# Acceleration of Affine Hybrid Transformations

## Summary (4 min read)

### 1 Introduction

- Hybrid automata [14] are a powerful formalism for modeling systems that combine discrete and continuous features, in particular those depending on physical processes that involve undiscretized time.
- Accelerating a cyclic path, which corresponds to a loop in a program, amounts to computing in one step all the configurations that can be reached by iterating this cycle arbitrarily many times [2].
- The transformations undergone by variables along control paths of linear hybrid automata3 correspond to Linear Hybrid Relations (LHR), the acceleration of which is studied in [5, 6].
- The results of [5] nevertheless suffer from two weaknesses.
- First, when this acceleration method is applied to purely integer transformations, which can be seen as a particular case of LHR, it is not able to handle all instances covered by an acceleration procedure that has been specifically developed for such transformations [2, 3].

### 2.1 Algebra Basics

- Rn that satisfy a given finite conjunction of equality constraints forms an affine space.
- The affine space of smallest dimension that contains a given set is unique, and known as the affine hull of this set.
- For each constraint in this set, there exists at least one point that saturates this constraint, and that satisfies all the other ones without saturating them.

### 2.2 Linear Hybrid Relations

- The authors refer the reader to [5, 6, 14] for further details and formal definitions.
- The current configuration can change in two ways.
- The first one (time step) is to let time elapse, in which case the control location remains constant, and the variable values evolve according to the invariant and evolution law of this location.
- Those are expressed as linear constraints over respectively the variable values, and their first time derivative.
- For the sake of simplicity, the authors assume that all inequality constraints that appear in LHR are non-strict, i.e., that stands for ≤m, and that LHR are characterized by their pair (P, q).

### 2.3 Representation of Convex Polyhedra

- In the following sections, the authors study the effect and repeated effect of LHR on sets.
- Following the discussion in Section 2.1, the authors consider w.l.o.g.
- Otherwise, the initial node q0 is an additional special node in which all constraints are considered to be saturated (yielding an empty affine hull).
- The procedure ends upon reaching a node labeled by equality constraints satisfied by v, which then represents the component to which v belongs.
- This data structure has been generalized to non-convex polyhedra in [4, 13].

### 2.4 Cycle Acceleration

- The cycle acceleration problem consists in checking, within a symbolic representation system, whether the image of any representable set by unbounded iterations of a given data transformation is representable as well.
- One also needs an algorithm for computing symbolically the image of represented sets by iterable transformations.
- This decision does not have to be precise: a sufficient criterion can be used provided that it handles practically relevant transformations.
- In the next section, the authors recall two iterability criteria, one developed for linear transformations over integer variables and one for linear hybrid relations, and show that they can be combined into a criterion that has a broader scope.

### 3.1 Discrete and Hybrid Periodic Transformations

- Over the domain Zn, it has been established that transformations of the form x 7→.
- This criterion can be decided using only integer arithmetic, and a suitable value of p can be computed whenever one exists [2, 3].
- A natural idea is therefore to study hybrid transformations that have a periodic behavior, but with a period that may be greater than one.
- The iterability criterion obtained for linear integer transformations straightforwardly extends to AHT.
- If A2p = Ap, this simplifies into θkp(v) = Apv+ ∑2p−1 i=0 A iΠ+(k−2) ∑2p−1 i=p A iΠ.

### 3.2 Detecting Affine Hybrid Transformations

- Ax+Π, and of computing the corresponding matrix A and convex polyhedron Π. Ax, where A ∈ Qn×n is identical for each point, and then adding a constant convex polyhedron Π to the result.
- Let us assume that this polyhedron has at least one vertex, i.e., a geometrical component of dimension 0.
- The same reasoning applied to other vertices will yield the same matrix A. Recall that the constraints defining θ are expressed over the variables x and x′, the value of which is respectively considered before and after applying the transformation.
- The reduction consists in performing a linear variable change operation onto the largest number of distinct variables that are not statically constrained.
- Finally, note that the acceleration method for AHT discussed in this section is able to successfully process all linear integer transformations that are handled by [2, 3].

### 4.1 Principles

- Affine hybrid transformations θ have the property that the authors can compute from their set of constraints a value p ∈ N>0 such that θp has an ultimately periodic behavior.
- This sufficient condition for iterability is not at all necessary: A possible acceleration procedure thus consists in computing such a value p by inspecting the geometrical components of Θ, computing p as the least common multiple of their detected periodicities pi, and then checking whether θ p reduces to a periodic transformation that is iterable within 〈R,Z,+,≤〉.
- This inspection does not necessarily have to be carried out for all geometrical components:.
- In Section 4.2, the authors establish a connection between the acceleration technique presented in this paper and the one proposed for MCS in [5], by showing that the periodicities that are captured by the graph analysis method can also be detected by the inspection of geometrical components.
- This problem is addressed in Section 4.3.

### 4.2 Multiple Counters Systems

- Let us briefly describe the method introduced in [5] for computing the periodicity of a MCS θ.
- Since the constraint represented by σ can be saturated, there exist values v,v′ ∈ Let S′ denote the set of constraints of θ that are necessarily saturated when S is saturated, i.e., that are saturated by every v and v′ that saturate S. The set S′ contains only constraints that are either saturated for all v,v′ ∈.
- Recall that these simple cycles are all of depth ±k.
- Akx preserves the values of the variables in X and assigns the value 0 to the other variables.
- The inspection of such components may produce matrices A that do not yield a periodicity pi, or yield a spurious one.

### 4.3 Checking Periodicity

- Such cycles correspond to periodic constraints, which are captured in θ′.
- The transformation θp therefore satisfies two properties.
- (1) The second property states that, in compositions of constraints of θp, periodic constraints do not need to be repeated at more than one place.
- This is illustrated in Section 5.1 below.
- In practical applications, these conditions can be decided by operations over CPDD representations of the transformations, as discussed in Section 2.3.

### 5.1 Periodic LHR

- Note that the affine hulls α2 and α3 produce the same matrix A, which hints at the property that θ is affine.
- Checking whether θ is affine, also known as Second step.
- Checking the candidate period, also known as Ax+Π. Alternative second step.
- Alternatively, the authors may avoid computing Π and directly use the technique of Section 4.3 for checking that the candidate periodicity p = 3 is valid.
- The reflexive and transitive closure of θ3k can be obtained by quantification over k.

### 5.2 Linear Hybrid Automaton

- The effect of the cycle in H, starting from the leftmost location and preceding each transition by the passage of time, is described by the LHR θH below.
- The variable x has been eliminated using the reductions of [5] since, after the first iteration, the cycle starts and ends with x = 0.
- Following the approach of Section 4.3 confirms that θ2H is periodic.
- Note that the computation of θ∗H was out of scope of the techniques of [5, 6], which cannot handle periodicities greater than one.

### 6 Conclusions

- This paper introduces an original method for accelerating the data transformations that label control cycles of linear hybrid automata.
- Given such a transformation θ, the idea consists in constructing a convex polyhedron from its linear constraints, and then inspecting the geometrical components of this polyhedron in order to compute a value p such that θp is periodic.
- This method is able to accelerate all transformations that can be handled by the specialized algorithms developed in [3, 5, 6, 11], in particular Multiple Counters Systems, to which the reachability analysis of timed automata can be reduced.
- Compared with those solutions, their method has the advantage of being closed under linear changes of coordinates, which naturally do not affect the geometrical features of polyhedra.
- In all their case studies, considering the minimal non-empty components for which a non-trivial matrix A can be extracted turned out to be sufficient, but the authors do not know whether this property holds in all cases.

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### Cites background or methods from "Acceleration of Affine Hybrid Trans..."

...Our motivation for studying convex polyhedra is to use them for representing the reachable sets produced during symbolic state-space exploration of linear hybrid systems and temporal automata [18, 9, 7, 1]....

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...The former drawback is alleviated by the Implicit Real Vector Automaton (IRVA) [14] and the Convex Polyhedron Decision Diagram (CPDD)[7], in which parts of the decision graph are encoded by more efficient algebraic structures....

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...Intuitively, a CPDD can be understood as a compact representation of a deterministic finite automaton accepting the points of a convex polyhedron [8, 7]....

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...A Convex Polyhedron Decision Diagram (CPDD) [7] representing a convex polyhedron P is a directed acyclic graph (Q,T, q0) such that:...

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##### References

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### "Acceleration of Affine Hybrid Trans..." refers background in this paper

...A possible workaround would be to introduce approximations, such as widening operators [12], in order to force termination....

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...One reason is that some linear hybrid automata have configurations that are only reached after an unbounded number of exploration steps; a typical example is the leaking gas burner studied in [15]....

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### "Acceleration of Affine Hybrid Trans..." refers methods in this paper

...This method is able to accelerate all transformations that can be handled by the specialized algorithms developed in [3, 5, 6, 11], in particular Multiple Counters Systems, to which the reachability analysis of timed automata can be reduced....

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...Furthermore, our approach is not limited to handling MCS, unlike the acceleration method developed in [11]....

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...The cycle acceleration method proposed in [5] is able to handle a broad class of LHR, in particular all Multiple Counters Systems (MCS) [11]....

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...This approach shares similarities with the solution proposed in [5] for accelerating Multiple Counters Systems (MCS) [11], which are a subclass of LHR in which all constraints are of the form zi#zj + c, with zi, zj ∈ {x1, ....

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