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Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Technical Report on Work Package 1: Theory of Hierarchical Structure

University of Birmingham Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe January 19, 2014

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

1

Work Package 1: Objectives and Achievements

2

Coarse Grainings Definitions Continuous vs discrete The kernel condition Aggregation coarse grainings

3

Lattices of Coarse Grainings

4

Random Heuristic Search Aggregations Tournament examples Reaction examples NP-completeness

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Objectives of Work Package 1

Develop a formal theory of the hierarchical structure of complex systems.

1 Theoretical characterisation of the hierarchical decomposition. 2 Topological characterisation of property convergence.

• Analyse the algebraic structure inherent in the hierarchical

decomposition of arbitrary complex systems.

• Determine appropriate topologies for considering the map

between complex system structure and space of information to be continuous.

• Develop a general theory of how sub-systems of a complex

system determine the whole.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Summary of Key Achievements for Work Package 1

Collaboration between UoB (Maths), UoB (Comp Sci), Chalmers and Jena has lead to: significant developments in linear CGs of non-linear systems (aggregation of polynomial systems on Rn) explicit clarification of the relation between CGs and invariant subspaces glimpse into the hierarchical structure of systems through different lattices of coarse grainings

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Summary of Key Achievements for Work Package 1

These results feed directly into the objectives of WP3, resulting in: a reformulation into the current setting of the bisimulation minimisation algorithm (to find the coarsest CG finer than a given partition), emphasises links to probabilistic verification in Comp Sci; a novel algorithm for the dual problem (to find CGs coarser than a given partition); a novel algorithm for general functions (and example to show caution is needed). In turn work on WP3 has fed back to WP1: Aggregation coarse graining for Markov chains is NP-complete.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Coarse Graining Dynamical Systems

A dynamical system consists of: a state space X (usually compact metric or a subset of Rn) and a map Φ: J × X → X (usually continuous) such that

• Φ(0, x) = x for all x ∈ X,
• Φ(t + s, x) = Φ(t, Φ(s, x)) for all t, s ∈ J and x ∈ X.

where J = [0, ∞) (continuous time) or J = {0, 1, 2, 3, · · · } (discrete time). We often write Φt(x) for Φ(t, x) in the continuous case Φn(x) for Φ(n, x) in the discrete case.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Coarse Graining Dynamical Systems

Let (X, Φ) and (Y , Ψ) be systems with the same time set J. Definition Ξ: X → Y is a coarse graining (CG) of Φ if, for all t ∈ J, the following diagram commutes X Y X Y Ξ Φt Ξ Ψt i.e. Ξ(Φt(x)) = Ψt(Ξ(x)) for all t ∈ J and x ∈ X.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Coarse Graining Dynamical Systems (Continuous vs Discrete)

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Coarse Graining Dynamical Systems (Continuous vs Discrete)

• ˙

x ˙ y

• =
• 2 + sin(x − y)
• Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe

Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

The Kernel Condition

Theorem (Rowe et al [2]) Let: V be open in Rn and W be open in Rm; Φ1 be continuously differentiable on V ; Ξ: V → W be a smooth map with level sets that are connected by smooth paths. Ξ coarse grains Φ1 if and only if, for all x ∈ V , (DΦ1)x · Tx ⊆ ker (DΞ)Φ1(x), (1) where Tx ⊆ ker (DΞ)x is a tangent space at x defined as a linear span of a set of all velocities realised by smooth paths passing through x and attaining values within the same level set of Ξ.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Kernel Condition for Linear CGs

When Ξ is a linear map, equation (1) simplifies to (DΦ1)x · ker Ξ ⊆ ker Ξ (2) Since CGs with the same kernel are considered equivalent, the following are equivalent approaches: linear coarse grainings; common right invariant subspaces of the differential; common left invariant subspaces.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Aggregation Coarse Grainings

Definition Let C = {C1, . . . , Cm} be a partition of the set Ω = {1, 2, . . . , n}. The aggregation map associated with the partition C is the map Ξ : Rn → Rm defined by Ξ(p1, . . . , pn) =

 

i∈C1

pi, . . . ,

• i∈Cm

pi

  .

This definition readily extends subsets of Rn. Typically we might look for aggregation maps that coarse grain a heuristic map G : Λn → Λn, where Λn is the probability distribution simplex {p ∈ Rn : p1 + · · · + pn = 1}.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Aggregation Coarse Grainings

partition ∼ matrix whose columns span right invariant kernel ∼ CG matrix Ξ (rows span left invariant space) {{1, 3}, {2}, {4, 5}} ∼       1 −1 1 −1       ∼   1 1 1 1 1  

This formulation forms the basis for the novel algorithms developed in Work Package 3.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Hierarchical Structure of Coarse Grainings: Lattices

Key join (∨) = sup = bigger kernel = coarser partition meet (∧) = inf = smaller kernel = finer partition The following form complete lattices:

partitions of Ω ⊇ aggregation CG ⊆ linear CGs ⊆ ⊆ differentiable CGs ⊆ partitions of the underlying set

Note that the lattice of linear CGs is also modular. In general, these are subsets and not sublattice. However: the lattices of in the first line share the join operation; the lattices in the second line share the meet operation.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Partition Lattice Meet = Aggregation Lattice Meet

Consider the following Markov chain on a 5 state space {1, 2, . . . , 5}: M =

      

.5 .5 .5 .5 .5 .5 .5 .5 .5 .5

      

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Partition Lattice Meet = Aggregation Lattice Meet

NOTE! Ordering is reversed (sup is at the bottom)

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Random Heuristic Search

For r ∈ N, let X r

n = {x ∈ Zn ≥0 : x1 + · · · + xn = r}, so that 1 r X r n is

a subset of Λn and has size

n+r−1

r

.

Definition (Vose [3]) Let G : Λn → Λn be a heuristic. A Random Heuristic Search (RHS) with population r is a Discrete Time Markov Chain (DTMC) with state space 1

r X r n and transition probabilities given by

P

1

r v → 1 r w

• = r!

w!

• G

1

r v

w

. (Note, the vector arithmetic is computed componentwise.)

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Induced Aggregations

Let Ξ : Λn → Λm be an aggregation. For any r ∈ N, the induced aggregation ΘΞ,r : Λ(n+r−1

r

) → Λ(m+r−1

r

) is the aggregation generated by the relation of equivalence on the set 1

r X r n

v r ≡ w r ⇐ ⇒ Ξ

v

r

• = Ξ

w

r

• .

Theorem (Vose [3]) An aggregation Ξ of coarse grains the heuristic G if and only if, for any population size r, the induced aggregation ΘΞ,r coarse grains the associated DTMC. Observation An aggregation CG of a DTMC is a DTMC. A linear CG of a DTMC need not be a DTMC.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

The Classic Tournament Example

Tournaments: two agents each with a state meet, the result is an agent or agents in one of these states State space Ω = {1, 2, . . . , n}. Each agent in a population of size r has a state from Ω Randomly choose 2 agents. Save the one with largest state. Repeat to produces r agents forming the next generation. The DTMC with parameter r and heuristic (G(p))j = p2

j + 2

• k<j

pkpj models precisely the above procedure.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

The Classic Tournament Example

Observation There are other ways to model such binary interactions that have the same heuristic G. In particular, after the first drawing we can simply substitute ’loser’ with another copy of the winner. This model again has the same aggregation CGs as the map G.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Tournament example

The differential of G is dG(p) = 2

       

p1 . . . p2 p1 + p2 . . . p3 p3 p1 + p2 + p3 . . . . . . . . . . . . . . . ... . . . pn pn pn pn . . . p1 + · · · + pn

       

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Tournament example

Theorem ([1], Theorem 7) An equivalence relation on Ω induces an aggregation CG iff it is contiguous, i.e. for all i, j, k ∈ Ω we have i ≡ k and i > j > k = ⇒ i ≡ j ≡ k. (3) Proof in the current context: If states i, j are aggregated, then the vector ei −ej is in ker Ξ. The invariance implies that dG(p)(ei − ej) = (p1 + · · · + pi)ei + pi+1ei+1 + · · · + pj−1ej−1 − (p1 + . . . pj−1)ej ∈ ker Ξ Fix l ∈ {i, . . . , j − 1}. Set pl=1 and ps = 0 for s = l. The above vector becomes el − ej, so that l and j must be aggregated.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Probabilistic Tournament

The above now extends to the following case. If two agents in states i and j meet, then the outcome is state i with probability P(i, j) and j with probability P(j, i) = 1 − P(i, j). In this case, the heuristic is (G(p1, . . . , pn))i = 2pi

• k∈Ω

P(i, k)pk. (4) Contiguacy is now defined as: Definition The equivalence relation ≡ on a search space is contiguous with respect to selection map P if for all i, j, k ∈ Ω we have i ≡ k and P(i, j) = P(k, j) = ⇒ i ≡ j ≡ k. (5)

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

General Polynomial Model

Reactions: two or more agents meet, each in a given state, the

• utcome is an agent or agents in possibly different states.

The following theorem characterizes aggregation coarse grainings for such models. (T(p))i =

• v:|v|≤d

1 v!αi,vpv, Theorem Let T be a polynomial map on Rn as above. An aggregation of variables Ξ: Rn → Rm is a valid coarse graining if and only if Ξ(v) = Ξ(w) implies Ξ(αv) = Ξ(αw) for all v, w ∈ Zn

≥0.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

n-type reaction model of degree d

T(p) =

d

• k=0

ρk

 

v:|v|=k

• k

v

• τvpv

  .

p ∈ Λn denotes proportion of n different reactants. Up to d reactants may react at any time. (ρ1, . . . , ρd) ∈ Λd and ρk denotes the rate of reaction with k reactants. v = (v1, . . . , vn) ∈ Λn denotes a particular reaction involving vi particles

• f type i.

τv ∈ Λn denotes the distribution of the products of reaction given by v. Theorem Let G be an n-type reaction model of the degree d. A partition A of the set {1, 2, . . . , n} is a valid aggregation CG if and only if Ξ(v) = Ξ(w) implies Ξ(τv) = Ξ(τw) for all v, w ∈ Zn

≥0 where Ξ is an aggregation associated to

partition A.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Modulus Example

Tournament on the set Ω = {0, 1, 2, . . . n − 1} where the clash between i and j produces (i + j) mod n (2-degree reaction model). A partition will then be a valid aggregation if and only for any two pieces in it, it is well defined which piece they produce. So assume that one of the pieces is C. Then for any other element d ∈ Ω a set (C + d) mod n must be a piece as well. Since pieces form a partition of Ω it immediately follows that all the pieces are the translates of the piece {0, l, 2l, . . . , (n − l)} for some divisor l

• f n. It immediately follows that lattice of aggregations in this

case is the same as the lattice of divisors of a number n where

• rdering relation is defined by a ≤ b if and only if ’a divides b’.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Modulus example (the lattice of aggregations for n = 12)

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Polynomial Model Example

Consider the map: G(x, y, u, v) =

    u2 + 4uv + x3 − uy − vy + 2x2y + xy2 u2 + 2v2 − ux − vx + x2y + y3 + 2xy2 −3u2 − uv − 5u2x − 3uvx − 5v2x + 2x2 − 5u2y + 2xy −5uv − 3v2 − 7uvx − 10uvy − 5v2y + 2y2 + 2xy    

Separating into homogeneous parts yields G2(x, y, u, v) =

    u2 + 4uv − uy − vy u2 + 2v2 − ux − vx −3u2 − uv + 2x2 + 2xy −5uv − 3v2 + 2y2 + 2xy    

G3(x, y, u, v) =

    x3 + 2x2y + xy2 x2y + y3 + 2xy2 −5u2x − 3uvx − 5v2x − 5u2y −7uvx − 10uvy − 5v2y    

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Polynomial model example

G2(x, y, u, v) =

     2 u2

2 + 4uv − uy − vy

2 u2

2 + 4 v2 2 − ux − vx

−6 u2

2 − uv + 4 x2 2 + 2xy

−5uv − 6 v2

2 + 4 y2 2 + 2xy

    

G3(x, y, u, v) =

     6 x3

6 + 4 x2y 2 + 2 xy2 2

2 x2y

2 + 6 y3 6 + 4 xy2 2

−10 u2x

2 − 3uvx − 10 v2x 2 − 10 u2y 2

−7uvx − 10uvy − 10 v2y

2

    

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Polynomial model example

Adding up the first two and the last two rows in both matrices results in:

• 4 u2

2 + 4uv + 4 v2 2 − ux − vx − uy − vy

−6 u2

2 − 6 v2 2 − 6uv + 4 x2 2 + 4xy + 4 y2 2

• =
• 2(u + v)2 − (u + v)(x + y)

−3(u + v)2 + 2(x + y)2

• 6 x3

6 + 6 x2y 2 + 6 xy2 2 + 6 y3 6

−10 u2x

2

− 10uvx − 10 v2x

2 − 10 u2y 2

− 10uvy − 10 v2y

2

• =
• (x + y)3

−5(u + v)2(x + y)

• We can see that the coefficients corresponding to coarser

partition of variables {{x, y}, {u, v}} are the same and hence higher level description of the system is possible (well defined) due to the previous theorem.

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

Polynomial model example

Higher level map is hence ˜ G(˜ x, ˜ u) =

  

2˜ u2 − ˜ u˜ x + ˜ x3 −3˜ u2 + 2˜ x2 − 5˜ u2˜ x

  

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

SSP and NP-completeness

Subset sum problem (SSP) can be reduced to a decision problem

• f existence of a non-trivial aggregation of a certain Markov chain

Theorem The existence of a non-trivial aggregation coarse graining for a Markov chain is an NP-complete problem.

• This is not bad news as there are good approximative algorithms

for solving SSP in many cases.

• As we have seen in previous examples, any kind of structure

within a system usually leads to the simplification of detection of the hierarchical structure so that it can be solved in polynomial time (cf. Probabilistic tournament)

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

The End!

Thank you!

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure

Overview WP1: Objectives and Achievements Coarse Grainings Lattices Random Heuristic Search References

References

[1] Jonathan E. Rowe, Michael D. Vose, and Alden H. Wright, State aggregation and population dynamics in linear systems, Artif. Life 11 (December 2005), no. 4, 473–492. [2] , Differentiable coarse graining, Theoret. Comput. Sci. 361 (2006),

• no. 1, 111–129. MR2254227 (2007h:68171)

[3] Michael D. Vose, The simple genetic algorithm, Complex Adaptive Systems, MIT Press, Cambridge, MA, 1999. Foundations and theory, A Bradford Book. MR1713436 (2000h:65024)

Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure