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...e inequality is useful in temporal and spatial reasoning problems. For instance, Figure 2 gives an Echidna program for scheduling tasks using some of the relations on temporal intervals described in (=-=Allen, 1983-=-). A task is represented by a term task(S, D) where S is the start time of the task and D is the duration of the task. The predicate, in(Task, SuperTask), is true if the interval for SuperTask contain...

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...f the paper is organized as follows: Section 2 describes how Echidna augments a logic programming language with real number constraints. Section 3 briefly describes the SLD-resolution theorem prover (=-=Lloyd, 1984-=-) and arc consistency algorithms (Mackworth, 1977) used in Echidna. Section 4 specifies Echidna's adaptation of hierarchical arc consistency (Mackworth, et al., 1985) for real number constraints. Sect...

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...y exponential time complexity in the worst case (Buchberger, 1985). The CHIP CLP language (Van Hentenryck, 1989) and BNR Prolog (Older and Vellino, 1990) use consistency and case analysis algorithms (=-=Mackworth, 1977-=-) for solving constraints. Consistency algorithms require that variables be associated with domains which are sets of candidate values for a variable. For consistency algorithms, a domain must be repr...

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...y) is open while the other is closed. There are several reasonable definitions for mid(x,y). If an unbounded precision (eg. rational) number system is used, then the mean ((x+y)/2) or the mediant 13 (=-=Graham et al., 1989-=-) can be used. If a floating point number system is used, then the number nearest to the mean or the median number in the system between x and y can be used. Cleary (1987) studies the efficiency of th...

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...on There have been many proposals for adding sound implementations of numeric processing to Prolog. Constraint logic programming (CLP) languages like CLP(IR) (Jaffar and Michaylov, 1987), Prolog III (=-=Colmerauer, 1990-=-), and CAL (Aiba et al., 1988) use symbolic constraint solving techniques. However, CLP(IR) and Prolog III can only actively process 1 linear constraints using linear programming algorithms. CAL activ...

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... algorithms can be very expensive when domain sizes are large. For instance, the running time of AC-3 is proportional to the square of the domain size in the best case and the cube in the worst case (=-=Mackworth & Freuder, 1985-=-). An extensional representation for real domains is impossible. Instead, we introduce a hierarchical and intensional domain representation. HACR is based on the hierarchical arc consistency algorithm...

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...tions in the CSP. Arc consistency algorithms only consider constraints in isolation. Higher levels of consistency can be achieved using path consistency and k-consistency algorithms (Mackworth, 1977; =-=Freuder, 1978-=-), but these general algorithms cannot compete with specialized algorithms for linear constraints. 5.4 Scheduling Given the scheduling program introduced earlier in Figure 2, consider the query: [39] ...

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...variable domains (the "label language") and constraints (the "constraint language"). Echidna implements label and constraint languages which are more general than the real number s=-=ystems surveyed in (Davis, 1987-=-) and Echidna's label language is more general than BNR Prolog's. Our label language is a collection of disjoint real interval sets and it contains real intervals as a subset. This makes it possible f...

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...cerned with real number processing. For a description of other aspects of the language, see (Havens, et al., 1990) and (Havens, 1991). In this presentation, we augment the syntax of Edinburgh Prolog (=-=Sterling and Shapiro, 1986-=-) as necessary for exposition. Echidna provides domain constraints, equalities, inequalities, and disjunctions of inequalities on real numeric expressions. A domain constraint is a unary constraint of...

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...ctions. Unlike BNR Prolog, our algorithms also support disjunctions of inequalities as atomic constraints. Echidna handles such general systems of constraints by using partial consistency algorithms (=-=Nadel, 1989-=-). Partial consistency algorithms approximate the set of solutions to a constraint satisfaction problem (CSP) by computing a superset of the solutions. Good partial consistency algorithms compute a su...

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...ls Minor: Theorem Proving 3 1. Introduction There have been many proposals for adding sound implementations of numeric processing to Prolog. Constraint logic programming (CLP) languages like CLP(IR) (=-=Jaffar and Michaylov, 1987-=-), Prolog III (Colmerauer, 1990), and CAL (Aiba et al., 1988) use symbolic constraint solving techniques. However, CLP(IR) and Prolog III can only actively process 1 linear constraints using linear pr...

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...ange(f E ) and max range(f E ), can be calculated from the respective minima and maxima of the function arguments by analyzing the monotonicity and continuity properties of the function (Bundy, 1984; =-=Ratschek and Rokne, 1984-=-). This analysis is applied in Echidna for the arithmetic, exponential, logarithmic, root extraction, and trigonometric functions. 4.4.1 Equalities For simple equalities of the form iso(T, C) = `T = E...

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...aints using linear programming algorithms. CAL actively processes polynomial constraints using algorithms from polynomial ideal theory which have doubly exponential time complexity in the worst case (=-=Buchberger, 1985-=-). The CHIP CLP language (Van Hentenryck, 1989) and BNR Prolog (Older and Vellino, 1990) use consistency and case analysis algorithms (Mackworth, 1977) for solving constraints. Consistency algorithms ...

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...straints using algorithms from polynomial ideal theory which have doubly exponential time complexity in the worst case (Buchberger, 1985). The CHIP CLP language (Van Hentenryck, 1989) and BNR Prolog (=-=Older and Vellino, 1990-=-) use consistency and case analysis algorithms (Mackworth, 1977) for solving constraints. Consistency algorithms require that variables be associated with domains which are sets of candidate values fo...

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...y describes the SLD-resolution theorem prover (Lloyd, 1984) and arc consistency algorithms (Mackworth, 1977) used in Echidna. Section 4 specifies Echidna's adaptation of hierarchical arc consistency (=-=Mackworth, et al., 1985-=-) for real number constraints. Section 5 gives some sample runs using Echidna (version 1.0) and compares it with other CLP languages. Finally, Section 6 draws some conclusions about this research. 2. ...

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...osals for adding sound implementations of numeric processing to Prolog. Constraint logic programming (CLP) languages like CLP(IR) (Jaffar and Michaylov, 1987), Prolog III (Colmerauer, 1990), and CAL (=-=Aiba et al., 1988-=-) use symbolic constraint solving techniques. However, CLP(IR) and Prolog III can only actively process 1 linear constraints using linear programming algorithms. CAL actively processes polynomial cons...

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...aints is missed although sometimes answers contain no solutions. This paper describes an approach to real number constraint processing which has been implemented in a new CLP language called Echidna (=-=Havens, et al., 1990-=-; Havens, 1991). Like both CHIP and BNR Prolog, Echidna uses consistency algorithms which can actively 1 By active constraint processing, we mean using constraints to at least partially test their sat...

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...bounds, min range(f E ) and max range(f E ), can be calculated from the respective minima and maxima of the function arguments by analyzing the monotonicity and continuity properties of the function (=-=Bundy, 1984-=-; Ratschek and Rokne, 1984). This analysis is applied in Echidna for the arithmetic, exponential, logarithmic, root extraction, and trigonometric functions. 4.4.1 Equalities For simple equalities of t...

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...gh sometimes answers contain no solutions. This paper describes an approach to real number constraint processing which has been implemented in a new CLP language called Echidna (Havens, et al., 1990; =-=Havens, 1991-=-). Like both CHIP and BNR Prolog, Echidna uses consistency algorithms which can actively 1 By active constraint processing, we mean using constraints to at least partially test their satisfiability wh...

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...rivial for constraints involving only invertible functions. A full presentation of how to compute projections of constraints involving more functions with open and closed intervals is in preparation (=-=Sidebottom, 1992-=-). We consider here only computing p T (C) for an arbitrary constraint C subject to these two restrictions. Echidna satisfies restriction 2 by introducing intermediate variables to decompose complex c...