a 3-manifold (which is permitted to have empty boundary) and asks whether it is irreducible.
Theorem 1.6 The decision problem Irreducibility of a compact orientable 3-manifold with toroidal boundary and is in NP.
Since a compact orientable 3-manifold M is irreducible and has incompressible boundary if and only if its double is irreducible, we can use Theorem 1.6 to detect the incompressibility of ∂M. The decision problem Incompressible boundary takes as its input a triangulation of a compact orientable 3-manifold M and it asks whether ∂M is incompressible. This was shown to be in co-NP by Ivanov (Theorem 4 in [12]). We show that it is NP, and hence we have the following.
Theorem 1.7 Incompressible boundary is in NP ∩ co-NP.
Theorem 1.7 has a consequence for the generalisation of unknot recognition to 3-manifolds other than the 3-sphere. The decision problem Knottedness in 3-manifolds takes, as its input, a triangulation of a compact orientable 3-manifold M, and a knot K given as a subcomplex of the 1-skeleton, and it asks whether K is knotted. This just means that K does not bound an embedded disc. Since the knottedness of K is closely related to whether has compressible boundary, Theorem 1.7 can be used to establish the following result.
Theorem 1.8 Knottedness in 3-manifolds is in NP ∩ co-NP.
In order to prove our main result, Theorem 1.5, one needs a method for certifying efficiently the Thurston norm of a class in . By use of a doubling argument, we show in Section 14 that it suffices to consider the case where M is closed and irreducible. In fact, our methods work just as well when ∂M is toroidal. So we now suppose that M is a compact orientable irreducible 3-manifold with boundary a (possibly empty) collection of tori.
The method that both we and Agol use to certify the Thurston norm of a class in is based on work of Thurston [27] and Gabai [3]. Thurston showed that if S is a compact leaf of some taut foliation of M, then S minimises Thurston complexity in its class in . Gabai showed that, conversely, if S is a compact orientable incompressible surface that minimises Thurston complexity in its class and that intersects each component of ∂M in a (possibly empty) collection of coherently oriented essential curves, then there is some taut foliation of M in which S appears as a compact leaf. Gabai's construction used a type of hierarchy for M, known as a sutured manifold hierarchy. A sutured manifold structure on M is a decomposition of ∂M into two subsurfaces and , which meet along some simple closed curves γ. It is denoted . The surface is given a transverse orientation pointing into M and is transversely oriented outwards. When M is cut along a transversely oriented properly embedded surface S in general position with respect to γ, the new manifold inherits a sutured manifold structure. A sutured manifold hierarchy is a sequence of decompositions where is a collection of 3-balls, each of which intersects in a single simple closed curve, and which satisfies some mild extra conditions. Gabai used these hierarchies to construct taut foliations on the exteriors of many knots [4], [5], and was thereby able to determine their genus. It was Scharlemann [25] who realised that much of Gabai's theory could work without any reference to taut foliations; just the sutured manifold hierarchies are enough to determine the Thurston norm of a homology class. For example, it is straightforward to verify that the sequence of sutured manifold decompositions given in Fig. 1 forms a taut sutured manifold hierarchy, and hence the first decomposing surface minimises Thurston complexity in its homology class. In particular, the existence of this hierarchy proves the knot 52 is not the unknot.
It is sutured manifold hierarchies, such as the one in Fig. 1, that we (and Agol) use to certify the Thurston norm of a homology class. The existence of such a hierarchy was proved by Gabai, but crucially, our certificate needs to be verifiable in polynomial time, as a function of the size of the input. Essentially, the sutured manifold hierarchy needs to be efficiently describable. The key to Agol's proof was to achieve this by placing some such hierarchy into ‘normal’ form with respect to a given triangulation for M. This normalisation procedure was based on work on Gabai [6], who considered the related problem of normalising essential laminations. In the present paper, we follow a similar approach, but instead of using triangulations, we focus on handle structures. The machinery for placing a sutured manifold hierarchy into ‘normal’ form with respect to a handle structure was developed by the author in [21]. This predates Agol's announcement of Theorem 1.1, Theorem 1.3, Theorem 1.5, and plays a central role in this paper.
Normal surfaces that minimise Thurston complexity in their homology class were studied by Tollefson and Wang [28]. So the first thing that we do is use their theory to realise the first surface in the hierarchy as a normal surface. However, one difficulty with normal surfaces in triangulated 3-manifolds is that, when one cuts along them, the resulting 3-manifold does not naturally inherit a triangulation. So we dualise the given triangulation of the initial manifold M to form a handle structure . There is a well-established theory of normal surfaces in handle structures [8], [13]. The next 3-manifold in the hierarchy then inherits a handle structure . Unfortunately, this may be much more complicated than , in two different senses. For a start, may simply have many more handles than . This happens if intersects some handle of in many discs, and then this handle of is divided into many handles of . Fortunately, all but a bounded number of these handles will be very simple copies of , lying between two parallel normal discs. These product regions patch together to form an I-bundle in , known as its ‘parallelity bundle’. It was shown in [21] how this parallelity bundle may be removed, primarily by decomposing along the annuli that form its intersection with the remainder of . So, a key part of our argument is the analysis of this I-bundle and an algorithmic method of removing it from .
However, there is another reason why may be more complicated than . It is not at all clear that the local structure of the 0-handles of is simpler than that of . This issue was faced right at the very first use of hierarchies by Haken [8] and Waldhausen [29]. They defined a notion of complexity for a handle structure of a 3-manifold, and showed that, when it is decomposed along a normal surface, then the complexity does not go up. Unfortunately, their notion of complexity does not fit well with sutured manifolds, primarily because it does not take account of the sutures. Fortunately, this issue was resolved in [21]. A variation of normal form more suited to sutured manifolds was introduced. In this paper, we call such surfaces ‘regulated’. Also, a notion of complexity for a handle structure of a sutured manifold was defined in [21]. It was also shown that this does not go up when the manifold is decomposed along a regulated surface. Moreover, it was shown that sutured manifold hierarchies can always be found where each decomposing surface is regulated.
Therefore, it is regulated surfaces that are used in this paper. Unfortunately, they come with their own complications. Although it is the case that we may arrange for the decomposing surfaces to be regulated, they may fail to satisfy one of the key technical requirements for a sutured manifold hierarchy. Some curves of intersection with the surface may be simple closed curves bounding discs in . Such curves are called ‘trivial’. (See Remark 5.6 for an explanation for why it seems hard to arrange that regulated decomposing surfaces have non-trivial boundary curves.) Because surfaces with trivial boundary components are not permitted to be part of a sutured manifold hierarchy, it is not clear that they can be used as part of a certificate for Thurston norm. However, we develop a theory of ‘allowable hierarchies’, where the decomposing surfaces may have trivial boundary curves, but which can nonetheless be used to certify Thurston norm.
So, allowable hierarchies of regulated surfaces will be used as part of our certificate. But these surfaces must be describable in an efficient way. In particular, it is important that our surfaces intersect each handle in at most discs, where c is a universal constant and h is the number of handles in the initial handle structure. To be able to establish such a bound, we use methods from linear algebra, that go back to Haken [8]. We show how regulated surfaces S can be described by means of a solution (S) to a system of linear equations, much in the same way that normal surfaces can be. When S, and are regulated surfaces and , then we say that S is a ‘sum’ of and . Just as in the normal surface case, one can place and into general position, and then obtain S by resolving the arcs and curves of intersection. It is crucial for our purposes that we may find a decomposing surface that is ‘fundamental’, which means that it cannot then be expressed as a sum of other non-trivial surfaces. This is because fundamental surfaces have a bounded number of discs of intersection with each handle, by methods that go back to Hass and Lagarias [9].
Therefore, we must analyse the case when S is a sum of surfaces and . Now, and need not inherit orientations from S, and indeed they need not even be orientable. But when they do inherit orientations, then the situation is fairly easy to analyse. It turns out that we can generally show that decomposition along or is taut, and so we can decompose along one of these instead. These are ‘simpler’ surfaces, and so in this way, one may arrange for decomposing surfaces to be fundamental. When and do not inherit transverse orientations, then the aim is to show that S was not as simple as it could have been. One can perform an ‘irregular switch’ along one of the arcs or curves of , creating a new transversely oriented surface . We show that decomposition along is also taut. This is possible when the irregular switch takes place along a curve of . However, the argument does not work in the case of an arc of , because the orientations of at its endpoints may be problematic. Fortunately, regulated surfaces rescue us here, because their boundaries are very tightly controlled, and in fact, it is possible to show that the regular switch along arcs of always respects the orientation of the surface.
Thus, we may arrange that the decomposing surfaces are fundamental, and hence have an exponential bound on complexity. But decomposing along these surfaces is not straightforward, because we never want to deal with handle structures having exponentially many handles. They are too unwieldy to be efficiently describable within our certificate. Fortunately, there is technology due to Agol, Hass and Thurston [2] which is applicable. Using their methods, we show that, given a surface with an exponential bound on its complexity, it is possible to determine the topological types of the components of the parallelity bundle for the manifold obtained by decomposing along . So, we never need to construct the handle structure for . Instead, we can go directly to the handle structure for the manifold that is obtained by removing the parallelity bundle.
Thus, our certificate for Thurston norm is comprised (primarily) of the following pieces of information: handle structures for a sequence of 3-manifolds, and regulated surfaces within these manifolds, expressed as solutions to a system of equations. We verify this certificate by checking that the next manifold is indeed obtained from the previous one by cutting along the regulated surface and then purging the resulting manifold of its parallelity bundle.
The strategy of our proof, as explained in Section 1.1, is very similar to the one outlined by Agol [1]. However, the details, as given in Section 1.2, are very different. Instead of using handle structures, Agol used triangulations. Since the result of decomposing a triangulation along a normal surface is not in general a triangulation, Agol had to work hard to build a triangulation for each manifold in the hierarchy. His technique was based on placing the surface into ‘spun’ normal form. The method of doing this was based on Gabai's method for normalising taut foliations, as explained in [6], which used ‘sutured manifold evacuation’. In Agol's argument, the surface still needed to be made ‘fundamental’ in some way. Moreover, an analogue of the removal of parallelity bundles would still need to have been achieved, again by the use of the algorithm of Agol, Hass and Thurston [2]. Agol's strategy certainly has some advantages, but it seems to us that the extensive use of the established techniques from [21], as followed in this paper, is also very convenient.
In Section 2, we recall some of the basic theory of sutured manifolds. In Section 3, we introduce ‘decorated’ sutured manifolds and ‘allowable’ hierarchies. The key result here is Theorem 3.1 which implies that allowable hierarchies can be used certify Thurston norm. In Section 4, we show that if a surface extends to an allowable hierarchy, then certain modifications can be made to it, and the resulting surface still extends to an allowable hierarchy. In Section 5, we introduce handle structures for sutured manifolds, and regulated surfaces. We also introduce the complexity of a handle structure, and explain how it behaves when a decomposition along a regulated surface is performed. In Section 6, we give some simplifications that can be made to a handle structure. We also explain how we need only to work with a finite universal list of 0-handle types. In Section 7, we develop an algebraic theory of regulated surfaces. We introduce a minor variant, known as ‘boundary-regulated’ surfaces, and show how they may expressed as solutions to certain equations. We also explain how summation of boundary-regulated surfaces can be interpreted topologically. Finally, we show that decompositions can always be made along fundamental regulated surfaces. This central result is Theorem 7.9. In Section 8, we bound the complexity of fundamental surfaces, using tools from linear algebra. In Section 9, we recall an algorithm of Agol, Hass and Thurston [2], and show how it can be used to determine the parallelity bundle for the manifold obtained by decomposing along a surface. In Section 10, we show to remove this parallelity bundle algorithmically. In Section 11, we show how it suffices to focus our attention on 3-manifolds that are irreducible and atoroidal. This convenient hypothesis occurs at several points in the preceding argument. In Section 12, we show how to certify efficiently that a sutured manifold is a product. This is useful because the hierarchies that we use terminate in products rather than 3-balls, for mostly technical reasons. We then go on to prove the main theorem in the special case where the 3-manifold is Seifert fibred. In Section 13, we complete the proof of Theorem 1.5 in the case of compact orientable irreducible 3-manifolds with (possibly empty) toroidal boundary. We describe in detail the certificate for Thurston norm, we show why it always exists, and how it may be verified in polynomial time. In Section 14, we show how to deal with the general case of 3-manifolds that might be reducible or have non-toroidal boundary components. In Section 15, we give the proofs of Theorem 1.7, Theorem 1.8. In Section 16, we provide a proof of Theorem 1.4.
Acknowledgements
The author would like to thank the referee for many helpful suggestions which substantially improved the paper. In particular, the referee suggested that one should double manifolds with non-toroidal boundary components. This led to the versions of Theorem 1.5, Theorem 1.7, Theorem 1.8 that appear in this paper, and that are improvements over the original versions. The author would also like to thank Mehdi Yazdi for helpful conversations regarding the some of the material in Section 11.
Section snippets
Sutured manifolds
A sutured manifold is a compact orientable 3-manifold M, with its boundary decomposed into two compact subsurfaces and , in such a way that is a collection of simple closed curves γ. These curves are called sutures. The surfaces and are assigned transverse orientations, with pointing into M and pointing outwards. The sutured manifold is usually denoted .
A compact oriented surface S embedded in a 3-manifold M, with ∂S in ∂M, is called taut if S
Decorated sutured manifolds
Unfortunately, we must consider decompositions where some curves of ∂S bound discs in ∂M disjoint from γ. This leads to some complications, because Theorem 2.3 does not apply, and so it is not obvious that S can be used as part of a certificate for Thurston norm. To get around this problem, we keep track of the curves of ∂S that bound discs in ∂M disjoint from γ. These give rise to ‘special’ sutures of . We want to be able to distinguish these sutures, and so we now introduce a
Surfaces that extend to an allowable hierarchy
In this section, we will consider surfaces S that form the first surface in an allowable hierarchy. We will show that certain modifications can be made to S that preserve this property.
Handle structures and their complexity
In this paper, handle structures on 3-manifolds will play a central role. If is a handle structure, then denotes the union of the i-handles. We will always insist that our handle structures satisfy the following requirements:
- (i)
The intersection between each i-handle and is .
- (ii)
Any two i-handles are disjoint.
- (iii)
The intersection between any 2-handle and any 1-handle is in the 2-handle, where α is a collection of disjoint arcs in , and of the form in the
Simplification modifications
According to Theorem 5.5, if a taut sutured manifold admits an allowable hierarchy, then one may find such a hierarchy where the first decomposing surface is regulated. However, the theorem has various technical hypotheses, including that the initial handle structure must be positive. Recall from Section 5.1 that this means that each 0-handle of must have positive index and, for each 0-handle of , is connected. In this subsection, we explain some of the modifications that one
Boundary-regulated surfaces
A key feature of normal surfaces is that they can be encoded by a finite list of numbers, which form a vector. This introduction of linear algebra is very useful. Unfortunately, it does not seem to be possible to create such a theory for regulated surfaces. The main difficulty is that regulated surfaces are required to have a transverse orientation, which is hard to incorporate algebraically.
Therefore, in this section, we introduce a new type of surface, which we call boundary-regulated. These
Bounding the complexity of normal and boundary-regulated surfaces
We have been encoding normal surfaces and boundary-regulated surfaces using vectors, which count the number of elementary discs of each type in the surface. In this section, we find upper bounds on the weights of these surfaces, by exploiting algebraic methods. Much of the material here is fairly well known, and goes back to Haken [8] and Hass-Lagarias [9].
We consider the following general set-up. Suppose that A is an matrix with integer entries. We will examine solutions to the system
Determining the components of the parallelity bundle
We saw in Theorem 7.9 that when is a taut sutured manifold with a positive handle structure satisfying some natural conditions, then a taut sutured manifold decomposition may be performed along a regulated surface S that is fundamental as a boundary-regulated surface. Hence, by Theorem 8.2, there is an upper bound to the number of elementary discs of S, that is an exponential function of the number of 0-handles of . Examples due to Hass, Snoeyink and Thurston [11] demonstrate that one
An algorithm to simplify handle structures
In Theorem 9.2, we showed that, when a taut sutured manifold decomposition is performed along a regulated surface, the components and structure of the parallelity bundle of the resulting sutured manifold may be efficiently determined. In this section, we utilise this information to produce a simplified handle structure for a sutured manifold obtained from .
Theorem 10.1 Let be a positive handle structure of a connected decorated sutured manifold , of uniform type, with h handles. Let S
Reduction to the atoroidal and Seifert fibred cases
At certain points in the previous sections, we made the assumption that the manifold is irreducible and atoroidal. In this section, we show why it suffices to consider this case and also the case where the manifold is Seifert fibred.
Products
Throughout this paper, we have used allowable hierarchies, which terminate in product sutured manifolds. It will be important that we can certify that a given sutured manifold is indeed a product. The existence of such a certificate is presented in this subsection.
Theorem 12.1 Let be a product sutured manifold with handle structure of uniform type. Let h be the number of 0-handles of . Suppose that is a product sutured manifold. Then there is a certificate that proves that is indeed a
The certificate and its verification
In this section, we complete the proof of Theorem 1.5, and of Theorem 1.1, Theorem 1.3, in the case of compact orientable irreducible 3-manifolds with (possibly empty) toroidal boundary. We describe the NP algorithm for determining the Thurston norm of a homology class. We also give the proof of Theorem 1.6 by showing how to certify that a compact orientable irreducible 3-manifold with toroidal boundary and positive first Betti number is indeed irreducible.
The case of closed irreducible 3-manifolds
In this section, we show that, in the proof of Theorem 1.5, it suffices to focus on the case of closed orientable irreducible 3-manifolds. In conjunction with the argument in the previous section, this will complete the proof of Theorem 1.5. We define the decision problem Thurston norm in closed orientable irreducible 3-manifolds to be the restriction of Thurston norm of a homology class to closed orientable irreducible 3-manifolds.
Theorem 14.1 Suppose that Thurston norm in closed orientable irreducible
A certificate for incompressibility
Theorem 1.7 Incompressible boundary is in NP ∩ co-NP.
Proof It was shown by Ivanov (Theorem 4 in [12]) that Incompressible boundary is in co-NP. So we focus on showing that it is in NP. Let be a triangulation of a compact orientable 3-manifold M with t tetrahedra. Our certificate for establishing the incompressibility of ∂M is as follows: a vector (S) of a normal surface S in with weight at most ; this will in fact be a collection of disjoint embedded 2-spheres; a triangulation of a 3-manifold with
The genus of knots in 3-manifolds
In this section, we give a proof of Theorem 1.4.
Theorem 1.4 If Knot genus in compact orientable 3-manifolds in is NP, then NP = co-NP.
This is a well known consequence of work of Agol, Hass and Thurston [2], but we give a proof here because it does not appear to have been put into print before, and because of its great relevance to the theme of this paper.
Recall that co-NP consists of the class of decision problems for which a negative answer can be certified in polynomial time. Since in practice, there is
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