Implicit Real Vector Automata^††thanks: This work is supported by the Interuniversity Attraction Poles program MoVES of the Belgian Federal Science Policy Office, and by the grant 2.4530.02 of the Belgian Fund for Scientific Research (F.R.S.-FNRS).

Let $n\in\mathbb{N}_{>0}$ be a dimension. A linear constraint over vectors $\vec{x}\in\mathbb{R}^{n}$ is a constraint of the form $\vec{a}.\vec{x}\,\#\,b$, with $\vec{a}\in\mathbb{Z}^{n}$, $b\in\mathbb{Z}$, and $\#\in\{<,\leq,=,\geq,$ $>\}$. A finite Boolean combination of such constraints forms a polyhedron. If a polyhedron can be expressed as a finite conjunction of linear constraints, it is said to be convex. A polyhedron that can be expressed as a conjunction of linear equalities, i.e., constraints of the form $\vec{a}.\vec{x}=b$, is an affine space. An affine space that contains $\vec{0}$ is a vector space. The dimension $\mbox{dim}(\mbox{VS})$ of a vector space VS is the size of the largest set of linearly independent vectors it contains.

Finally, given a convex polyhedron $D\subseteq\mathbb{R}^{n}$, a polyhedron $P\subseteq\mathbb{R}^{n}$, and a vector $\vec{v}\in D$, we say that $P$ is conical in $D$ with respect to the apex $\vec{v}$ iff for all $\vec{x}\in D$ and $\lambda\in]0,1[$, we have $\vec{x}\in P\,\Leftrightarrow\,\lambda(\vec{x}-\vec{v})+\vec{v}\in P$. (Intuitively, this condition expresses that within $D$, the polyhedron $P$ is not affected by a scaling centered on $\vec{v}$.) It is shown in [8] that the set of the vectors $\vec{v}$ with respect to which $P$ is conical in $D$ necessarily coincides with an affine space over $D$.

This section is adapted from [7, 9, 8]. Let $r\in\mathbb{N}_{>1}$ be a numeration base. In the positional number system in base $r$, a number $z\in\mathbb{R}_{\geq 0}$ can be encoded by an infinite word $a_{p-1}a_{p-2}\ldots a_{0}\star a_{-1}a_{-2}a_{-3}\ldots$, where $\forall i:\,a_{i}\in\{0,1,\ldots,r-1\}$, such that $z=\sum_{i<p}a_{i}r^{i}$. (The distinguished symbol “$\star$” separates the integer from the fractional part of the encoding.) Negative numbers are encoded by using the $r$’s-complement method, which amounts to representing a number $z\in\mathbb{R}_{<0}$ by the encoding of $z+r^{p}$, where $p$ is the length of its integer part. This length $p$ does not have to be fixed, but must be large enough for the constraint $-r^{p-1}\leq z\leq r^{p-1}$ to hold, in order to reliably discriminate the sign of encoded numbers. Under this scheme, every real number admits an infinite number of encodings in base $r$. Note that some numbers admit different encodings with the same integer-part length, for instance, the base-$2$ encodings of $1/4$ form the language $0^{+}\star 010^{\omega}\,\cup\,0^{+}\star 001^{\omega}$. Such encodings are then called dual.

The positional encoding of numbers generalizes to vectors in $\mathbb{R}^{n}$, with $n\in\mathbb{N}_{>0}$. A vector is encoded by first choosing encodings of its components that share the same integer-part length. Then, these component encodings are combined by repeatedly and synchronously reading one symbol in each component. The result takes the form of an infinite word over the alphabet $\{0,1,\ldots,r-1\}^{n}\,\cup\,\{\star\}$ (since the separator is read simultaneously in all components, it can be denoted by a single symbol). It is also worth mentioning that the exponential size of the alphabet can be avoided if needed by serializing the symbols, i.e., reading the components of each symbol sequentially in a fixed order rather than simultaneously [6].

This encoding scheme maps any set $S\subseteq\mathbb{R}^{n}$ onto a language of infinite words. If this language is $\omega$-regular, then it can be accepted by an infinite-word automaton, which is then known as a Real Vector Automaton (RVA) representing the set $S$.

Some classes of infinite-word automata are notoriously difficult to handle algorithmically [26]. A weak automaton is a Büchi automaton such that each strongly connected component of its transition graph contains either only accepting or only non-accepting states. The advantage of this restriction is that weak automata admit efficient manipulation algorithms, comparable in cost to those suited for finite-word automata [27]. The following result is established in [9].

In the sequel, we will only consider weak and deterministic RVA. These structures can efficiently be minimized into a canonical form [22], and combining them by Boolean operators amounts to performing similar operations on the languages they accept. Implementations of RVA are available as parts of the tools LASH [19] and LIRA [21].

It has been observed in [15] that, for every polyhedron $P\subseteq\mathbb{R}^{n}$ and point $\vec{v}\in\mathbb{R}^{n}$, the set $P$ is conical in all sufficiently small convex neighborhoods of $\vec{v}$. We now formalize this property, and prove it by reasoning about the structure of RVA representing $P$. This will provide valuable insight into the principles of operation of automata-based representations of polyhedra.

For every $\vec{v}=(v_{1},v_{2},\ldots,v_{n})\in\mathbb{R}^{n}$ and $\varepsilon\in\mathbb{R}_{>0}$, let $N_{\varepsilon}(\vec{v})$ denote the $n$-cube of size $[\varepsilon]^{n}$ centered on $\vec{v}$, that is, the set $[v_{1}-\varepsilon/2,v_{1}+\varepsilon/2]\times[v_{2}-\varepsilon/2,v_{2}+\varepsilon/2]\times\cdots\times[v_{n}-\varepsilon/2,v_{n}+\varepsilon/2]$.

Proof: Let $\cal A$ be a RVA representing $P$ in a base $r\in\mathbb{N}_{>1}$, which exists thanks to Theorem 1. We assume w.l.o.g. that $\cal A$ is weak, deterministic, and has a complete transition relation. Consider a word $w$ encoding $\vec{v}$ in base $r$. For each $k\in\mathbb{N}$, let $w_{k}$ denote the finite prefix of $w$ with $k$ symbols in its fractional part, i.e., such that $w_{k}=u\star u^{\prime}$ with $|u^{\prime}|=k$. The set of all vectors that admit an encoding of prefix $w_{k}$ forms a $n$-cube $C_{w_{k}}$ of size $[r^{-k}]^{n}$. For every $k\in\mathbb{N}$, we have $\vec{v}\in C_{w_{k}}$ and $C_{w_{k+1}}\subset C_{w_{k}}$, leading to $\bigcap_{k\in\mathbb{N}}C_{w_{k}}=\{\vec{v}\}$. Intuitively, each symbol read by $\cal A$ reduces by a factor $r^{n}$ the size of the set of possibly recognized vectors.

Consider $\varepsilon\in\mathbb{R}_{>0}$ with $\varepsilon<1$. The set $N_{\varepsilon}(\vec{v})$ is covered by the union of the sets $C_{w_{k}}$ for all $w$ encoding $\vec{v}$, choosing $k$ such that $r^{-k}\geq\varepsilon$. It is thus sufficient to prove that for every word $w$ encoding $\vec{v}$ and sufficiently large $k$, the set $P$ is conical in $C_{w_{k}}$ with respect to the apex $\vec{v}$. This property has been proved in [8], where it is additionally shown that the suitable values of $k$ include those for which $w_{k}$ reaches the last strongly connected component of $\cal A$ visited by $w$. $\Box$

In the previous proof, the strongly connected components of $\cal A$ turn out to be connected to conical structures present in $P$. This can be explained as follows. Consider two finite prefixes $w_{k}$ and $w_{k+d}$ of $w$, with $d>0$, such that $w_{k+d}$ only differs from $w_{k}$ by additional iterations of cycles in the last strongly connected component of $\cal A$ visited by $w$. Since both $w_{k}$ and $w_{k+d}$ lead to the same state of $\cal A$, the sets of suffixes that can be appended to them so as to obtain words accepted by $\cal A$ are identical. In order to be able to compare such sets of suffixes, we introduce the following notation. For each $k\in\mathbb{N}$, let $\vec{c}_{w_{k}}=(c_{w_{k},1},c_{w_{k},2},\ldots,c_{w_{k},n})$ denote the vector encoded by $w_{k}(0^{n})^{\omega}$, in other words the vector such that $C_{w_{k}}=[c_{w_{k},1},c_{w_{k},1}+r^{-k}]\times[c_{w_{k},2},c_{w_{k},2}+r^{-k}]\times\cdots\times[c_{w_{k},n},c_{w_{k},n}+r^{-k}]$. Given a $n$-cube $C\subset\mathbb{R}^{n}$ of size $[\lambda]^{n}$ and a vector $\vec{c}\in\mathbb{R}^{n}$, we then define the normalized view of $P$ with respect to $C$ and $\vec{c}$ as the set $P[C,\vec{c}]=(1/\lambda)((P\,\cap\,C)-\vec{c})$. In other words, this normalized view is obtained by a translation bringing $\vec{c}$ onto the origin $\vec{0}$, followed by a scaling that makes the size of the $n$-cube in which $P$ is observed become equal to $[1]^{n}$.

Observe that the set $P[C_{w_{k}},\vec{c}_{w_{k}}]$ is precisely characterized by the language accepted from the state of $\cal A$ reached by $w_{k}$. Since this state is identical to the one reached by $w_{k+d}$, we obtain $P[C_{w_{k}},\vec{c}_{w_{k}}]=P[C_{w_{k+d}},\vec{c}_{w_{k+d}}]$. Recall that we have $\vec{v}\in C_{w_{k}}$ and $C_{w_{k+d}}\subset C_{w_{k}}$. The previous property shows that $P$ is self-similar in the vicinity of $\vec{v}$: Following additional cycles in the last strongly connected component visited by $w$ amounts to increasing the “zoom level” at which the set $P$ is viewed close to $\vec{v}$, without influencing this view. It is shown in [8] that this self-similarity entails the conical structure of $P$ around $\vec{v}$, which intuitively means that the zoom levels that preserve the local structure of $P$ are not restricted to integer powers of $r^{d}$.

In addition, we have established that the structure of $P$ in a small neighborhood $N_{\varepsilon}(\vec{v})$ of $\vec{v}$ is uniquely determined by the state of $\cal A$ reached by $w_{k}$. Since there are only finitely many such states, we have the following result.

Proof: The proof follows the same lines as the one of Theorem 2. Let $\cal A$ be a weak, deterministic, and complete RVA representing $P$ in a base $r\in\mathbb{N}_{>1}$. To every word $w$ encoding a given vector $\vec{v}$ in base $r$, we associate the integer $k(w)$ such that the path of $\cal A$ recognizing $w$ reads the finite prefix $w_{k(w)}$ before reaching the last strongly connected component that it visits. From the previous developments, we have that $P$ is conical in $C_{w_{k(w)}}$ with respect to the apex $\vec{v}$. Furthermore, the set $P[C_{w_{k(w)}},\vec{c}_{w_{k(w)}}]$ only depends on the state of $\cal A$ reached after reading $w_{k(w)}$, which are in finite number. It follows that, in arbitrarily small neighborhoods of $\vec{v}$, the polyhedron $P$ has a conical structure with respect to the apex $\vec{v}$, and that there are only finitely many such structures over all vectors $\vec{v}$. $\Box$

Theorem 3 shows that a polyhedron $P\subseteq\mathbb{R}^{n}$ partitions $\mathbb{R}^{n}$ into finitely many equivalence classes, each of which corresponds to a unique conical set in the $n$-cube $[-1/2,1/2]^{n}$ with respect to the apex $\vec{0}$. For each $\vec{v}\in\mathbb{R}^{n}$, let $P_{\vec{v}}\subseteq[-1/2,1/2]^{n}$ denote the conical set associated to $\vec{v}$ by $P$. We call $P_{\vec{v}}$ the component of $P$ associated to $\vec{v}$. Recall that, as discussed in Section 2.1, the set of apexes according to which $P_{\vec{v}}$ is conical coincides with a vector space over $[-1/2,1/2]^{n}$. The dimension $\mbox{dim}(P_{\vec{v}})$ of the component $P_{\vec{v}}$ is defined as the dimension of this vector space. Finally, we say that a component $P_{\vec{v}}$ is in if $\vec{v}\in P$, and out if $\vec{v}\not\in P$.

An example is given in Figure 1. The triangle $x_{1}\geq 1\,\wedge\,x_{2}<2\,\wedge\,x_{1}-x_{2}\leq 1$ in $\mathbb{R}^{2}$ has three components of dimension $0$ corresponding to its vertices $(1,0)$ (in), $(1,2)$ (out) and $(3,2)$ (out), three components of dimension $1$ associated to its sides (two in and one out), and two components of dimension $2$ corresponding to its interior (in) and exterior (out) points.

In Section 3.1, we have established a link between the components of a polyhedron $P\subseteq\mathbb{R}^{n}$ and the strongly connected components (SCC) of a RVA $\cal A$ representing $P$. We know that there exists a hierarchy between the SCC of an automaton: That a SCC $S_{2}$ is reachable from a SCC $S_{1}$ implies that every finite prefix that reaches a state of $S_{1}$ can be followed by a suffix that ends up visiting $S_{2}$, while the reciprocal property does not hold. In a similar way, we can define an incidence relation between the components of a polyhedron.

Remark that the incidence relation between the components of a polyhedron is a partial order, and that $Q_{1}\prec Q_{2}$ implies $\mbox{dim}(Q_{1})<\mbox{dim}(Q_{2})$. As an example, in the triangle depicted in Figure 1, each side is incident to the vertices it links, since every neighborhood of a vertex contains points from its adjacent sides. The reverse property does not hold. The interior and exterior components of the triangle are incident to each of its sides and vertices.

We are now able to explain the mechanism employed by a RVA $\cal A$ in order to check whether the vector encoded by a word $w$ belongs or not to a polyhedron $P\subseteq\mathbb{R}^{n}$. After reading an integer part and a separator symbol, the word $w$ follows some transitions in the fractional part of $\cal A$, reaching a first non-trivial strongly connected component $S_{1}$ (that is, a component containing at least one cycle). At this location in $w$, inserting arbitrary iterations of cycles within $S_{1}$ would not affect the accepting status of $w$. This intuitively means that the prefix $w_{k}$ of $w$ read so far has led us to a point that belongs to a component $Q_{1}$ of $P$, and that the decision can now be carried out further in an arbitrarily small neighborhood of this point. Reading additional symbols from $w$, one either stays within $S_{1}$, or follows transitions that eventually lead to another non-trivial strongly connected component $S_{2}$. Once again, this means that the decision can now take place in an arbitrarily small neighborhood of a point belonging to a component $Q_{2}$ of $P$, such that either $Q_{1}=Q_{2}$ or $Q_{1}\prec Q_{2}$. The same procedure repeats itself until $w$ reaches a strongly connected component that it does not leave anymore.

In other words, in order to decide whether to accept or not a word $w$, the RVA $\cal A$ first chooses deterministically a component $Q_{1}$ of $P$ in the vicinity of which this decision can be carried out. Then, it checks whether the vector $\vec{v}$ encoded by $w$ belongs or not to $Q_{1}$. If yes, the decision is taken according to whether $Q_{1}$ is in or out. If no, the RVA chooses deterministically a component $Q_{2}$ incident to $Q_{1}$, from which the same procedure is then repeated.

Let us now study more finely the mechanism used for moving from a component $Q_{1}$ that does not contain the vector $\vec{v}$ to another component $Q_{2}$ from which $\vec{v}\in P$ can be decided. One follows a path of $\cal A$ that leaves a strongly connected component associated to $Q_{1}$, travels through an acyclic structure of transitions, and finally reaches a SCC associated to $Q_{2}$. Recall that, as discussed in Section 3.1, at each step in this path, the prefix $w_{k}$ of $w$ read so far determines a $n$-cube $C_{w_{k}}$. This $n$-cube covers some subset $U_{w_{k}}=\{P_{\vec{u}}\mid\vec{u}\in C_{w_{k}}\}$ of the components of $P$. If $U_{w_{k}}$ contains a single minimal component with respect to the incidence order $\prec$, then this component is necessarily equal to $Q_{2}$, and its associated SCC is the only possible destination of $w_{k}$. Indeed, all components in $U_{w_{k}}$ are then either equal or incident to $Q_{2}$. If, on the other hand, $U_{w_{k}}$ contains more than one minimal component, then further transitions have to be followed in order to discriminate between them.

An easy way of managing the choice of an initial component is to consider only polyhedra in which this component is unique. This can be done without loss of generality thanks to the following definition.

For every polyhedron $P\subseteq\mathbb{R}^{n}$, the polyhedron $\overline{P}$ is conical in $\mathbb{R}^{n+1}$ with respect to the apex $\vec{0}$, from which it can be inferred that every neighborhood of $\vec{0}$ contains a unique minimal component $Q_{0}$ with respect to the incidence order $\prec$. It follows that for every $\vec{v}\in\mathbb{R}^{n+1}$, the decision $\vec{v}\in\overline{P}$ can be started from $Q_{0}$. Remark that $\overline{P}$ describes $P$ without ambiguity, since $P$ can be reconstructed from $\overline{P}$ by computing its intersection with the constraint $x_{n+1}=1$, and projecting the result over the first $n$ vector components. In the sequel, we assume w.l.o.g. that the polyhedra that we consider are conical with respect to the apex $\vec{0}$. A similar mechanism is employed in [20].

Consider a polyhedron $P\subseteq\mathbb{R}^{n}$ that is conical with respect to the apex $\vec{0}$. As explained in Section 3.2, a component of such a polyhedron is characterized by a vector space, a Boolean polarity (either in or out), and its incident components. Checking whether a given vector $\vec{v}\in\mathbb{R}^{n}$ belongs or not to the component reduces to deciding whether $\vec{v}$ belongs to its associated vector space. This is a simple algebraic operation if, for instance, the vector space is represented by a vector basis $\{\vec{b}_{1},\vec{b}_{2},\ldots,\vec{b}_{m}\}$: One simply has to check whether $\vec{v}$ is linearly dependent with $\{\vec{b}_{1},\vec{b}_{2},\ldots,\vec{b}_{m}\}$. This approach leads to a much more concise representation of polyhedral components than the one used in RVA.

We now address the problem of leaving a component $Q_{1}$ of a polyhedron $P\subseteq\mathbb{R}^{n}$ that does not contain a vector $\vec{v}\in\mathbb{R}^{n}$, and moving to a component $Q_{2}$ that is incident to $Q_{1}$, and from which $\vec{v}\in P$ can be decided.

A first solution would be to borrow from a RVA representing $P$ the acyclic structure of transitions leaving the strongly connected components $S_{1}$ associated to $Q_{1}$. However, this would negate the advantage in conciseness obtained in Section 4.2, since this acyclic structure of transitions is generally as large as $S_{1}$ itself.

The solution we propose consists in performing a variable change operation. Let $\{\vec{y}_{1},\vec{y}_{2},\ldots,\vec{y}_{m}\}$, with $0<m\leq n$, be a basis of the vector space associated with the component $Q_{1}$. If $m=n$, then $Q_{1}$ is universal and there is no possibility of leaving it. If $m<n$, then we introduce $n-m$ additional vectors $\vec{z}_{1}$, $\vec{z}_{2}$, …, $\vec{z}_{n-m}$, such that $\{\vec{y}_{1},\ldots,\vec{y}_{m},\vec{z}_{1},\ldots,\vec{z}_{n-m}\}$ forms a basis of $\mathbb{R}^{n}$. These additional vectors can be chosen in a canonical way by selecting among $(1,0,\ldots,0),(0,1,\ldots,0),\ldots,(0,0,\ldots,1)$, considered in that order, $n-m$ vectors that are linearly independent with $\{\vec{y}_{1},\vec{y}_{2},\ldots,\vec{y}_{m}\}$.

We then express the vector $\vec{v}$ in the coordinate system $\{\vec{y}_{1},\ldots,\vec{y}_{m},\vec{z}_{1},\ldots,$ $\vec{z}_{n-m}\}$, obtaining a vector $(y_{1},\ldots,y_{m},z_{1},\ldots,z_{n-m})$. That $\vec{v}$ leaves $Q_{1}$ simply means that we have $(z_{1},\ldots,z_{n-m})\neq\vec{0}$. As a consequence, we associate $Q_{1}$ with an acyclic structure ${\cal D}_{1}$ of outgoing transitions, recognizing prefixes of encodings of non-zero vectors $(z_{1},\ldots,z_{n-m})$, in order to map these vectors to the polyhedral components (incident to $Q_{1}$) to which they lead.

A difficulty is that, from Theorem 2, the set $P$ has a conical structure in arbitrary small neighborhoods of points in $Q_{1}$. If follows that the structure ${\cal D}_{1}$ has to map onto the same polyhedral component two vectors $\vec{z}$ and $\vec{z}^{\prime}$ such that $\vec{z}^{\prime}=\lambda\vec{z}$ for some $\lambda\in\mathbb{R}_{>0}$. An efficient solution is to normalize the vectors handled by ${\cal D}_{1}$: Given a vector $\vec{z}=(z_{1},\ldots,z_{n-m})$ such that $\vec{z}\neq\vec{0}$, we define its normalized form as $[\vec{z}]=(1/{(2.\mbox{max}_{i}|z_{i}|)})\vec{z}$. In other words, $[\vec{z}]$ is obtained by turning $\vec{z}$ into the half-line $\{\lambda\vec{z}\mid\lambda\in\mathbb{R}_{>0}\}$, and computing the intersection of this half-line with the faces of the normalization cube $[-1/2,1/2]^{n-m}$. In this way, two vectors that only differ by a positive factor share the same normalized form, and will thus be handled identically.

The purpose of the structure ${\cal D}_{1}$ is thus to recognize normalized forms of vectors, and map them onto the polyhedral components to which they lead. In order to define the transition graph of ${\cal D}_{1}$, one therefore needs a suitable encoding for normalized forms of vectors. Using the standard positional encoding of vectors in a base $r\in\mathbb{N}_{>1}$ is possible, but inefficient. We instead use the following scheme. An encoding of a normalized vector $[\vec{v}]=([v]_{1},[v]_{2}\,\ldots,[v]_{n-m})$ starts with a leading symbol $a\in\{-1,+1,-2,+2,\ldots,-(n-m),+(n-m)\}$ that identifies the face of the normalization cube $[-1/2,1/2]^{n-m}$ to which $[\vec{v}]$ belongs: If $a=-i$, with $1\leq i\leq n-m$, then $[v]_{i}=-1/2$; if $a=+i$, then $[v]_{i}=+1/2$. This prefix is followed by a suffix $w\in\{0,1\}^{\omega}$ that encodes the position of $[\vec{v}]$ within the face of the normalization cube defined by $a$. This suffix is obtained as follows. Assume that we have $a\in\{-i,+i\}$, with $1\leq i\leq n-m$ (which implies $[v]_{i}\in\{-1/2,1/2\}$). We turn $[\vec{v}]$ into $[[\vec{v}]]=([v_{1}],\ldots,[v_{i-1}],[v_{i+1}],\ldots,[v_{n-m}])+(1/2,1/2,\ldots,1/2)$, i.e., we remove the $i$-th vector component, and offset the result in order to obtain $[[\vec{v}]]\in[0,1]^{n-m-1}$. We then define $w\in\{0,1\}^{\omega}$ as a word such that $0\star w$ is a serialized binary encoding of $[[\vec{v}]]$. Note that some vectors $\vec{v}$ may belong to several faces of the normalization cube, hence their normalized form may admit multiple encodings. This is not problematic, provided that the structure ${\cal D}_{1}$ handles these encodings consistently.

In summary, the structure ${\cal D}_{1}$ is an acyclic decision graph that partitions the space of normalized vectors according to their destination components. Each prefix $w_{k}$ of length $k$ read by ${\cal D}_{1}$ corresponds to a convex region $R_{w_{k}}\subset\mathbb{R}^{n}$ that is conical in every neighborhood of any element of $Q_{1}$, with this element as apex. The situation is similar to that discussed in Section 3.4: If in a sufficiently small neighborhood of any point of $Q_{1}$, the set of components of $P$ covered by $R_{w_{k}}$ contains a unique minimal component $Q_{2}$ with respect to the incidence order $\prec$, then $w_{k}$ leads to $Q_{2}$. Otherwise, the decision process is not yet complete, and additional transitions have to be followed in ${\cal D}_{1}$.

We are now ready to describe our proposed data structure for representing arbitrary polyhedra of $\mathbb{R}^{n}$, with $n\in\mathbb{N}_{>0}$. Recall that we assume w.l.o.g. that the polyhedra we consider are conical in $\mathbb{R}^{n}$ with respect to the apex $\vec{0}$.

A procedure for checking whether a given vector belongs to a polyhedron represented by an IRVA has already been outlined in Section 4.4.2. In the case of a polyhedron $P\subseteq\mathbb{R}^{n}$ that is not conical, an IRVA can be obtained for its representing cone $\overline{P}\subset\mathbb{R}^{n+1}$, as discussed in Section 4.1. In this case, checking whether a vector $(v_{1},v_{2},\ldots,v_{n})\in\mathbb{R}^{n}$ belongs to $P$ simply reduces to determining whether $(v_{1},v_{2},\ldots,v_{n},1)$ belongs to $\overline{P}$, which is done by the algorithm of Section 4.4.2.

An IRVA $(n,S_{I},S_{E},s_{0},\Delta)$ can be minimized in order to reduce its number of implicit and explicit states. Since the transition relation $\Delta$ is acyclic, the explicit and implicit states can be processed in a bottom-up order, starting from the implicit states with the largest vector spaces. At each step, reduction rules are applied in order simplify the current structure. A first rule is aimed at merging states that are indistinguishable: If two explicit states share the same successors, they can be merged. In the case of two implicit states, one additionally has to check that their associated vector spaces are equal, and that their polarities match. The purpose of the second rule is to get rid of unnecessary decisions. Consider a state $s$ (either implicit or explicit) with an outgoing transition that leads to an implicit state $s_{1}$, representing a polyhedron component $Q_{1}$. If all the implicit states $s_{i}$ that are reachable from $s$ are also reachable from $s_{1}$, then these implicit states represent polyhedral components $Q_{i}$ such that either $Q_{i}=Q_{1}$ or $Q_{1}\prec Q_{i}$. The state $s$ can then be absorbed into $s_{1}$, provided that $s$ is not an implicit state with a different polarity from the one of $s_{1}$. Note that this reduction rule correctly handles the case of a state $s$ that is implicit and does not correspond to a polyhedral component, but to a proper subset of the component $Q_{1}$ represented by $s_{1}$. For example, in $\mathbb{R}^{2}$, $s$ may correspond to a unidimensional line $x_{1}-x_{2}=0$ covered by the larger universal component $\mathbb{R}^{2}$ represented by $s_{1}$.

Proof sketch: The canonicity of a minimized IRVA ${\cal A}$ representing a polyhedron $P\subseteq\mathbb{R}^{n}$ is the consequence of two properties. First, the minimization algorithm is able to identify and merge together implicit states that correspond to identical polyhedral components, as well as to remove the implicit states that do not represent such components. This yields a one-to-one relationship between the implicit states of $\cal A$ and the polyhedral components of $P$. Second, the transition structure leaving an explicit state $s$ of $\cal A$ satisfies the following constraints. As discussed in Section 4.3, the state $s$ corresponds to a component $Q$ of $P$, and every prefix $w_{k}$ of length $k$ read from $s$ defines a convex conical region $R_{s,w_{k}}\subset\mathbb{R}^{n}$. If, in all sufficiently small neighborhoods of $Q$, the region $R_{s,w_{k}}$ covers a unique component $Q^{\prime}$ of $P$ that is minimal with respect to the incidence order, then the path reading $w_{k}$ from $s$ leads to the implicit state $s^{\prime}$ corresponding to $Q^{\prime}$. Provided that explicit states that have identical successors are merged, this property characterizes precisely the decision structure leaving $s$. Such structures will then be isomorphic in all minimized IRVA representing the same polyhedron. $\Box$

In order to apply a Boolean operator to two polyhedra $P_{1}$ and $P_{2}$ respectively represented by IRVA ${\cal A}_{1}$ and ${\cal A}_{2}$, one builds an IRVA ${\cal A}$ that simulates the concurrent behavior of ${\cal A}_{1}$ and ${\cal A}_{2}$. The procedure is analogous to the computation of the product of two finite-state automata. The initial implicit state of ${\cal A}$ is obtained by combining the initial states of ${\cal A}_{1}$ and ${\cal A}_{2}$, which amounts to intersecting their associated vector spaces, and applying the appropriate Boolean operator to their polarities. Each time an implicit state $s$ is added to ${\cal A}$, representing a polyhedron component $Q$, its successors are recursively explored. As explained in Section 4.3, each finite prefix $w_{k}$, of length $k$, read from $s$ corresponds to a convex conical region $R_{s,w_{k}}\subset\mathbb{R}^{n}$. The idea is to check, in a sufficiently small neighborhood $R$ of $Q$, whether $R_{s,w_{k}}$ covers unique minimal components $Q_{1}$ of $P_{1}$ and $Q_{2}$ of $P_{2}$, with respect to their respective incidence orders. In the positive case, one computes the intersection of the underlying vector spaces of $Q_{1}$ and $Q_{2}$. If the resulting vector space has a higher dimension than $\mbox{dim}(\mbox{VS}(s))$, as well as a non-empty intersection with $R_{s,w_{k}}$, a corresponding new implicit state is added to ${\cal A}$. In all other cases, the decision structure leaving $s$ has to be further developed, which amounts to creating new explicit states and new transitions between them, in order to read prefixes longer than $w_{k}$.

A key operation in the previous procedure is thus to compute, from an IRVA representing a polyhedron $P$, a component $Q$ of $P$, and a given convex conical region $C$, the unique minimal component of $P$ (if it exists) covered by $C$ in the neighborhood of $Q$, with respect to the incidence order $\prec$. This is done by exploring the IRVA starting from the implicit state representing $Q$. From a given implicit state $s$, the exploration only has to consider the paths labeled by words $w_{k}$ such that $C\,\cap\,R_{s,w_{k}}\neq\emptyset$, until they reach another implicit state. Let $S$ be the set of the implicit states reached this way. For each state in $S$, one checks whether its underlying vector space has a non-empty intersection with $C$. If this check succeeds for some nonempty subset of $S$, then the procedure returns its minimal component, or fails when such a component does not exist. Otherwise, it can be shown that the exploration can be continued from a single state chosen arbitrarily in $S$. The regions of space that are manipulated by this procedure are convex polyhedra, and can be handled by specific data structures [4].