Abstract Domains in Constraint Programming

Marie Pelleau

• Éditeur: ISTE Press - Elsevier
• Année: 2015
• ISBN:
  9780081004647
• Sujet: Computer Science
• Sous-sujet: Computer Engineering

Constraint Programming aims at solving hard combinatorial problems, with a computation time increasing in practice exponentially. The methods are today efficient enough to solve large industrial problems, in a generic framework. However, solvers are dedicated to a single variable type: integer or real. Solving mixed problems relies on ad hoc transformations. In another field, Abstract Interpretation offers tools to prove program properties, by studying an abstraction of their concrete semantics, that is, the set of possible values of the variables during an execution. Various representations for these abstractions have been proposed. They are called abstract domains. Abstract domains can mix any type of variables, and even represent relations between the variables.

In this work, we define abstract domains for Constraint Programming, so as to build a generic solving method, dealing with both integer and real variables. We also study the octagons abstract domain, already defined in Abstract Interpretation. Guiding the search by the octagonal relations, we obtain good results on a continuous benchmark. We also define our solving method using Abstract Interpretation techniques, in order to include existing abstract domains. Our solver, AbSolute, is able to solve mixed problems and use relational domains.

• Exploits the over-approximation methods to integrate AI tools in the methods of CP
• Exploits the relationships captured to solve continuous problems more effectively
• Learn from the developers of a solver capable of handling practically all abstract domains

1

State of the Art

Abstract

In this chapter, we present the notions upon which abstract interpretation (AI) is based and the principles of constraint programming (CP). We do not provide an exhaustive presentation of both areas, but rather give the notions needed for the understanding of this book. The concepts discussed include those of partially ordered sets, lattice and fixpoint, which are at the basis of the underlying theories in both fields. It also includes the in-place tools, such as narrowing and widening operators in AI or consistency and splitting operators in CP. Finally, the chapter presents an analysis of the similitudes between an AI and CP upon which rely the works presented in this book.

Keywords

Abstract interpretation (AI)

Concrete semantics

Constraint programming (CP)

Constraint satisfaction problem (CSP)

Large Hadron Collider (LHC)

Narrowing operator

Propagation loop

Transfer function

Widening operator

In this chapter, we present the notions upon which abstract interpretation (AI) is based and the principles of constraint programming (CP). We do not provide an exhaustive presentation of both areas, but rather give the notions needed for the understanding of this book. The concepts discussed include those of partially ordered sets, lattice and fixpoint, which are at the basis of the underlying theories in both fields. It also includes the in-place tools, such as narrowing and widening operators in AI or consistency and splitting operators in CP. Finally, the chapter presents an analysis of the similitudes between an AI and CP upon which rely the works presented in this book.

1.1 Abstract interpretation

The founding principles of AI were introduced in 1976 by Patrick and Cousot [COU 76]. In this section, we only present some aspects of AI that will be needed afterward. For a more complete presentation, see [COU 92a, COU 77a].

1.1.1 Introduction to abstract interpretation

One of the applications of AI is to automatically prove that a certain type of bug does not exist in a program and that there is no error during a program execution. Let us see an example.

Example 1.1

Backtrace

Consider the following program:

1: real x, y

2: x ← 2

3: y ← 5

4: y ← y ∗ (x − 2)

5: y ← x/y

The backtrace for this program is:

line	x	y
1	?	?
2	2	?
3	2	5
4	2	0
5	2	NaN Error: division by zero

In toy examples like this one, the backtrace allows us to quickly dete...