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 Work Package 1: Objectives and Achievements 1 Coarse Grainings 2 Definitions Continuous vs discrete The kernel condition Aggregation coarse grainings Lattices of Coarse Grainings 3 Random Heuristic Search 4 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 R n ) 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 R n ) 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 Φ t ( x ) for Φ( t , x ) in the continuous case Φ n ( 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 Φ t X X Ξ Ξ Ψ t Y Y i.e. Ξ(Φ t ( x )) = Ψ t (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) � � � � ˙ 2 + sin( x − y ) x = y ˙ 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 The Kernel Condition Theorem (Rowe et al [2]) Let: V be open in R n and W be open in R m ; Φ 1 be continuously differentiable on V ; Ξ: V → W be a smooth map with level sets that are connected by smooth paths. Ξ coarse grains Φ 1 if and only if, for all x ∈ V , ( D Φ 1 ) x · T x ⊆ ker ( D Ξ) Φ 1 ( x ) , (1) where T x ⊆ ker ( D Ξ) 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 ( D Φ 1 ) 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 = { C 1 , . . . , C m } be a partition of the set Ω = { 1 , 2 , . . . , n } . The aggregation map associated with the partition C is the map Ξ : R n → R m defined by � � Ξ( p 1 , . . . , p n ) = p i , . . . , p i . i ∈ C 1 i ∈ C m This definition readily extends subsets of R n . Typically we might look for aggregation maps that coarse grain a heuristic map G : Λ n → Λ n , where Λ n is the probability distribution simplex { p ∈ R n : p 1 + · · · + p n = 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 matrix whose columns CG matrix Ξ partition ∼ span right ∼ (rows span left invariant kernel invariant space) 1 0 0 0 1 0 1 0 0 {{ 1 , 3 } , { 2 } , { 4 , 5 }} ∼ − 1 0 ∼ 0 1 0 0 0 0 1 0 0 0 1 1 0 − 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 } : 0 . 5 0 . 5 0 0 0 . 5 0 . 5 M = 0 . 5 0 . 5 0 0 0 . 5 0 . 5 0 . 5 0 . 5 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 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 ≥ 0 : x 1 + · · · + x n = r } , so that 1 For r ∈ N , let X r n = { x ∈ Z n r X r n is � n + r − 1 � . a subset of Λ n and has size r Definition (Vose [3]) Let G : Λ n → Λ n 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 �� w � 1 r v → 1 = r ! � 1 � � P r w G r v . w ! (Note, the vector arithmetic is computed componentwise.) Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Theory of Hierarchical Structure
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