Solving Monotone Polynomial Equations

Transcription

1 Solving Monotone Polynomial Equations Javier Esparza, Stean Kieer, and Michael Luttenberger Institut ür Inormatik, Technische Universität München, Garching, Germany Abstract. We survey some recent results on iterative methods or approximating the least solution o a system o monotone ixed-point polynomial equations. Introduction Consider the ollowing problem ormulated by Francis Galton in the (politically incorrect) 9th century [26], and quoted by Thomas Harris in his classical text on branching stochastic processes [20]: Let p 0, p, p 2...be the respective probabilities that a man has 0,, 2,... sons, let each son have the same probability or sons o his own, and so on. What is the probability that the male line is extinct ater r generations, and more generally what is the probability or any given number o descendants in the male line in any given generation? We are interested here in the probability that the male line eventually becomes extinct. A little thought shows that this probability is a solution o the ixed-point equation X = p n X n () n 0 and ater some more thought one concludes that it is in act the smallest solution. Consider now the ollowing stochastic context-ree grammar (i.e., a grammar whose productions are annotated with probabilities) with axiom X: X 0.4 XY, X 0.6 a Y 0.3 XY, Y 0.4 YZ, Y 0.3 b Z 0.3 XZ, Z 0.7 b What is the probability that the grammar eventually generates a word, i.e., a string o non-terminals? Again, it is not to diicult to show that it is equal to the X-component o the least solution o the ollowing system o equations. X = 0.4XY Y = 0.3XY + 0.4YZ+ 0.3 (2) Z = 0.3XZ

2 286 J. Esparza, S. Kieer, M. Luttenberger Notice that the vector (,,) is a solution o the system. We will later investigate whether it is the least solution or not. Equations () and (2) are two examples o monotone systems o polynomial equations (MSPEs or short). MSPEs are systems o the orm X = (X,...,X n ). X n = n (X,...,X n ) where,..., n are polynomials with positive real coeicients. In vector orm we denote an MSPE by X = (X). We call the vector (X) o polynomials a monotone system o polynomials, or MSP. Obviously, a solution o X = (X) is a ixed-point o (X), and vice versa. Further, any solution o X = (X) can be visualized as a point o intersection o the submaniolds deined by the n implicit unctions i (X) X i = 0. In particular, when the polynomials o (X) are quadratic the solutions o X = (X) correspond to the intersection o n quadrics. Figure shows the graph o such a quadratic MSPE with n = X X = (X,X 2 ) µ X 2 = 2 (X,X 2 ) X Fig. Graphs o the equations X = (X,X 2 ) and X 2 = 2 (X,X 2 ) with (X,X 2 )=X X and 2 (X,X 2 )= 6 X2 + 9 X X X There are two real solutions in R2 [0, ], the least one is labelled with µ. We call MSPEs and MSPs monotone because x x implies (x) (x ) or every x,x R n 0. This is a bit imprecise, because not every monotone polynomial has positive coeicients. Perhaps positive systems would be a better name, but since we have used the term monotone in several papers we stick to it. MSPEs appear naturally in the analysis o many stochastic models, such as stochastic context-ree grammars (with numerous applications to natural language processing

3 Solving Monotone Polynomial Equations 287 [23, 9], and computational biology [24, 5, 4, 22]), probabilistic programs with procedures [9, 2, 3,, 0, 2, 4], web-suring models with back buttons [6, 7], and branching processes [20], a topic in stochastic theory that can be traced back to Galton s problem. In the last years Etessami and Yannakakis [3] and ourselves [2, 8] have studied the problem o solving MSPEs. This paper gives a succinct and inormal overview o our results. 2 Some Deinitions and Facts Let R [0, ] denote the set o non-negative reals extended with. We extend the deinitions o sum and product as usual: + k = or every k R [0, ], 0 = 0, and k = or every k R [0, ] \{0}. The resulting algebraic structure is the real semiring. MSPEs are systems o ixed-point equations over the real semiring. Given two vectors u,v R n [0, ], we say that u v holds i u i v i holds or every i n, where u i,v i are the i-th components o u and v, respectively. This is the pointwise order on vectors o reals. The irst positive result on MSPEs is a direct consequence o Kleene s theorem: Theorem (Kleene s ixed-point theorem). Every MSP (X) has a least ixed-point µ in R n [0, ] with respect to the pointwise order. Moreover, the sequence (κ (k) ) k N given by κ (0) := 0 κ (k+) := (κ (k) )= k+ (0) is non-decreasing with respect to (i.e., κ (k) κ (k+) ) and converges to µ. We call (κ (k) ) k N the Kleene sequence, and its elements the Kleene approximants o µ. Example. For the system (2) we obtain: κ (0) =(0,0,0), κ () =(0.6, 0.3, 0.7), κ (2) =(0.672,0.438,0.826), κ (3) =(0.78, 0.533, 0.867), κ (4) =(0.753,0.600,0.887), κ (5) =(0.78, 0.648, 0.900),... The least solution o a system o linear equations (monotone or not) satisies some good properties that no longer hold or MSPEs. It is easy to show (using or instance Cramer s rule) that i the coeicients are rationals, then the least solution is also rational. However, using Galois theory one can prove that the least solution o a polynomial system may not be expressible by radicals. For instance: Fact. The least ixed-point o

4 288 J. Esparza, S. Kieer, M. Luttenberger is not expressible by radicals. X = 6 X X (3) This act also holds or quadratic systems, i.e., systems in which all polynomials have at most degree 2. Given an MSP over a set X o variables, it is easy to construct a quadratic MSP g over a larger set X Y such that the projection o µg onto X is equal to µ. The construction is very similar to the one that brings a context-ree grammar in Chomsky normal orm. For instance, it expands Equation (3) into the system X = 6 XX XX X n = XX n (or n = 5,4,3) X 2 = X 2 Since this expansion involves only a linear blowup, we can take quadratic MSPEs as a normal orm o MSPEs. The least solution o linear MSPEs is not only rational, but a succinct rational. Consider a system o dimension n (i.e., with n equations) whose coeicients are given as ratios o m-bit integers. It is easy to show using Cramer s rule that the least solution can be written as the quotient o two natural numbers with at most O(n 2 m + nlogn) bits. As a consequence, we get Fact 2. Let X = (X) be a linear MSPE o dimension n whose coeicients are given as ratios o m-bit integers. For every component µ i o the least ixed-point o : i 0 < µ i < then 2 O(n2 m+nlogn) µ i 2 O(n2 m+nlogn) (where the constant o the Big-Oh notation is independent o ). Since the least ixed-point o a MSP can be irrational, there is no bound on the number o digits needed to write it down. However, using results o [8] we can still give a lower and an upper bound: Fact 3. Let be a quadratic MSP o o dimension n whose coeicients are given as ratios o m-bit integers. For every component µ i o the least ixed-point o : i 0 < µ i < then 2 m 2O(n) µ i 2 m 2O(n) (where the constant o the Big-Oh notation is independent o ). So, loosely speaking, while the least ixed-point o a linear system is at most exponential in the dimension o the system, the least solution o a quadratic system is at most double exponential. It is easy to ind examples o quadratic MSPs in which the least ixed-point is rational and double exponential. The n-th component o the least solution o system

5 Solving Monotone Polynomial Equations 289 is equal to k 2(n ). X = k X 2 = X 2. X n = X 2 n 3 Computational complexity The undamental decision problem or MSPs is whether (µ ) i a holds or a given MSP and a component i, where a is some positive rational number and {,=, }. Let us call this problem MSP-DECISION. Little is known about its computational complexity. The problem lies in PSPACE: Consider e.g. a two-dimensional MSPE X = (X,X 2 ),X 2 = 2 (X,X 2 ). To decide whether (µ ) a holds one can equivalently decide i the ollowing ormula is true: x R,x 2 R : x = (x,x 2 ) x 2 = 2 (x,x 2 ) x,x 2 0 x a Such ormulas can be decided in PSPACE, because the irst-order theory o the reals is decidable, and its existential ragment is even in PSPACE [3]. For a lower bound, we introduce the problem SQUARE-ROOT-SUM: Given k + natural numbers n,...,n k and b, determine whether k i= ni b holds. The SQUARE-ROOT-PROBLEM is a natural subproblem o many questions in computational geometry. For instance, the length o the boundary o a polygon whose vertices lie in Z 2 is a sum o square roots o integers. It has been a major open problem since the 70s whether SQUARE-ROOT-SUM belongs to NP. The problem can easily be reduced to MSP-DECISION: Suppose we are given n = 2, n 2 = 3, and b = 3, and we want to decide i One would like to come up with an MSP (X) such that (µ ) = 2,(µ ) 2 = 3,(µ ) 3 = 2 + 3, so that deciding is equivalent to deciding (µ ) 3 3. One has to be careul though, because or instance the equation X = X 2 + X 2 is not an MSPE. It was shown in [3] how to overcome this problem: Instead o encoding e.g. 2 directly, it suices to encode a + b 2 or some rationals a,b. The least solution o the equation X = X 2 +( λ 2 n)/4 equals ( λ n)/2. So, by choosing or λ a small enough rational number we get a -dimensional MSP whose least solution is a + b n or some rationals a,b. In our example we can set λ = max(2,3) = 3 which leads to the ollowing MSPE.

6 290 J. Esparza, S. Kieer, M. Luttenberger Its least solution is µ = X = X X 2 = X X 3 = X + X 2 ( 2 2, 6 2 ( ) ) 3, So, the question whether holds can be translated into the question whether (µ ) = 2 holds. It ollows rom this reduction that proving membership o MSP-DECISION in NP would be a major breakthrough. An interesting issue is the complexity o MSP-DECISION in the Blum-Shub-Smale computational model, in which all operations on rationals take unit time independently o their size. SQUARE-ROOT-SUM can be decided in polynomial time in this model [25], but it is open whether the result extends to MSP-DECISION. 4 Approximating the Least Fixed-Point: Newton s Method For most practical purposes, the main computational problem concerning MSPs is the approximation o the least ixed-point up to a given accuracy. Kleene s method can be applied, and it is very robust: it always converges when started at 0, or any MSP. On the other hand, the convergence speed o the Kleene sequence can be very poor. Beore presenting an example, we deine a notion o convergence order that diers rom the one commonly used in numerical mathematics, but is particularly natural or computer science. Let (a k ) k 0 be a non-decreasing sequence o vectors over the real semiring such that lim k a k = a <. The convergence order o the sequence is the unction β : N N deined as ollows: β (k) is the greatest natural number i such that a a k a 2 i where is some norm. We say that a sequence has linear, exponential, logarithmic, etc. convergence order i the unction β (k) grows linearly, exponentially, or logarithmically in k, respectively. Notice that the asymptotic behaviour o β (k) is independent o the norm, because all norms are equivalent up to a constant. In the univariate case, β (k) is the number o bits o a k that coincide with the corresponding bits o a (the ormalization o this intuition requires some care, like identiying and ). For instance, or the sequence ( 2 k ) k 0 we have β (k) =k, i.e., the irst k bits o the k-th element o the sequence coincide with the irst k bits o the limit.

7 Solving Monotone Polynomial Equations 29 Consider now this very simple but at the same time very illustrative quadratic MSPE in one variable: X = /2 + /2X 2 (4) In Galton s problem, the least solution o this equation gives the extinction probability o an individual s descent line when every individual has 0 or 2 children with probability /2. The least solution is. We have: Fact 4. The i-th Kleene approximant o X = /2 + /2X 2 satisies κ (i) i+ or every i 0. So the Kleene sequence only has logarithmic convergence order. Example 2. Here are some o the Kleene iterates. κ (0) = 0, κ () = 0.5, κ (2) = κ (3) = 0.695, κ (4) = 0.742, κ (5) = κ (20) = 0.920,...,κ (200) = 0.990,...,κ (2000) = ,... Faster approximation techniques have been known or a long time. In particular, Newton s method, suggested by Isaac Newton more than 300 years ago, is a standard eicient technique or approximating a zero o a dierentiable unction. Since the least solution o a ixed-point equation X = (X) is a zero o g(x)= (X) X, the method can be applied to search or ixed-points o (X) g(x) ν (0) ν () ν (2) X 0.8 Fig. 2 Newton s method to ind a zero o a one-dimensional unction g(x) We briely recall the method or the case o one variable, see Fig. 2 or an illustration. Starting at some value ν (0) close enough to the zero o g(x), we proceed iteratively: given ν (i), we compute a value ν (i+) closer to the zero than ν (i). For that, we compute the tangent to g(x) passing through the point (ν (i),g(ν (i) )), and take ν (i+)

8 292 J. Esparza, S. Kieer, M. Luttenberger as the zero o the tangent (i.e., the X-coordinate o the point at which the tangent cuts the X-axis). A little arithmetic leads to: ν (i+) = ν (i) + (ν(i) ) ν (i) (ν (i) ) Newton s method can be easily generalized to the multivariate case: ν (i+) = ν (i) +(Id (ν (i) )) ( (ν (i) ) ν (i) ) where (X) is the Jacobian o, i.e., the matrix o partial derivatives o, and Id is the identity matrix. Notice that Newton s method is not restricted to the real semiring, it can be applied to any dierentiable unctions over the real ield. However, when applied with this generality it is ar less robust than Kleene s method: it may converge very slowly, converge only when started at a point very close to the zero which must be guessed or even not converge at all. However, i we apply Newton s method to (X)=/2+/2X 2, starting at ν (0) = 0, we obtain: Fact 5. The i-th Newton approximant o X = /2 + /2X 2 satisies ν (i) = or 2 every i 0. The i-th approximant has i correct bits, i.e., the Newton sequence i has linear convergence. So in this particular example the Newton sequence converges exponentially aster than the Kleene sequence. The number o arithmetic operations needed to compute i correct bits o the solution grows polynomially instead o exponentially in i. (Recall, however, that the operations have to be applied to rationals whose length may grow exponentially in the number o iterations.) One can ask whether the good behaviour on this example is just a coincidence, or whether perhaps Newton s method is robust on the real semiring. A number o recent results have shown that (with certain is and buts) the latter is the case, and we briely survey them in the next section. 5 Convergence Order and Thresholds or Newton s Method The irst positive result on the convergence o Newton s method was obtained by Etessami and Yannakakis in [3]. They showed that the method always converges to the least ixed-point starting rom ν (0) = 0, and that it converges at least as ast at the Kleene sequence. Inspired by this positive result, we started to study the convergence order. Given an MSPE X = (X) whose least solution µ is inite, it is well-known that the convergence order depends critically on the Jacobian matrix at the least ixed-point, i.e., on More precisely, Etessami and Yannakakis proved the result or a structured version o the method, and we showed in [8] that this additional structure is not required or convergence (although it is convenient or eiciency).

9 Solving Monotone Polynomial Equations 293 (µ ). Every textbook proves that the method perorms brilliantly when the matrix (Id (µ )) is non-singular: it exhibits exponential convergence order. So we ocused our attention on the singular case, o which (X)=/2 + /2X 2 is an example. By Fact 5 we can expect at most linear convergence. But perhaps the method converges more slowly on other examples? It is convenient to start with the special case o strongly connected MSPEs. Loosely speaking, an MSPE is strongly connected i every variable depends on any other variable, where dependence is deined as ollows. Given two variables X and Y, X depends on Y i either Y appears on the right-hand-side o the equation or X, or i there is a variable Z such that X depends on Z and Z depends on Y. 5. Strongly Connected MSPEs We proved the ollowing theorem in [2]. Theorem 2. Let (X) be a strongly connected MSP such that µ is inite. There is a number t such that or every i 0: β (t + i) i. In particular, the Newton sequence has linear convergence order. We call t the threshold o (X). Loosely speaking, the theorem states that ater crossing the threshold (i.e., rom the t -th approximant onwards) the Newton sequence gains at least one bit o accuracy per iteration. The threshold itsel is an upper bound on the number o iterations needed to obtain the irst bit o the least ixed-point. The proo o [2] was based on the ollowing topological property o R n : i the inimum o the distances between points o two sets is 0, then the two sets have at least one common point. As a consequence, it was a purely existential proo, and provided no inormation on the size o the threshold. In [7] we obtained the ollowing relation between the threshold and the minimal component o µ. Theorem 3. Let (X) be a quadratic strongly connected MSP o dimension n whose coeicients are given as ratios o m-bit integers. Let µ min be the minimal component o µ. The threshold t o Theorem 2 satisies t 3n 2 m + 2n 2 logµ min. Moreover, i i (0) > 0 holds or every i n, then t 3mn. Example 3. Consider again the ollowing MSPE, which was given as Equation (2) on page 285. X = 0.4XY Y = 0.3XY + 0.4YZ+ 0.3 Z = 0.3XZ+ 0.7

10 294 J. Esparza, S. Kieer, M. Luttenberger Using a result rom [8], slightly stronger than Theorem 3 but technically more diicult to state, one can prove that the threshold o this system satisies t 6 or the maximum-norm (i.e., the norm o a vector is the absolute value o its maximal component). So β (4) 8. Ater computing 4 Newton iterates we get ν (4) = (0.983,0.974,0.993). As we have computed at least β (4) 8 bits, we know that µ is at most ν (4) +(2 8,2 8,2 8 ) which is strictly less than in every component. Thereore, the stochastic context-ree grammar rom the introduction produces a terminal string with probability less than. Combining Theorem 3 with Fact 3 we obtain: Corollary. Let X = (X) be a quadratic strongly connected MSPE o dimension n whose coeicients are given as ratios o m-bit integers. The threshold t o Theorem 2 satisies t m2 O(n). This corollary gives an exponential bound on the number o iterations needed to compute the irst bit o the least ixed-point. It is open whether this bound is tight. 5.2 General MSPEs The ollowing example shows that an exponential number o iterations is sometimes needed or the irst bit, i the MSPE is not strongly connected. We give a amily o MSPEs in which the number o iterations needed to compute the irst bit grows exponentially in the dimension o the system. Example 4. Consider the ollowing amily o MSPEs. X = /2 + /2 X 2 X 2 = /4 X 2 + /2 X X 2 + /4 X 2 2. (5) X n = /4 X 2 n + /2 X n X n + /4 X 2 n The variable X i depends on X j i and only i j i. So the dependence graph contains n strongly connected components, one or each variable. The least ixed-point o the system is the vector (,,...,). We show in [2] that ν (2n ) n /2 holds, and so that at least 2 n iterations o Newton s method are needed to obtain the irst bit o X n. The proo goes as ollows. We consider a decomposed version o Newton s method, in which or a given k we perorm k iterations o the normal method on the irst equation, yielding a lower bound a (k) o the irst component o the least ixed-point. Then we perorm k iterations on the second equation ater setting X := a (k) ; by monotonicity, this yields a lower bound a (k) 2 o the second component. Repeating this procedure we inally obtain a lower bound a (k) n o the n-th component. It is easy to see that ν (k) i

11 Solving Monotone Polynomial Equations 295 holds, i.e, the decomposed method converges at least as ast as the method that perorms k iterations on the whole system. Now, let δ (k) i = a (k) i be the error o the decomposed method. A simple analysis reveals that δ (k) a (k) i i+ δ (k) i holds or every i < n. By Fact 5 we have δ (2n ) =(/2) 2n, and so we get δ (2n ) n /2, i.e., ν (2n ) n /2. So, intuitively, the problem o non-strongly connected systems is that the error gets ampliied when we move up the graph o strongly connected components. For MSPs that are not strongly connected, Newton s method still has linear convergence order, but a worse rate [8]: Theorem 4. Let (X) be a clean (see below) MSP such that µ is inite. There is a number t such that or every i 0: β (t + i (n + ) 2 n ) i. In particular, the Newton sequence has linear convergence order. In order to make sure that Newton s method stays well-deined (i.e. that the matrix inverses exist) Theorem 4 assumes that the MSP is clean, i.e., (µ ) i > 0 or all i. An MSP can easily be made clean in linear time by identiying and removing the components with (µ ) i = 0: (µ ) i = 0 holds i (κ (n) ) i = 0. The rate in Theorem 4 is worse than in the strongly connected case: Newton s method needs (in the worst case) about 2 n iterations per bit, instead o only as in the strongly connected case. This worst case is attained by the MSPE in Equation (5) above, so the exponential rate in Theorem 4 cannot be avoided. Unortunately, we do not have an upper bound on the threshold t in this general case. 5.3 min-max-mspes Theorem 4 orms the basis or the convergence analysis o a recent extension [6] o Newton s method to min-max-mspes, i.e., MSPEs where minimum and maximum are allowed as additional operators. Here is an example o a min-max-mspe: X = max{0.7y + 0.3, 0.6XY + 0.4} Y = min{x, 0.8Y } Such systems arise, or instance, in extinction games. Those games add two adversarial players to Galton s setting rom the beginning: There are n species, each o which is controlled by one o two players, the terminator and the rescuer. Each player can apply actions to the individuals controlled by her; an action transorms an individual (probabilistically) into zero or more individuals. The terminator tries to extinguish all individuals, whereas the rescuer tries to save them. Natural questions are: What are

12 296 J. Esparza, S. Kieer, M. Luttenberger optimal strategies 2 or the terminator and the rescuer, and what is the probability o extinction o all individuals, assuming that there is a single initial individual and the players ollow optimal strategies? The MSPE above can be thought o as an equation system or the extinction probabilities o two species X and Y. Species X is controlled by the terminator, whereas Y is controlled by the rescuer. The terminator can apply one o two possible actions to an X-individual: the irst one kills the X-individual with probability 0.3, but with probability 0.7 transorms it to a Y -individual; the second action kills the X-individual with probability 0.4, but, with probability 0.6, keeps the X-individual and creates a Y -individual. What can the rescuer do with a Y -individual? She can choose between transorming it to an X-individual and a second action which kills the Y -individual with probability 0.3 and adds another Y-individual with probability 0.7. It turns out that the X-component (resp. Y-component) o the least solution o the MSPE above equals the extinction probability assuming a single initial X-individual (resp. Y -individual) i both the terminator and the rescuer ollow optimal strategies. Such systems also arise in the analysis o recursive simple stochastic games [4, 5]. In order to approximate the least solution o a min-max-mspe, one could use Kleene iteration. But, as we have seen beore (Fact 4), Kleene iteration may converge very slowly even without minimum and maximum. Thereore, in [6] we propose two methods or approximating the least solution o a min-max-mspe. Both are iterative procedures based on Newton s method. The irst method linearizes each polynomial appearing in the system (possibly inside a minimum or a maximum expression) by computing the tangent at the current iterate. One obtains a min-max-msp whose polynomials have degree at most. Its least ixed-point can be computed exactly by a method rom [8] that uses strategy iteration and linear programming. The result is the next iterate. The second method linearizes each max-polynomial appearing in the system (possibly inside a minimum expression) by computing the tangent at the current iterate. (A special tie breaking policy must be adhered to i the current iterate is at the edge between two polynomials inside a maximum expression.) One obtains a min-msp whose polynomials have degree at most. Its least ixed-point can be computed exactly by solving a single linear program. The result is the next iterate. Both methods have at least linear convergence order [6]: Theorem 5. Let (X) be a min-max-msp such that µ is inite. There is a number t such that or every i 0: β (t + i m (n + ) 2 n ) i, where m is the number o possible strategies o the players. In particular, the two extensions o Newton s method have linear convergence order. The irst method converges somewhat aster whereas a single step o the second method is cheaper. The second method also computes ε-optimal strategies or the 2 A strategy tells a player which action to apply to the individuals controlled by her.

13 Solving Monotone Polynomial Equations 297 terminator, i.e., strategies that achieve as extinction probabilities at least the current iterate. We have used the second method to approximate the extinction probabilities assuming perect strategies: A population that starts with a single X-individual (resp. Y -individual) becomes extinct with probability (resp ). We have obtained those numbers ater perorming 3 iterations and then rounding, but in this case those numbers are already the exact solution. The optimal strategy or the terminator is to apply the irst action to the X-individuals. The rescuer should choose her second action or her Y -individuals. 6 Conclusions We have shown that Newton s method is not only eicient but also remarkably robust when applied to monotone systems o ixed-point equations (MSPEs). Unlike or arbitrary systems, the method always converges when started at 0. For strongly connected systems the method always reaches a point, the threshold, ater which it is guaranteed to gain at least one bit o accuracy per iteration (in avourable cases it doubles the number per iteration). In ewer words, ater crossing the threshold the method has linear convergence order with rate. I the system is not strongly connected the method still has linear convergence, but the rate deteriorates. The threshold o the strongly connected case is inversely proportional to the logarithm o the minimal component o the least ixed-point. Thereore, i some kind o analysis can establish that the least ixed-point is not very small, then the method quickly enters the one-bit-per-iteration zone. We still do not have any threshold or the general, non-strongly-connected case. Newton s method still works or MSPEs that are not strongly connected. We have shown that the convergence order is still linear, albeit the rate may deteriorate exponentially with the dimension. Newton s method can be extended to min-max-mspes, preserving its linear convergence order. MSPEs appear in a large number o stochastic systems. In [] we have designed a ormal system or establishing the reputation o the individuals o a social network. The reputation o the individuals (deined as the stationary distribution o a Markov chain) is the least solution o a MSPE. These case studies lead to very large MSPEs, and computing their least solutions is an exciting challenge or uture research. Reerences. Bouajjani, A., Esparza, J., Schwoon, S., Suwimonteerabuth, D.: SDSIrep: A reputation system based on SDSI. In: Proceedings o TACAS (Tools and Algorithms or the Construction and Analysis o Systems), LNCS, vol. 4963, pp Springer (2008)

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