Truth Table Invariant
Cylindrical Algebraic Decomposition

A Tarski formula $F(x_{1},\ldots,x_{n})$ is a Boolean combination ($\land,\lor,\neg,\rightarrow$) of statements about the signs, ($=0,>0,<0$, but therefore $\neq 0,\geq 0,\leq 0$ as well), of certain polynomials $f_{i}(x_{1},\ldots,x_{n})$ with integer coefficients. Such statements may involve the universal or existential quantifiers ($\forall,\exists$). We denote by QFF a quantifier-free Tarski formula.

Given a quantified Tarski formula

(where $Q_{i}\in\{\forall,\exists\}$ and $F$ is a QFF) the quantifier elimination problem is to produce $\psi(x_{1},\ldots,x_{k})$, an equivalent QFF to (1).

Collins developed CAD as a tool for quantifier elimination over the reals. He proposed to decompose $\mathbb{R}^{n}$ cylindrically such that each cell was sign-invariant for all polynomials $f_{i}$ used to define $F$. Then $\psi$ would be the disjunction of the defining formulae of those cells $c_{i}$ in $\mathbb{R}^{k}$ such that (1) was true over the whole of $c_{i}$, which due to sign-invariance is the same as saying that (1) is true at any one sample point of $c_{i}$.

A complete description of Collins’ original algorithm is given by Arnon et al. (1984a). The first phase, projection, applies a projection operator repeatedly to a set of polynomials, each time producing another set in one fewer variables. Together these sets contain the projection polynomials. These are used in the second phase, lifting, to build the CAD incrementally. First $\mathbb{R}$ is decomposed into cells which are points and intervals corresponding to the real roots of the univariate polynomials. Then $\mathbb{R}^{2}$ is decomposed by repeating the process over each cell in $\mathbb{R}$ using the bivariate polynomials at a sample point. Over each cell there are sections (where a polynomial vanishes) and sectors (the regions between) which together form a stack. Taking the union of these stacks gives the CAD of $\mathbb{R}^{2}$. This is repeated until a CAD of $\mathbb{R}^{n}$ is produced. At each stage the cells are represented by (at least) a sample point and an index: a list of integers corresponding to the ordered roots of the projection polynomials which locates the cell in the CAD.

To conclude that a CAD produced in this way is sign-invariant we need delineability. A polynomial is delineable in a cell if the portion of its zero set in the cell consists of disjoint sections. A set of polynomials are delineable in a cell if each is delineable and the sections of different polynomials in the cell are either identical or disjoint. The projection operator used must be defined so that over each cell of a sign-invariant CAD for projection polynomials in $r$ variables (the word over meaning we are now talking about an $(r+1)$-dim space) the polynomials in $r+1$ variables are delineable.

The output of this and subsequent CAD algorithms (including the one presented in this paper) depends heavily on the variable ordering. We usually work with polynomials in $\mathbb{Z}[{\bf x}]=\mathbb{Z}[x_{1},\ldots,x_{n}]$ with the variables, ${\bf x}$, in ascending order (so we first project with respect to $x_{n}$ and continue to reach univariate polynomials in $x_{1}$). The main variable of a polynomial (${\rm mvar}$) is the greatest variable present with respect to the ordering.

CAD has doubly exponential complexity in the number of variables (Brown and Davenport, 2007; Davenport and Heintz, 1988). There now exist algorithms with better complexity for some CAD applications (see for example Basu et al. (1996)) but CAD implementations often remain the best general purpose approach. There have been many developments to the theory since Collin’s treatment, including the following:

• •

  Improvements to the projection operator (Hong, 1990; McCallum, 1988, 1998; Brown, 2001; Han et al., 2014), reducing the number of projection polynomials computed.

• •

  Algorithms to identify the adjacency of cells in a CAD (Arnon et al., 1984b, 1988) and following from this the idea of clustering (Arnon, 1988) to minimise the lifting.

• •

  Partial CAD, introduced by Collins and Hong (1991), where the structure of $F$ is used to lift less of the decomposition of $\mathbb{R}^{k}$ to $\mathbb{R}^{n}$, if it is sufficient to deduce $\psi$.

• •

  The theory of equational constraints, (McCallum, 1999, 2001; Brown and McCallum, 2005), also aiming to deduce $\psi$ itself, this time using more efficient projections.

• •

  The use of certified numerics in the lifting phase to minimise the amount of symbolic computation required (Strzeboński, 2006; Iwane et al., 2009).

• •

  New approaches which break with the normal projection and lifting model: local projection (Strzeboński, 2014), the building of single CAD cells (Brown, 2013; Jovanovic and de Moura, 2012) and CAD via Triangular Decomposition (Chen et al., 2009b). The latter is now used for the CAD command built into Maple, and works by first creating a cylindrical decomposition of complex space.

Brown (1998) defined a truth-invariant CAD as one for which a formula had invariant truth value on each cell. Given a QFF, a sign-invariant CAD for the defining polynomials is trivially truth-invariant. Brown considered the refinement of sign-invariant CADs whilst maintaining truth-invariance, while some of the developments listed above can be viewed as methods to produce truth-invariant CADs directly. We define a new but related type of CAD, the topic of this paper.

A sign-invariant CAD for all polynomials occurring in a list of formulae would clearly be a TTICAD for the list. However, we aim to produce smaller TTICADs for many such lists. We will achieve this by utilising the presence of equational constraints, a technique first suggested by Collins (1998) with key theory developed by McCallum (1999).

In Sections 3 and 4 we will describe how TTICADs can be produced efficiently when there are ECs present in the list of formulae. There are two reasons to use this theory.

1. 1.

   As a tool to build a truth-invariant CAD efficiently: If a parent formula $\phi^{*}$ is built from the formulae $\{\phi_{i}\}$ then any TTICAD for $\{\phi_{i}\}$ is also truth-invariant for $\phi^{*}$.

   We note that for such a formula a TTICAD may need to contain more cells than a truth-invariant CAD. For example, consider a cell in a truth-invariant CAD for $\phi^{*}=\phi_{1}\lor\phi_{2}$ within which $\phi_{1}$ is always true. If $\phi_{2}$ changed truth value in such a cell then it would need to be split in order to achieve a TTICAD, but this is unnecessary for a truth-invariant CAD of $\phi^{*}$.

   Nevertheless, we find that our TTICAD theory is often able to produce smaller truth-invariant CADs than any other available approach. We demonstrate the savings offered via worked examples introduced in the next subsection.

2. 2.

   When given a problem for which truth table invariance is required: That is, a problem for which the list of formulae are not derived from a larger parent formula and thus a truth-invariant CAD for their disjunction may not suffice.

   For example, decomposing complex space according to a set of branch cuts for the purpose of algebraic simplification (Bradford and Davenport, 2002; Phisanbut et al., 2010). Here the idea is to represent each branch cut as a semi-algebraic set to give input admissible to CAD, (recent progress on this has been described by England et al. (2013)). Then a TTICAD for the list of formulae these sets define provides the necessary decomposition. Example 33 is from this class.

To demonstrate our ideas we will provide details for two worked examples. Assume we have the variable ordering $x\prec y$ (meaning 1-dimensional CADs are with respect to $x$) and consider the following polynomials, graphed in Figure 1.

Suppose we wish to find the regions of $\mathbb{R}{}^{2}$ where the following formula is true:

Both Qepcad (Brown, 2003) and Maple 16 (Chen et al., 2009b) produce a sign-invariant CAD for the polynomials with 317 cells. Then by testing the sample point from each region we can systematically identify where the formula is true.

At first glance it seems that the theory of ECs is not applicable to $\Phi$ as neither $f_{1}=0$ nor $f_{2}=0$ is logically implied by $\Phi$. However, while there is no explicit EC we can observe that $f_{1}f_{2}=0$ is an implicit constraint of $\Phi$. Using Qepcad with this declared (an implementation of (McCallum, 1999)) gives a CAD with 249 cells. Later, in Section 3.3 we demonstrate how a TTICAD with 105 cells can be produced.

We also consider the related problem of identifying where

is true. As above, we could use a sign-invariant CAD with 317 cells, but this time there is no implicit EC. In Section 3.3 we produce a TTICAD with 183 cells.

We review the projection operators of McCallum (1998, 1999) in Section 2. The former produces sign-invariant CADs¹¹1Actually order-invariant CADs (see Definition 3).and the latter CADs truth-invariant for a formula with an EC. The review is necessary since we use some of this theory to verify our new algorithm. It also allows us to compare our new contribution to these existing approaches. For this purpose we provide new complexity analyses of these existing theories in Section 2.3.

Sections 3 and 4 present our new TTICAD projection operator and verified algorithm. They follow Sections 2 and 3 of our ISSAC 2013 paper (Bradford et al., 2013a), but instead of requiring all QFFs to have an EC the theory here is applicable to all QFFs (producing savings so long as one has an EC). The strengthening of the theory means that a TTICAD can now be produced for $\Psi$ in Section 1.3 as well as $\Phi$. This extension is important since it means TTICAD theory now applied to cases where there can be no overall implicit EC for a parent formula. In these cases the existing theory of ECs is not applicable and so the comparative benefits offered by TTICAD are even higher.

In Section 5 we discuss how the theory of reduced projection operators also allows for improvements in the lifting phase. This is true for the existing theory also but the discovery was only made during the development of TTICAD. In Section 6 we present a complexity analysis of our new contributions from Sections 3 $-$ 5, demonstrating their benefit over the existing theory from Section 2. We have implemented the new ideas in a Maple package, discussed in Section 7. In particular, Section 7.3 summarises (Bradford et al., 2013b) on the choices required when using TTICAD and heuristics to help. Experimental results for our implementation (extending those in our ISSAC 2013 paper) are given in Section 8, before we finish in Section 9 with conclusions and future work.

Data access statement: Data directly supporing this paper (code, Maple and Qepcad input) is openly available from http://dx.doi.org/10.15125/BATH-00076.

Throughout the paper we let $\mathop{\rm cont}\nolimits,\mathop{\rm prim}\nolimits,\mathop{\rm disc}\nolimits,\mathop{\rm coeff}\nolimits$ and $\mathop{\rm ldcf}\nolimits$ denote the content, primitive part, discriminant, coefficients and leading coefficient of polynomials respectively (in each case taken with respect to a given main variable). Similarly, we let $\mathop{\rm res}\nolimits$ denote the resultant of a pair of polynomials. When applied to a set of polynomials we interpret these as producing sets of polynomials, so for example

The first improvements to Collins original projection operator were given by McCallum (1988) and Hong (1990). They were both subsets of Collins operator, meaning fewer projection polynomials, fewer cells in the CADs produced and quicker computation time. McCallum’s is actually a strict subset of Hong’s, however, it cannot be guaranteed correct (incorrectness is detected in the lifting process) for a certain class of (statistically rare) input polynomials, where Hong’s can.

Additional improvements have been suggested by Brown (2001) and Lazard (1994). The former required changes to the lifting phase while the latter had a flawed proof of validity (with current unpublished work suggesting it can still be safely used in many cases). In this paper we will focus on McCallum’s operators, noting that the alternatives could likely be extended to TTICAD theories too if desired. McCallum’s theory is based around the following condition, which implies sign-invariance.

Recall that a set $A\subset\mathbb{Z}[{\bf x}]$ is an irreducible basis if the elements of $A$ are of positive degree in the main variable, irreducible and pairwise relatively prime. Let $A$ be a set of polynomials and $B$ an irreducible basis of the primitive part of $A$. Then

defines the operator of McCallum (1988). We can assume some trivial simplifications such as the removal of constants and exclusion of entries identical to a previous one (up to constant multiple). The main theorem underlying the use of $P$ follows.

Theorem 4 means that we can use $P$ in place of Collins’ projection operator to produce sign-invariant CADs so long as none of the projection polynomials with main variable $x_{k}$ vanishes on a cell of the CAD of $\mathbb{R}^{k-1}$; a condition that can be checked when lifting. Input with this property is known as well-oriented. Note that although McCallum’s operator produces order-invariant CADs, a stronger property than sign-invariance, it is actually more efficient that the pre-existing sign-invariant operators. We examine the complexity of CAD using this operator in Section 2.3.

The main result underlying CAD simplification in the presence of an EC follows.

Figure 2 gives a graphical representation of the question answered by Theorem 5. Here we consider polynomials $f(x,y,z)$ and $g(x,y,z)$ of positive degree in $z$ whose resultant $r$ is non-zero, and a connected subset $S\subset\mathbb{R}^{2}$ in which $r$ is order-invariant. We further suppose that $f$ is delineable on $S$ (noting that Theorem 4 with $n=3$ and $A=\{f\}$ provides sufficient conditions for this). We ask whether $g$ is sign-invariant in the sections of $f$ over $S$. Theorem 5 answers this question affirmatively: the real variety of $g$ either aligns with a given section of $f$ exactly (as for the bottom section of $f$ in Figure 2), or has no intersection with such a section (as for the top). The situation at the middle section of $f$ cannot happen.

Theorem 5 thus suggests a reduction of the projection operator $P$ relative to an EC $f=0$: take only $P(f)$ together with the resultants of $f$ with the non-ECs. Let $A$ be a set of polynomials, $E\subset A$ contain only the polynomial defining the EC, $F$ be a square free basis of $A$, and $B$ be the subset of $F$ which is a square-free basis for $E$. The operator

was presented by McCallum (1999) along with an algorithm to produce a CAD truth-invariant for the EC and sign-invariant for the other polynomials when the EC was satisfied. It worked by applying first $P_{E}(A)$ and then building an order-invariant CAD of $\mathbb{R}^{n-1}$ using $P$. We call such CADs invariant with respect to an equational constraint. Note that as with McCallum (1999) the algorithm only works for input satisfying a well-orientedness condition. Full details of the verification are given by McCallum (1999) and a complexity analysis is given in the next subsection.

We provide complexity analyses of the algorithms from McCallum (1998, 1999) for comparison with our new contributions later. An analysis for the latter has not been published before, while the analysis for the former differs substantially from the one in (McCallum, 1985): instead of focusing on computation time, we examine the number of cells in the CAD of $\mathbb{R}^{n}$ produced: the cell count. We compare the dominant terms in a cell count bound for each algorithm studied. This focus avoids calculations with less relevant parameters, identical for all the algorithms. We note that all CAD experimentation shows a strong correlation between the number of cells produced and the computation time.

Our key parameters are the number of variables $n$, the number of polynomials $m$ and their maximum degree $d$ (in any one variable). Note that these are all restricted to positive integer values. We make much use of the following concepts.

So for example, the set $\{x^{2}+1,x^{2}+y^{3}\}$ has combined degree $4$ (since the product has degree $4$ in $x$ and degree $3$ in $y$).

So for example, the set of polynomials $\{xy^{3}-x,x^{4}-xy,x^{4}-y^{4}+1\}$ has combined degree $9$ and thus the $(1,9)$-property. However, by partitioning it into three sets of one polynomial each, it also has the $(3,4)$-property. Partitioning into 2 sets will show it to have the $(2,5)$, $(2,7)$ and $(2,8)$-properties also.

The following result follows simply from the definitions.

This contrasts with the facts that taking a square-free basis may not reduce the combined degree, but may cause exponential blow-up in the number of polynomials.

Note that in the case $\ell=2$ we have $\left\lceil\tfrac{m}{2}\right\rceil=\left\lfloor\tfrac{m+1}{2}\right\rfloor$.

Though not used in this paper, we note an advantage of the $(m,d)$-property over the $(1,md)$-property is a better bound on root separation: any two roots require $O(2d)$ subdivisions to isolate, rather than the $O(md)$ implied by considering the product of all polynomials.

We also recall the following classic identities for polynomials $f,g,h$:

where $d$ is the degree of $f$, $f^{\prime}$ its derivative and $a_{d}$ its leading coefficient (all taken with respect to the given main variable).

This concerns a single projection, and we must apply it recursively to consider the full set of projection polynomials. Weakening the bound as in the following allows for a closed form solution.

We consider the growth in projection polynomials and their degree when using the operator $P$ in Table 1. Here the column headings refer not to the number of polynomials and their degree, but to the number of sets and their combined degree when applying Definition 7. We start with $m$ polynomials of degree $d$ and after one projection have a set with the $(M,2d^{2})$ property, using $M$ from Lemma 11. We then use Corollary 12 to model the growth in subsequent projections, and a simple induction to fill in the table.

The size of the CAD produced depends on the number of real roots of the projection polynomials. We can hence bound the number of real roots in a set of polynomials with the $(m,d)$-property with $md$ (in practice many of them will be strictly complex). We can therefore bound the number of real roots of the univariate projection polynomials by the product of the two entries in the row of Table 1 for 1 variable. The number of cells in the CAD of $\mathbb{R}^{1}$ is bounded by twice this plus 1. Similarly, the total number of cells in the CAD of $\mathbb{R}^{n}$ is bounded by the product of $2K+1$ where $K$ varies through the Product column of Table 1, i.e. by

Omitting the $+1$ will leave us with the dominant term of the bound, which can be calculated explicitly as

where the inequality was introduced by omitting the floor function in (9). This may be compared with the bound in Theorem 6.1.5 of McCallum (1985), with the main differences explained by Remark 13(2).

We now turn our focus to CAD invariant with respect to an EC. Recall that we use operator $P_{E}(A)$ for the first projection only and $P(A)$ thereafter. Hence we use Corollary 12 for the bulk of the analysis, and the next lemma when considering the first projection.

We can use Table 1 to model the growth in projection polynomials for the algorithm in (McCallum, 1999) as well, since the only difference will be the number of polynomials produced by the first projection, and thus the value of $M$. Hence the dominant term in the bound on the total number of cells is given again by (11), which in this case becomes (upon omitting the floor)

Since $P_{E}(A)$ is a subset of $P(A)$ a CAD invariant with respect to an EC should certainly be simpler than a sign-invariant CAD for the polynomials involved. Indeed, comparing the different values of $M$ we see that

Comparing the dominant terms in the cell count bounds, (14) and (12), we see the main effect is a decrease in one of the double exponents by $1$.

In (McCallum, 1999) the central concept is the reduced projection of a set of polynomials $A$ relative to a subset $E$ (defining the EC). The full projection operator is applied to $E$ and then supplemented by the resultants of polynomials in $E$ with those in $E\setminus A$, since the latter group only effect the truth of the formula when they share a root with the former. We extend this idea to define a projection for a list of sets of polynomials (derived from a list of formulae), some of which may have subsets (derived from ECs).

For simplicity in (McCallum, 1999) the concept is first defined for the case when $A$ is an irreducible basis. We emulate this approach, generalising for other cases by considering contents and irreducible factors of positive degree when verifying the algorithm in Section 4. So let $\mathcal{A}=\{A_{i}\}_{i=1}^{t}$ be a list of irreducible bases $A_{i}$ and let $\mathcal{E}=\{E_{i}\}_{i=1}^{t}$ be a list of subsets $E_{i}\subseteq A_{i}$. Put $A=\bigcup_{i=1}^{t}A_{i}$ and $E=\bigcup_{i=1}^{t}E_{i}$. Note that we use the convention of uppercase Roman letters for sets of polynomials and calligraphic letters for lists of these.

Theorem 16 is the key tool for the verification of our TTICAD algorithm in Section 4. It allows us to conclude the output is correct so long as no $f\in E$ vanishes identically on the lower dimensional manifold, $S$. A polynomial $f$ in $r$ variables that vanishes identically at a point $\alpha\in\mathbb{R}{}^{r-1}$ is said to be nullified at $\alpha$.

The theory of this subsection appears identical to the work in (Bradford et al., 2013a). The difference is in the application of the theory in Section 4. We suppose that the input is a list of QFFs, $\{\phi_{i}\}$, with each $A_{i}$ defined from the polynomials in each $\phi_{i}$. In (Bradford et al., 2013a) there was an assumption (no longer made) that each of these formulae had a designated EC $f_{i}=0$ from which the subsets $E_{i}$ are defined. Instead, we define $E_{i}$ to be a basis for $\{f_{i}\}$ if there is such a designated EC and define $E_{i}=A_{i}$ otherwise. That is, we need to treat all the polynomials in QFFs with no EC with the importance usually reserved for ECs.

It is clear that in general the reduced projection $P_{\mathcal{E}}(\mathcal{A})$ will lead to fewer projection polynomials than using the full projection $P$. However, a comparison with the existing theory of equational constraints requires a little more care.

First, we note that the TTICAD theory is applicable to a sequence of formulae while the theory of McCallum (1999) is applicable only to a single formula. Hence if the truth value of each QFF is needed then TTICAD is the only option; a truth-invariant CAD for a parent formula will not necessarily suffice. Second we note that even if the sequence do form a parent formula then this must have an overall EC to use (McCallum, 1999) while the TTICAD theory is applicable even if this is not the case.

Let us consider the situation where both theories are applicable, i.e. we have a sequence of formulae (forming a parent formula) for which each has an EC and thus the parent formula an implicit EC (their product). In the context of Section 1.2 this corresponds to using $\prod_{i}f_{i}$ as the EC. The implicit EC approach would correspond to using the reduced projection $P_{E}(A)$ of (McCallum, 1999), with $E=\cup_{i}E_{i}$ and $A=\cup_{i}A_{i}$. We make the simplifying assumption that $A$ is an irreducible basis. In general $P_{\mathcal{E}}(\mathcal{A})$ will still contain fewer polynomials than $P_{E}(A)$ since $P_{E}(A)$ contains all resultants res$(f,g)$ where $f\in E_{i},g\in A_{j}$ (and $g\notin E$), while $P_{\mathcal{E}}(\mathcal{A})$ contains only those with $i=j$ (and $g\notin E_{i}$). Thus even in situations where the previous theory applies there is an advantage in using the new TTICAD theory. These savings are highlighted by the worked examples in the next subsection and the complexity analysis later.

In Section 4 we define an algorithm for producing TTICADs. First we illustrate the savings with our worked examples from Section 1.3, which satisfy the simplifying assumptions from Section 3.1.

We start by considering $\Phi$ from equation (2). In the notation above we have:

We construct the reduced projection sets for each $\phi_{i}$,

and the cross-resultant set

$P_{\mathcal{E}}(\mathcal{A})$ is then the union of these three sets. In Figure 3 we plot the polynomials (solid curves) and identify the 12 real solutions of $P_{\mathcal{E}}(\mathcal{A})$ (solid vertical lines). We can see the solutions align with the asymptotes of the $f_{i}$’s and the important intersections (those of $f_{1}$ with $g_{1}$ and $f_{2}$ with $g_{2}$).

If we were to instead use a projection operator based on an implicit EC $f_{1}f_{2}=0$ then in the notation above we would construct $P_{E}(A)$ from $A=\{f_{1},f_{2},g_{1},g_{2}\}$ and $E=\{f_{1},f_{2}\}$. This set provides an extra 4 solutions (the dashed vertical lines) which align with the intersections of $f_{1}$ with $g_{2}$ and $f_{2}$ with $g_{1}$. Finally, if we were to consider $P(A)$ then we gain a further 4 solutions (the dotted vertical lines) which align with the intersections of $g_{1}$ and $g_{2}$ and the asymptotes of the $g_{i}$’s. In Figure 4 we magnify a region to show explicitly that the point of intersection between $f_{1}$ and $g_{1}$ is identified by $P_{\mathcal{E}}(\mathcal{A})$, while the intersections of $g_{2}$ with both $f_{1}$ and $g_{1}$ are ignored.

The 1-dimensional CAD produced using $P_{\mathcal{E}}(\mathcal{A})$ has 25 cells compared to 33 when using $P_{E}(A)$ and 41 when using $P(A)$. However, it is important to note that this reduction is amplified after lifting (using Theorem 16 and and Algorithm 1). The 2-dimensional TTICAD has 105 cells and the sign-invariant CAD has 317. Using Qepcad to build a CAD invariant with respect to the implicit EC gives us 249 cells.

Next we consider determining the truth of $\Psi$ from equation (3). This time

and so $P_{E_{1}}(A_{1})$ is as above but $P_{E_{2}}(A_{2})$ contains an extra polynomial $x-4$ (the coefficient of $y$ in $g_{2}$). The cross-resultant set ${\rm RES}^{\times}(\mathcal{E})$ also contains an extra polynomial,

These two extra polynomials provide three extra real roots and hence the 1-dimensional CAD produced using $P_{\mathcal{E}}(\mathcal{A})$ this time has 31 cells.

In Figure 5 we again graph the four curves this time with solid vertical lines highlighting the real solutions of $P_{\mathcal{E}}(\mathcal{A})$. By comparing with Figure 3 we see that more points in the CAD of $\mathbb{R}^{1}$ have been identified for the TTICAD of $\Psi$ than the TTICAD of $\Phi$ (15 instead of 12) but that there is still a saving over the sign-invariant CAD (which had 20, the five extra solutions indicated by dotted lines). The lack of an EC in the second clause has meant that the asymptote of $g_{2}$ and its intersections with $f_{1}$ have been identified. However, note that the intersections of $g_{1}$ with $f_{2}$ and $g_{2}$ and have not been. Figure 6 magnifies a region of Figure 5. Compare with Figure 4 to see the dashed line has become solid, while the dotted line remains unidentified by the TTICAD.

Note that we are unable to use (McCallum, 1999) to study $\Psi$ as there is no polynomial equation logically implied (either explicitly or implicitly) by this formula. Hence there are no dashed lines and the choice is between the sign-invariant CAD with 317 cells or the TTICAD, which for this example has 183 cells.

We describe carefully Algorithm 1. This will create a TTICAD of $\mathbb{R}{}^{n}$ for a list of QFFs $\{\phi_{i}\}_{i=1}^{t}$ in variables ${\bf x}=x_{1}\prec x_{2}\prec\cdots\prec x_{n}$, where each $\phi_{i}$ has at most one designated EC $f_{i}=0$ of positive degree (there may be other non-designated ECs).

It uses a subalgorithm CADW, which was validated by McCallum (1998). The input of CADW is: $r$, a positive integer and $A$, a set of $r$-variate integral polynomials. The output is a boolean $w$ which if true is accompanied by an order-invariant CAD for $A$ (represented as a list of indices $I$ and sample points $S$).

Let $A_{i}$ be the set of all polynomials occurring in $\phi_{i}$. If $\phi_{i}$ has a designated EC then put $E_{i}=\{f_{i}\}$ and if not put $E_{i}=A_{i}$. Let $\mathcal{A}$ and $\mathcal{E}$ be the lists of the $A_{i}$ and $E_{i}$ respectively. Our algorithm effectively defines the reduced projection of $\mathcal{A}$ with respect to $\mathcal{E}$ in terms of the special case of this definition from the previous section. The definition amounts to

Here $C$ is the set of contents of all the elements of all $A_{i}$; $\mathcal{B}$ the list $\{B_{i}\}_{i=1}^{t}$ such that $B_{i}$ is the finest²²2A decomposition into irreducibles. This avoids various technical problems. squarefree basis for the set ${\rm prim}(A_{i})$ of primitive parts of elements of $A_{i}$ which have positive degree; and $\mathcal{F}$ is the list $\{F_{i}\}_{i=1}^{t}$, such that $F_{i}$ is the finest squarefree basis for ${\rm prim}(E_{i})$. (The reader may notice that this notation and the definition of $P_{\mathcal{E}}(\mathcal{A})$ here is analogous to the work in Section 5 of (McCallum, 1999).)

We shall prove that, provided the input satisfies the condition of well-orientedness given in Definition 18, the output of Algorithm 1 is indeed a TTICAD for $\{\phi_{i}\}$. We first recall the more general notion of well-orientedness from (McCallum, 1998). The boolean output of CADW is false if the input set was not well-oriented in this sense.

This condition is required for CADW since the validity of this algorithm relies on Theorem 4 which holds only when polynomials do not vanish identically. The conditions allows for a finite number of these nullifications since this indicates a problem on a zero cell, that is a single point. In such cases it is possible to replace the nullified polynomial by a so called delineating polynomial which is not nullified and can be used in place to ensure the delineability of the other. The use of these is part of the verified algorithm CADW (McCallum, 1998) and they are studied in detail by Brown (2005).

We now define our new notion of well-orientedness for the lists of sets $\mathcal{A}$ and $\mathcal{E}$.