Scott McCallum

On projection in CAD-based quantifier elimination with equational constraint

Collins [4] observed t.hat quant.ifirr climimtion problen~s oft,en have cquat,ional constraints. and he asserted that. such const,raints ~a.11 be used to reduce the projection wt,s required for cylindrical algebraic decomposition (cad) lmwd qumtifier climinat.ion. This paper provides a. dct,ailed partial validity proof for t,lic: restricted equatiom.1 projection method out,lirwd by Collins [Ll]. The proof is sufficicmt t.0 valitMo the use of t.he nwt~hod for the first. projection of the projection phase. A further consequence is t,hat the 111cth0d cm 1.~ used for both t,he first, and second projections in the case of a trivarintc qua.nt.ificr eliniination prolAcu~ llil.viIlg sufficient e~UatiOlli~l constraints.

Solving polynomial strict inequalities using cylindrical algebraic decomposition

We consider the problem of determining the consistency over the real numbers of a system of integral polynomial strict inequalities. This problem hasapplications in geometric modelling. The cylindrical algebraic decomposition (cad) algorithm [2] can be used to solve this problem, though not very efficiently. In this paper we present a less powerful version of the cad algorithm which can be used to solve the consistency problem for conjunctions of strict inequalities, and which runs considerably faster than the original method applied to this problem. In the case that a given conjunction of strict inequalities is consistent, the modified cad algorithm constructs solution points with rational coordinates

Cylindrical algebraic decompositions for boolean combinations

This article makes the key observation that when using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is not always the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This motivates our definition of a Truth Table Invariant CAD (TTICAD). We generalise the theory of equational constraints to design an algorithm which will efficiently construct a TTICAD for a wide class of problems, producing stronger results than when using equational constraints alone. The algorithm is implemented fully in Maple and we present promising results from experimentation.

On propagation of equational constraints in CAD-based quantifier elimination

Collins [4] observed that quantifier elimination problems often have equational constraints, and he asserted that such constraints can be used to reduce the projection sets required for cylindrical algebraic decomposition (cad) based quantifier elimination. This paper follows on from [11], and validates the use of a semi-restricted equational projection scheme throughout the projection phase of cad. The fully restricted projection scheme as originally proposed in [4] is proved valid for four variable problems under certain conditions.

Truth table invariant cylindrical algebraic decomposition

When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD).In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallums theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We also explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint.The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our contributions.

On using bi-equational constraints in CAD construction

This paper introduces an improved method for constructing cylindrical algebraic decompositions (CADs) for formulas with two polynomial equations as implied constraints. The fundamental idea is that neither of the varieties of the two polynomials is actually represented by the CAD the method produces, only the variety defined by their common zeros is represented. This allows for a substantially smaller projection factor set, and for a CAD with many fewer cells.In the current theory of CADs, the fundamental object is to decompose n-space into regions in which a polynomial equation is either identically true or identically false. With many polynomials, one seeks a decomposition into regions in which each polynomial equation is identically true or false independently. The results presented here are intended to be the first step in establishing a theory of CADs in which systems of equations are fundamental objects, so that given a system we seek a decomposition into regions in which the system is identically true or false --- which means each equation is no longer considered independently. Quantifier elimination problems of this form (systems of equations with side conditions) are quite common, and this approach has the potential to bring large problems of this type into the scope of what can be solved in practice. The special case of formulas containing two polynomial equations as constraints is an important one, but this work is also intended to be extended in the future to the more general case.

Deciding polynomial-exponential problems

This paper presents a decision procedure for a certain class of sentences of first order logic involving integral polynomials and the exponential function in which the variables range over the real numbers. The inputs to the decision procedure are prenex sentences in which only the outermost quantified variable can occur in the exponential function. The decision procedure has been implemented in the computer logic system REDLOG. Closely related work is reported in [2, 7, 16, 20, 24].

Iterated discriminants

It is shown that the discriminant of the discriminant of a multivariate polynomial has the same irreducible factors as the product of seven polynomials each of which is defined as the GCD of the generators of an elimination ideal. Under relatively mild conditions of genericity, three of these polynomials are irreducible and generate the corresponding elimination ideals, while the other four are equal to one. Moreover the irreducible factors of two of these polynomials have multiplicity at least two in the iterated discriminant and the irreducible factors of two others of the seven polynomials have multiplicity at least three. The proof involves an extended use of the notion of generic point of an algebraic variety and a careful study of the singularities of the hypersurface defined by a discriminant, which may be interesting by themselves.

On Testing a Bivariate Polynomial for Analytic Reducibility

LetKbe an algebraically closed field of characteristic zero. We present an efficient algorithm for determining whether or not a given polynomialf(x,y) inKx,y is analytically reducible overKat the origin. The algorithm presented is based upon an informal method sketched by Kuo (1989) which is in turn derived from ideas of Abhyankar (1988). The presentation contained herein emphasises the proofs of the algorithms correctness and termination, and is suitable for computer implementation. A polynomial worst case time complexity bound is proved for a partial version of the algorithm.

On delineability of varieties in CAD-based quantifier elimination with two equational constraints

Let <i>V</i> ⊂ R^<i>r</i> denote the real algebraic variety defined by the conjunction <i>f</i> = 0 ∧ <i>g</i> = 0, where <i>f</i> and <i>g</i> are real polynomials in the variables <i>x</i>₁, ..., <i>x</i>_<i>r</i> and let <i>S</i> be a submanifold of R^<i>r</i>-2. This paper proposes the notion of the <i>analytic delineability of V on S with respect to the last 2 variables</i>. It is suggested that such a notion could be useful in solving more efficiently certain quantifier elimination problems which contain the conjunction <i>f</i> = 0 ⊂ <i>g</i> = 0 as subformula, using a variation of the CAD-based method. Two bi-equational lifting theorems are proved which provide the basis for such a method.