Quantifier elimination for real closed fields by cylindrical algebraic decompostion

Collins, George E.

doi:10.1007/3-540-07407-4_17

Cited by 615 publications

(234 citation statements)

References 17 publications

(15 reference statements)

Supporting

Mentioning

232

Contrasting

Unclassified

Order By: Relevance

“…One of the main advantages of such constraint-based approaches is that they are goal-oriented. But, on the other hand, they still require the computation of several Gröbner Bases [13] or require first-order quantifier elimination [14,15], and known algorithms for those problems are, at least, of double exponential complexity. Alternatively, SAT Modulo Theory decision procedures and polynomial systems [16,17,9,18,19] could also, eventually, lead to decision procedures for invariant generation.…”

mentioning

confidence: 99%

Generating invariants for non-linear hybrid systems

Rebiha

Moura

Matringe

2015

Theoretical Computer Science

View full text Add to dashboard Cite

We describe powerful computational techniques, relying on linear algebraic methods, for generating ideals of non-linear invariants of algebraic hybrid systems. We show that the preconditions for discrete transitions and the Lie-derivatives for continuous evolution can be viewed as morphisms, and so can be suitably represented by matrices. We reduce the non-trivial invariant generation problem to the computation of the associated eigenspaces or nullspaces by encoding the consecution requirements as specific morphisms represented by such matrices. Our methods are the first to establish very general sufficient conditions that show the existence and allow the computation of invariant ideals. Our approach also embodies a strategy to estimate certain degree bounds, leading to the discovery of rich classes of inductive invariants. By reducing the problem to related linear algebraic manipulations we are able to address various deficiencies of other state-of-the-art invariant generation methods, including the efficient treatment of non-linear hybrid systems. Our approach avoids first-order quantifier eliminations, Gröbner basis computations or direct system resolutions, thereby circumventing difficulties met by other recent techniques.

show abstract

mentioning

confidence: 99%

Generating invariants for non-linear hybrid systems

Rebiha

Moura

Matringe

2015

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…Other techniques for eliminating existentially quantified variables can be used. For instance, one might use cylindrical algebraic decomposition [9] for specifications with non-linear arithmetic. In our case, whenever the specification σ does not belong to linear arithmetic, the FM theory solver is not called.…”

Section: Cegis(t ) With a Theory Solver Based On Fm Eliminationmentioning

confidence: 99%

Counterexample Guided Inductive Synthesis Modulo Theories

Abate

David

Kesseli³

et al. 2018

Computer Aided Verification

View full text Add to dashboard Cite

Abstract. Program synthesis is the mechanised construction of software. One of the main difficulties is the efficient exploration of the very large solution space, and tools often require a user-provided syntactic restriction of the search space. We propose a new approach to program synthesis that combines the strengths of a counterexample-guided inductive synthesizer with those of a theory solver, exploring the solution space more efficiently without relying on user guidance. We call this approach CEGIS(T ), where T is a first-order theory. In this paper, we focus on one particular challenge for program synthesizers, namely the generation of programs that require non-trivial constants. This is a fundamentally difficult task for state-of-the-art synthesizers. We present two exemplars, one based on Fourier-Motzkin (FM) variable elimination and one based on first-order satisfiability. We demonstrate the practical value of CEGIS(T ) by automatically synthesizing programs for a set of intricate benchmarks.

show abstract

“…Moreover, at each iteration, the solver produces increasingly better configurations as it attempts to converge on the solution; we can directly study and visualise these intermediate configurations as a spatial history of configurations [17,18] giving further insight into the nature of the problem at hand. However, a key limitation is that 2 Tarski famously proved that the theory of real-closed fields is decidable via quantifier elimination (see [2,12,13] for an overview and algorithms); i.e. in a finite amount of time we can determine the consistency (or inconsistency) of any formula consisting of quantifiers (∀, ∃) over the reals, and polynomial equations and inequalities combined using logical connectors (∧, ∨, ¬).…”

Section: A Motivations For Utilising Geometric Constraint Solvingmentioning

confidence: 99%

Encoding Relative Orientation and Mereotopology Relations with Geometric Constraints in CLP(QS)

Schultz

Bhatt

2015

Proceedings of the LQMR 2015 Workshop

View full text Add to dashboard Cite

Abstract-One approach for encoding the semantics of spatial relations within a declarative programming framework is by systems of polynomial constraints. However, solving such constraints is computationally intractable in general (i.e. the theory of realclosed fields), and thus far the proposed symbolic algebraic methods do not scale to real world problems. In this paper we address this intractability by investigating the use of constructive geometric constraint solving (in combination with numerical optimisation) within the context of constraint logic programming over qualitative spaces, CLP(QS). We present novel geometric encodings for relative orientation and mereotopology relations and show that our encodings are significantly more robust than other proposed approaches for directly encoding inequalities, due to our encodings being based on a standard, well known set of relations encoded as quadratic equations. Our encodings are general (i.e. not implementation specific) and can thus be directly employed in all standard geometric constraint solvers, particularly solvers that are prominent within the Computer Aided Design and Manufacturing communities. We empirically evaluate our approach on a range of benchmark problems from the Qualitative Spatial Reasoning community, and show that our method outperforms other symbolic algebraic approaches to spatial reasoning by orders of magnitude on those benchmark problems (such as Cylindrical Algebraic Decomposition).

show abstract

Quantifier elimination for real closed fields by cylindrical algebraic decompostion

Cited by 615 publications

References 17 publications

Generating invariants for non-linear hybrid systems

Generating invariants for non-linear hybrid systems

Counterexample Guided Inductive Synthesis Modulo Theories

Encoding Relative Orientation and Mereotopology Relations with Geometric Constraints in CLP(QS)

Contact Info

Product

Resources

About