### Citations

1921 |
Theory of Linear and Integer Programming.
- Schrijver
- 1986
(Show Context)
Citation Context ...ork and conclusion follow in Section 6 and Section 7. For readability reasons, all proofs of the paper are reported in Appendix A. 1.1 Preliminaries We adhere to standard notation for linear algebra [=-=Schrijver 1986-=-], linear programming [Murty 1983] and (constraint) logic programming [Apt 1997; Jaffar and Maher 1994; Jaffar et al. 1998]. Linear algebra. R denotes the set of real numbers. Small capital letters (a... |

872 | Constraint logic programming, in:
- Jaffar, Lassez
- 1987
(Show Context)
Citation Context ... Appendix A. 1.1 Preliminaries We adhere to standard notation for linear algebra [Schrijver 1986], linear programming [Murty 1983] and (constraint) logic programming [Apt 1997; Jaffar and Maher 1994; =-=Jaffar et al. 1998-=-]. Linear algebra. R denotes the set of real numbers. Small capital letters (a, b, . . . ) denote column vectors, while capital letters (A, B, . . . ) denote matrices. 0 and 1 are column vectors with ... |

869 | Constraint logic programming: a survey,
- Jaffar, Maher
- 1994
(Show Context)
Citation Context ...e paper are reported in Appendix A. 1.1 Preliminaries We adhere to standard notation for linear algebra [Schrijver 1986], linear programming [Murty 1983] and (constraint) logic programming [Apt 1997; =-=Jaffar and Maher 1994-=-; Jaffar et al. 1998]. Linear algebra. R denotes the set of real numbers. Small capital letters (a, b, . . . ) denote column vectors, while capital letters (A, B, . . . ) denote matrices. 0 and 1 are ... |

487 |
A polynomial algorithm for linear programming,
- Khachiyan
- 1979
(Show Context)
Citation Context ...the issue of computational complexity of the type checking problem. First, we observe that the LPInfer procedure has a polynomial time complexity when a polynomial time algorithm (such as the one in [=-=Khachiyan 1979-=-]) is adopted for solving the required linear programming problems. By Theorem 3.7, Check(LPInfer) is then a polynomial time decision procedure for type assertions d1 ` c→ d2 such that types in d1 bel... |

409 |
Programming with constraints: an introduction.
- Marriott, Stuckey
- 1998
(Show Context)
Citation Context ...ding one, due to the presence of a worst-case constraint in it (see later on the worst testbed). The term testbed. It includes programs mixing linear constraints and terms, mainly from CLP textbooks [=-=Marriott and Stuckey 1998-=-]. As a special case, when clauses contain no constraint, pure Prolog programs belong to this testbed. Table II shows very low execution times, better than in the folk testbed. In fact, we observe tha... |

337 |
R.H.C.: The CLP(R) language and system.
- Jaffar, Michaylov, et al.
- 1992
(Show Context)
Citation Context ... lower bounds have no direct counterpart. Type assertions are at the basis of moding programs in constraint logic programming languages with linear constraints over reals and rationals, as in CLP(R) [=-=Jaffar et al. 1992-=-; Holzbaur 1995], ECLiPSe, SICStus Prolog, SWI Prolog, and many others. We conservatively extend the notion of well-moding [Apt 1997] from pure logic programming to CLP(R), proving useful properties i... |

230 | On the combinatorial and algebraic complexity of quantifier elimination’,
- Basu, Pollack, et al.
- 1996
(Show Context)
Citation Context ... core of the procedure in the case of linear polynomials over reals is the Fourier-Motzkin projection method [Schrijver 1986]. Although in the worst case quantifier elimination is doubly exponential [=-=Basu et al. 1996-=-; Davenport and Heintz 1988], approaches efficient-in-practice have been proposed and successfully applied to theorem proving and program verification. We mention partial cylindrical algebraic decompo... |

227 | Partial Cylindrical Algebraic Decomposition for Quantier Elimination,” - Collins, Hong - 1991 |

213 |
On the computational complexity and geometry of the first-order theory of the reals. Parts I–III.
- Renegar
- 1992
(Show Context)
Citation Context ...ions Let us concentrate now on the problem of checking the validity of a type assertion. In principle, formulas as in (1) can be checked by real quantifier elimination methods [Dolzmann et al. 1998b; =-=Renegar 1992-=-]. Quantifier elimination traces back to Tarski’s decision procedure [Van Den Vries 1988] for first order formula over real polynomials. The core of the procedure in the case of linear polynomials ove... |

212 | The execution algorithm of Mercury, an efficient purely declarative logic programming language.
- Somogyi, Henderson, et al.
- 1996
(Show Context)
Citation Context ...ed answer characterization. They are at the basis of several techniques for program analysis, such as Typing Linear Constraints · 27 termination [Etalle et al. 1999], transformation and optimization [=-=Somogyi et al. 1996-=-]. The next result shows that the mentioned properties hold for the proposed extension of well-moding to CLP(R). By a left-derivation we mean a derivation via the leftmost selection rule. Theorem 4.5.... |

210 |
From logic programming to Prolog
- Apt
- 1997
(Show Context)
Citation Context ...epresent a sophisticated scheme for programming with constraints. Modern constraint logic programming languages, such as Mercury [Somogyi et al. 1996; Becket et al. 2006], offer the notion of moding [=-=Apt 1997-=-] as program annotations allowing the programmer to specify the input-output behavior of predicate arguments. Modes are at the basis of compiler optimizations, program transformations and termination ... |

130 | Redlog: Computer algebra meets computer logic.
- Dolzmann, Sturm
- 1997
(Show Context)
Citation Context ...B systems [Collins and Hong 1991; Brown 2003] and available in the Mathematica tool [Strzebonski 2000] and virtual substitution of test terms [Dolzmann et al. 1998a] as provided in the REDLOG system [=-=Dolzmann and Sturm 1997-=-] and specialized for low-degree polynomials. While quantifier elimination represents a direct solution to the checking problem and it allows for generalizing to the non-linear case, we observe that f... |

123 |
Real quantifier elimination is doubly exponential.
- Davenport, Heintz
- 1988
(Show Context)
Citation Context ...nomials over 8 · S. Ruggieri and F. Mesnard reals is the Fourier-Motzkin projection method [Schrijver 1986]. Although in the worst case quantifier elimination is doubly exponential [Basu et al. 1996; =-=Davenport and Heintz 1988-=-], approaches efficient-in-practice have been proposed and successfully applied to theorem proving and program verification. We mention partial cylindrical algebraic decomposition as provided in the Q... |

119 | A library for doing polyhedral operations
- Wilde
(Show Context)
Citation Context ...is is the “double” description) of Theorem 3.16. In addition, they reduces to consider homogeneous systems by transforming forth and back Ax ≤ b into the form A′x′ ≤ 0 with x ⊆ x′ (see [Goldman 1956; =-=Wilde 1993-=-] for details on the transformation). For homogeneous systems, the additional condition cTV = b1T of Lemma 3.20 is trivially satisfied, since b = 0 and V = 0. The implementation that later on we will ... |

108 |
Linear Programming.
- Murty, G
- 1983
(Show Context)
Citation Context ...6 and Section 7. For readability reasons, all proofs of the paper are reported in Appendix A. 1.1 Preliminaries We adhere to standard notation for linear algebra [Schrijver 1986], linear programming [=-=Murty 1983-=-] and (constraint) logic programming [Apt 1997; Jaffar and Maher 1994; Jaffar et al. 1998]. Linear algebra. R denotes the set of real numbers. Small capital letters (a, b, . . . ) denote column vector... |

106 | The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. - Bagnara, Hill, et al. - 2008 |

98 | QEPCAD B: a program for computing with semialgebraic sets using CADs,” - Brown - 2003 |

93 | Incremental Analysis of Constraint Logic Programs.
- Hermenegildo, Puebla, et al.
- 2000
(Show Context)
Citation Context ...tial [Murty 1983]. Definiteness analysis for CLP(R), also called groundness analysis in logic programming, has been investigated in several works [Baker and Søndegaard 1993; Codish et al. 2001; de la =-=Banda et al. 1996-=-; Howe and King 2000], and it is used as a basic tool in CLP(R) compiler optimizations [Kelly et al. 1998]. Here we have extended the concept from definite values, namely the ! type, to ranges, namely... |

82 |
R.M.: The double description method
- Motzkin, Raiffa, et al.
- 1953
(Show Context)
Citation Context ...vertices, namely the smallest convex set which contains all vertices. A procedure to extract minimal R and V is the double description method, also known as the Motzkin-Chernikova-Le Verge algorithm [=-=Motzkin et al. 1953-=-; Chernikova 1965; Le Verge 1992]. Turning back to the LPInfer procedure, the satisfiability test at Step 2 is performed as part of the construction of the explicit representation of the polyhedron. T... |

56 | A Note on Chernikova’s Algorithm - Verge - 1994 |

51 |
A geometric algorithm for multi-parametric linear programming.
- Borrelli, Bemporad, et al.
- 2003
(Show Context)
Citation Context ...near programming. The solution of the problem can be expressed as a piecewise linear function of the parameters [Gal and Nedoma 1972; Gal 1995], or as the maximum of a finite set of linear functions [=-=Borrelli et al. 2003-=-; Keerthi and Sridharan 1990; Schechter 1987]. Therefore, an approach alternative to procedure Check(POLYInfer) is to compute (for each variable to be typed) the max and min functions of a parameteriz... |

49 | Parameterized polyhedra and their vertices.
- Loechner, Wilde
- 1997
(Show Context)
Citation Context ...the last two examples). Let us consider now Check(POLYInfer) and ParCheck. They require to compute the pairs (va(1),C1a ≤ c1), . . ., (va(k),Cka ≤ ck) as from Theorem 3.24. The algorithm proposed in [=-=Loechner and Wilde 1997-=-] for the purpose consists of first computing the vertices of Sol(P) where P is the linear system (2) in the space of variables plus parameters; then to project each vertex over the parameter space. T... |

45 |
lpsolve: Open source (mixed-integer) linear programming system,”
- Berkelaar, Eikland, et al.
- 2011
(Show Context)
Citation Context ...nstance using a Simplex-based linear programming solver, which we call Check(LP2Infer). We have implemented the Check(LP2Infer) procedure as part of the clpt system by relying on the lpsolve library [=-=Berkelaar et al. 2008-=-]. Table V reports the execution times for some programs from the previously introduced testbeds moded with types in BT2. Apart from the worst-case scenario (we recall that send+more=money contains a ... |

42 | Real quantifier elimination in practice.
- Dolzmann, Sturm, et al.
- 1998
(Show Context)
Citation Context ...d Inferring Type Assertions Let us concentrate now on the problem of checking the validity of a type assertion. In principle, formulas as in (1) can be checked by real quantifier elimination methods [=-=Dolzmann et al. 1998-=-b; Renegar 1992]. Quantifier elimination traces back to Tarski’s decision procedure [Van Den Vries 1988] for first order formula over real polynomials. The core of the procedure in the case of linear ... |

40 |
Algorithm for finding a general formula for the non-negative solutions of system of linear inequalities
- Chernikova
- 1965
(Show Context)
Citation Context ...smallest convex set which contains all vertices. A procedure to extract minimal R and V is the double description method, also known as the Motzkin-Chernikova-Le Verge algorithm [Motzkin et al. 1953; =-=Chernikova 1965-=-; Le Verge 1992]. Turning back to the LPInfer procedure, the satisfiability test at Step 2 is performed as part of the construction of the explicit representation of the polyhedron. The maximization p... |

39 | Solving systems of strict polynomial inequalities. - Strzebonski - 2000 |

38 | A new approach for automatic theorem proving in real geometry
- Dolzmann, Sturm, et al.
- 1998
(Show Context)
Citation Context ...d Inferring Type Assertions Let us concentrate now on the problem of checking the validity of a type assertion. In principle, formulas as in (1) can be checked by real quantifier elimination methods [=-=Dolzmann et al. 1998-=-b; Renegar 1992]. Quantifier elimination traces back to Tarski’s decision procedure [Van Den Vries 1988] for first order formula over real polynomials. The core of the procedure in the case of linear ... |

35 |
OFAI clp(q,r) Manual, Edition 1.3.3.
- Holzbaur
- 1995
(Show Context)
Citation Context ...o direct counterpart. Type assertions are at the basis of moding programs in constraint logic programming languages with linear constraints over reals and rationals, as in CLP(R) [Jaffar et al. 1992; =-=Holzbaur 1995-=-], ECLiPSe, SICStus Prolog, SWI Prolog, and many others. We conservatively extend the notion of well-moding [Apt 1997] from pure logic programming to CLP(R), proving useful properties in support of st... |

32 | Polylib: A library for manipulating parameterized polyhedra - Loechner - 1999 |

29 | Optimizing compilation for CLP(R
- Kelly, Macdonald, et al.
- 1998
(Show Context)
Citation Context ..., has been investigated in several works [Baker and Søndegaard 1993; Codish et al. 2001; de la Banda et al. 1996; Howe and King 2000], and it is used as a basic tool in CLP(R) compiler optimizations [=-=Kelly et al. 1998-=-]. Here we have extended the concept from definite values, namely the ! type, to ranges, namely the 2r type. The cited papers adopt abstract interpretation techniques to infer boolean expressions rela... |

28 | Termination of well-moded programs
- Etalle, Bossi, et al.
- 1999
(Show Context)
Citation Context ... derivations, call pattern characterization and computed answer characterization. They are at the basis of several techniques for program analysis, such as Typing Linear Constraints · 27 termination [=-=Etalle et al. 1999-=-], transformation and optimization [Somogyi et al. 1996]. The next result shows that the mentioned properties hold for the proposed extension of well-moding to CLP(R). By a left-derivation we mean a d... |

23 |
Multiparametric linear programming,”
- Gal, Nedoma
- 1972
(Show Context)
Citation Context ...a parameterized system of linear inequalities is the subject of (multi)parameterized linear programming. The solution of the problem can be expressed as a piecewise linear function of the parameters [=-=Gal and Nedoma 1972-=-; Gal 1995], or as the maximum of a finite set of linear functions [Borrelli et al. 2003; Keerthi and Sridharan 1990; Schechter 1987]. Therefore, an approach alternative to procedure Check(POLYInfer) ... |

19 | On proving left termination of constraint logic programs - MESNARD, RUGGIERI |

18 | Definiteness Analysis for CLP(R
- Baker, Søndergaard
- 1993
(Show Context)
Citation Context ...re defined on. It is worth mentioning that, even for a single parameter, the number of breaks can be exponential [Murty 1983]. Definiteness analysis for CLP(R) has been investigated in several works [=-=Baker and Søndegaard 1993-=-; Codish et al. 2001; Garcia de la Banda et al. 1996; Howe and King 2000], and it is used as a basic tool in CLP(R) compiler optimizations [Kelly et al. 1998]. Here we have extended the concept from d... |

16 | Termination Analysis with Types Is More Accurate - Lagoon, Mesnard, et al. |

15 |
Polyhedral functions and multiparametric linear programming,”
- Schechter
- 1987
(Show Context)
Citation Context ...be expressed as a piecewise linear function of the parameters [Gal and Nedoma 1972; Gal 1995], or as the maximum of a finite set of linear functions [Borrelli et al. 2003; Keerthi and Sridharan 1990; =-=Schechter 1987-=-]. Therefore, an approach alternative to procedure Check(POLYInfer) is to compute (for each variable to be typed) the max and min functions of a parameterized linear programming problem and then to co... |

14 |
Resolution and separation theorems for polyhedral convex sets
- Goldman
- 1956
(Show Context)
Citation Context ...licit form (this is the “double” description) of Theorem 3.16. In addition, they reduces to consider homogeneous systems by transforming forth and back Ax ≤ b into the form A′x′ ≤ 0 with x ⊆ x′ (see [=-=Goldman 1956-=-; Wilde 1993] for details on the transformation). For homogeneous systems, the additional condition cTV = b1T of Lemma 3.20 is trivially satisfied, since b = 0 and V = 0. The implementation that later... |

14 | Consistency, redundancy and implied equalities in linear systems
- Greenberg
- 1996
(Show Context)
Citation Context ...straints only. In the presence of inequalities, as in x :! ` x ≤ y, y ≤ x→ y :!, Gaussian elimination alone is not enough. We recall the following well-known result (see [Stuckey 1991] or the survey [=-=Greenberg 1996-=-]). Theorem 3.16. (implicit equalities) Assume that Sol(Ax ≤ b) 6= ∅. There exists an effective procedure that given Ax ≤ b yields an equivalent system A=x = b=,A+x ≤ b+ such that for every cTx ≤ b fr... |

14 | Analysis of Nonlinear Constraints in CLP(R
- Hanus
- 1993
(Show Context)
Citation Context ...so, it is worth noting that groundness inference for logic programs is shown to be exponential in the worst case [Genaim et al. 2001]. Finally, we include in this stream of research also the work of [=-=Hanus 1995-=-] which adopts abstract interpretation to detect non-linear constraints that become linear at run-time. A survey of applications of polyhedra and their Minkowski’s form to the analysis and verificatio... |

13 | Not necessarily closed convex polyhedra and the double description method
- Bagnara, Hill, et al.
(Show Context)
Citation Context ...lyhedra and their Minkowski’s form to the analysis and verification of hardware and software systems is reported in [Bagnara et al. 2009]. The definition of the Minkowski’s form has been extended in [=-=Bagnara et al. 2005-=-] to explicitly take into account strict inequalities. Finally, we refer the reader to [Bagnara et al. 2008] for an experimental comparison of several libraries, including polylib, for reasoning about... |

13 | A Termination Analyser for Java Bytecode Based on Path-Length
- Spoto, Mesnard, et al.
- 2009
(Show Context)
Citation Context ...etation technique used by some termination analysers [Lagoon et al. 2003; Mesnard and Ruggieri 2003]. Java bytecode programs are transformed into CLP(R) programs by applying the Julia+BinTerm system [=-=Spoto et al. 2009-=-]. This analyzer combines information from sharing, cyclicity, and path-length analysis to generate binary CLP programs, the termination of which ensures termination of the original Java programs. The... |

7 |
Applications of polyhedral computations to the analysis and verification of hardware and software systems.
- Bagnara, Hill, et al.
- 2009
(Show Context)
Citation Context ...n-linear constraints that become linear at run-time. A survey of applications of polyhedra and their Minkowski’s form to the analysis and verification of hardware and software systems is reported in [=-=Bagnara et al. 2009-=-]. The definition of the Minkowski’s form has been extended in [Bagnara et al. 2005] to explicitly take into account strict inequalities. Finally, we refer the reader to [Bagnara et al. 2008] for an e... |

6 |
Practical tools for reasoning about linear constraints.
- Huynh, Joskowicz, et al.
- 1991
(Show Context)
Citation Context ...bc computed by Check(LPInfer) to test the satisfiability of the input constraint c. 36 · S. Ruggieri and F. Mesnard 6. RELATED WORK A class of formulas, called parametric queries, is investigated in [=-=Huynh et al. 1991-=-]. It includes formulas ∃a∀v c→ x ∼ a, where ∼ ∈ {≤,=,≥}, or, with our notation, type assertions of the form ` c→ x : τ with τ ∈ BT2. The approach switches from the problem of checking max{cTx | Acv ≤... |

4 | Higher-precision groundness analysis
- Codish, Genaim, et al.
- 2001
(Show Context)
Citation Context ...r of breaks can be exponential [Murty 1983]. Definiteness analysis for CLP(R), also called groundness analysis in logic programming, has been investigated in several works [Baker and Søndegaard 1993; =-=Codish et al. 2001-=-; de la Banda et al. 1996; Howe and King 2000], and it is used as a basic tool in CLP(R) compiler optimizations [Kelly et al. 1998]. Here we have extended the concept from definite values, namely the ... |

4 |
Approaches to the incremental detection of implicit equalities with the revised simplex method
- Refalo
- 1998
(Show Context)
Citation Context ... the set of (implicit) equalities of c. Again, we have two alternatives. One is a Simplex-based algorithm for detecting implicit equalities, such as the one proposed in [Stuckey 1991] and refined in [=-=Refalo 1998-=-]. Another choice is to rely, again, on the Minkowski’s form of polyhedra. Lemma 3.20. Assume Sol(Ax ≤ b) 6= ∅, and let R and V be the generating and vertex matrices of Ax ≤ b. The system A=x = b= can... |

3 | Abstracting numeric constraints with boolean functions
- Howe, King
- 2000
(Show Context)
Citation Context ...Definiteness analysis for CLP(R), also called groundness analysis in logic programming, has been investigated in several works [Baker and Søndegaard 1993; Codish et al. 2001; de la Banda et al. 1996; =-=Howe and King 2000-=-], and it is used as a basic tool in CLP(R) compiler optimizations [Kelly et al. 1998]. Here we have extended the concept from definite values, namely the ! type, to ranges, namely the 2r type. The ci... |

2 | Average-case analysis of the double description method and the beneath-beyond algorithm - Borgwardt |

2 | Worst-case groundness analysis using definite boolean functions. Theory and Practice of Logic Programming
- Genaim, Codish, et al.
- 2001
(Show Context)
Citation Context .... However, the mentioned approaches restrict to consider equality constraints only. Also, it is worth noting that groundness inference for logic programs is shown to be exponential in the worst case [=-=Genaim et al. 2001-=-]. Finally, we include in this stream of research also the work of [Hanus 1995] which adopts abstract interpretation to detect non-linear constraints that become linear at run-time. A survey of applic... |

2 | Typing linear constraints for moding CLP(R) programs - Ruggieri, Mesnard |

1 |
Solution of parameterized linear inequalities by fourier elimination and its applications
- Keerthi, Sridharan
- 1990
(Show Context)
Citation Context ...c→ x : 2r is valid iff d|B ` c→ x : 2r is valid. Let us consider now an example which illustrates the Fourier-Motzkin elimination method for linear inequalities applied in the presence of parameters [=-=Keerthi and Sridharan 1990-=-]. Example 3.23. Consider the constraint c defined as y + x ≤ z, y − x ≤ z, z ≤ y, 0 ≤ z, w ≤ z, and the type declaration z :!. We start by isolating variable y in φ(z :!) ∧ c, as shown at (a) in the ... |

1 |
Typing Linear Constraints · 39
- Lassez, McAllon
- 1992
(Show Context)
Citation Context ...3, x ≥ 3, y ≤ 0→ y : u follows. Checking satisfiability of g can be easily accomplished in the framework of computing the explicit form of polyhedra. In fact, by independence of negative constraints [=-=Lassez and McAllon 1992-=-], it reduces to show that Sol(Acv ≤ bc) 6= ∅, and that for every disequality e 6= α in g, e '0 α over Acv ≤ bc does not hold. Lemma 3.27 provides us with a procedure to show that. The same result can... |

1 |
Incremental linear constraint solving and implicit equalities
- Stuckey
- 1991
(Show Context)
Citation Context ...ar we considered equality constraints only. In the presence of inequalities, as in x :! ` x ≤ y, y ≤ x→ y :!, Gaussian elimination alone is not enough. We recall the following well-known result (see [=-=Stuckey 1991-=-] or the survey [Greenberg 1996]). Theorem 3.16. (implicit equalities) Assume that Sol(Ax ≤ b) 6= ∅. There exists an effective procedure that given Ax ≤ b yields an equivalent system A=x = b=,A+x ≤ b+... |

1 | Alfred Tarski’s elimination theory for closed fields - Vries - 1988 |

1 | Theorem A.3. Let Sol(Ax ≤ b) be a non-empty polyhedron. We have: max {cTx | Ax ≤ b} ∈ R iff max {cTx | Ax ≤ 0} = 0 Proof. See [Murty 1983, Corollary 3.1]. Lemma 3.5. Proof. Consider the if part. By Lemma 3.4, there exists some u such that Sol(P,u) 6= ∅. B - Ruggieri, Mesnard |

1 | Theorem 3.19 and Lemma 2.17, Check(IEInfer) is a decision procedure, complete for BT !. Its complexity is polynomial by observing that: —LPInfer has polynomial time complexity when adopting a polynomial time algorithm for linear programming [Khachiyan 197 - By - 1986 |