Implicit Real Vector Automata Bernard Boigelot, Julien Brusten, - - PowerPoint PPT Presentation

Implicit Real Vector Automata

Bernard Boigelot, Julien Brusten, Jean-Fran¸ cois Degbomont Universit´ e de Li` ege

Introduction

Goal : Representing symbolically polyhedra in Rn, i.e., sets of the form

• x ∈ Rn | Bi
• ai.

x

• <

≤

• bi
• ,

with ∀i : ai ∈ Zn, bi ∈ Z. Wish list :

• Concise data structure.
• Efficient manipulation algorithms : ∩, ∪, \, ×, . . .
• Efficient decision procedures : ⊆, =, empty?, . . .
• Possible conversions to and from other representations.

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Closed Convex Polyhedra

In the particular case of closed convex polyhedra

• i
• ai.

x ≤ bi, suitable representations can be :

• a set of bounding constraints,
• sets of vertices and extremal rays, or
• a combination of both.

y ≥ 0 x ≥ y y ≥ −x + 1

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Representing Non-Convex Polyhedra

Classical solutions :

• Represent a polyhedron by a formula.
• Decompose a polyhedron into an explicit union of convex polyhedra.

Main drawback : Set comparison operations become difficult. Better approach : Look for a canonical representation of polyhedra.

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Real Vector Automata

Principles :

• Vectors in Rn are encoded as infinite words over the alphabet

{0, 1, . . . , r − 1, ⋆}, where r > 1 is a base, and ⋆ a separator symbol. Example (r = 2) : Enc

1

3

• = 0+ ⋆ (01)ω.
• A Real Vector Automaton (RVA) representing a set S ⊆ Rn is an

infinite-word automaton accepting the encodings of the elements of S. Key properties :

• All the sets definable in R, Z, <, + can be represented by weak

deterministic RVA.

• These automata are easily manipulated algorithmically, and can be

minimized into a canonical form.

• But the representation of linear constraints is not efficient enough . . .

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Understanding Real Vector Automata : Conical Sets

Definition : Let D be a convex subset of Rn. A set S ⊆ D is conical in D if there exists v ∈ D such that ∀ x ∈ D − v, λ ∈ ]0, 1] : x ∈ S − v ⇔ λ x ∈ S − v. Property of weak deterministic RVA representing polyhedra : For each state q that belongs to a non-trivial strongly connected component, the language 0 ⋆ L(q) encodes a conical set. . . . . . . . . .

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Application to Polyhedra

Theorem: Let P be a polyhedron in Rn. For every x ∈ Rn and sufficiently small convex neighborhood D of x, P is conical in D.

x2 x2 < 2 x1 − x2 ≤ 1 x1 ≥ 1 x1

Application:

• The conical structure of arbitrarily small convex neighborhoods of

points partitions Rn into a finite equivalence relation ∼.

• The equivalence classes of ∼ correspond to the characteristic elements
• f P (vertices, edges, faces, . . . ).
• An incidence relation ≺ between those elements can be defined.

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The Representation of Polyhedra by RVA

Properties:

• Non-trivial strongly connected components correspond to

characteristic elements.

• The acyclic structures between components represent the incidence

relation. Illustration:

1

2 3 1 3 1 3 2 3

1

1 2 * 3 1,1 10 1,0 14 0,1 18 0,0 4 0,0 11 0,1 15 1,0 19 1,1 1,1 5 1,0 8 0,1 7 0,0 6 0,1 0,0 9 1,0 0,0 0,0 0,1 1,0 1,1 1,0 1,1 0,1 0,0 0,1 1,1 0,0 1,0 1,0 1,1 0,1 12 0,0 13 1,1 0,1 0,1 1,1 0,0 1,0 0,1 1,1 1,0 16 0,0 17 1,1 1,0 1,0 1,1 0,0 0,1 0,0 1,1 1,0 0,1

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Improving the Conciseness of RVA

Principles:

• A characteristic element of a polyhedron is characterised by:

– an affine space { v0 + λ1 v1 + . . . + λp vp | ∀i : λi ∈ R}, with v0, v1, . . . ,

• vp ∈ Qn.

(Note: By a technical argument, one can assume w.l.o.g. v0 = 0.) – its successors by the incidence relation, and – its polarity (in/out).

• Real Vector Automata can be represented implicitly by:

– Replacing their non-trivial strongly connected components by canonical algebraic descriptions of vector spaces (“implicit states”), – Keeping an acyclic transition relation for representing the incidence relation, and – Encoding the polarity of elements by a Boolean acceptation condition.

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Problem : In a RVA, the structure of transitions leaving a strongly connected component can be as large as the component itself. Solution : At each component, perform a variable change operation : The coordinate system ( e1, e2, . . . , em) becomes ( y1, y2, . . . , yp, z1, z2, . . . , zm−p), where

• The dimension of the corresponding characteristic element is p,

y1, y2, . . . , yp move inside that element, and

z1, z2, . . . , zm−p explore the directions along which one can leave the element.

• e1
• z1
• e3
• y1
• e2
• z2

Remark : The vectors z1, z2, . . . , zm−p can be chosen canonically.

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Encoding Directions

Problem : Encoding the directions in which one can leave an implicit state, in a canonical way. Solution :

• 1. Apply a normalization function to the direction vectors.

We use norm( v) = 1 2d v, with v = (v1, v2, . . . , vq) and d = maxi|vi|.

−1

2 1 2 1 2

−1

2

−1

2

−1

2 1 2 1 2

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• 2. Encode a normalized vector by :
• A prefix identifying the face of the hypercube to which it belongs.

Example (q = 3) : {1

2} × [−1 2, 1 2] × [−1 2, 1 2]

− → +1 [−1

2, 1 2] × {1 2} × [−1 2, 1 2]

− → +2 [−1

2, 1 2] × [−1 2, 1 2] × {1 2}

− → +3 {−1

2} × [−1 2, 1 2] × [−1 2, 1 2]

− → −1 [−1

2, 1 2] × {−1 2} × [−1 2, 1 2]

− → −2 [−1

2, 1 2] × [−1 2, 1 2] × {−1 2}

− → −3

• A suffix encoding the vector inside this face (i.e., in a restricted

domain of the form [−1

2, 1 2]q−1).

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Examples of Implicit Real Vector Automaton

x x − 2z = 0 y − 3z = 0 y z

+1, −1 +1, −1

• 1
• ,

1

• ,

1

• 3
• 1

3/2 1/2

• 1

1

• ,
• 1

1/3

• 2
• 1

1/2

• ,

1

• +2, −2

+1, −1

13

• 1

2 1

• 1

1

• 1

2/3 1/3

• 1

1

• ,
• 1

−1

• 1
• ,
• 1

1/2

• 1

1

• ,
• 1
• 1
• ,
• 1
• ,
• 1
• 1
• ,
• 1
• ,
• 1
• x2

x2 < 2 x1 − x2 ≤ 1 x1 ≥ 1 x1

+1 +3 −1,−2,−3 1 1 1 1 1 1 1 1 1 +2 1 1 −2 −1, +2 +1 +2 +1 −1, −2 −1 +1, +2 −2 +1 +1 −1 −1 −1 +1

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Overview of Main Results

• The representation of a polyhedron can easily be minimized into a

canonical form.

• Membership of a given vector can be decided by following a single

path in a IRVA.

• An algorithm for computing the product of two IRVA is available.
• The following operations are currently being investigated:

– Projection/determinization, – Import/export to and from other structures, – Visualization of sets, – Efficient ray tracing computations, – Efficient handling of large dimensions, – . . .

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