Search algorithms in type theory

Caldwell, James L.; Gent, Ian P.; Underwood, Judith

doi:10.1016/s0304-3975(99)00170-x

Cited by 7 publications

(9 citation statements)

References 31 publications

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“…See [CGU00] for applications to classical search algorithms. The work of Makarov [Mak06] may also be useful here, as it gives ways to optimize program extraction to make it feasible for practical programming.…”

Section: Corollary 410 If R ∈ Sn and Umentioning

confidence: 99%

A call-by-value lambda-calculus with lists and control

Krebbers

2012

Electron. Proc. Theor. Comput. Sci.

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Calculi with control operators have been studied to reason about control in programming languages and to interpret the computational content of classical proofs. To make these calculi into a real programming language, one should also include data types.As a step into that direction, this paper defines a simply typed call-by-value λ -calculus with the control operators catch and throw, a data type of lists, and an operator for primitive recursion (à la Gödel's T). We prove that our system satisfies subject reduction, progress, confluence for untyped terms, and strong normalization for well-typed terms.

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Section: Corollary 410 If R ∈ Sn and Umentioning

confidence: 99%

A call-by-value lambda-calculus with lists and control

Krebbers

2012

Electron. Proc. Theor. Comput. Sci.

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“…In its usual imperative formulation this algorithm is notoriously difficult to understand or prove correct. While we have relied on the analysis of Caldwell, et al [4] for our understanding of conflict sets, we are unaware of any description of the algorithm as a form of labeling.…”

Section: Conflict-directed Backjumpingmentioning

confidence: 99%

“…The term "conflict set" is very common in the literature, but a precise definition is difficult to achieve; we base ours on that of Caldwell, et al [4].…”

Section: Related Workmentioning

confidence: 99%

“…(Thinking imperatively, we might say a conflict set contains variables at least one of which 'must be changed' to reach a solution.) Although use of conflict sets is very common in the literature, a precise definition is difficult to achieve; we base ours on that of Caldwell et al (1997). If a conflict set exists for a state, then evidently no extension of that state can be a solution.…”

Section: Definitionmentioning

confidence: 99%

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Modular lazy search for Constraint Satisfaction Problems

Nordin¹,

Tolmach²

2001

J. Funct. Prog.

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We describe a unified, lazy, declarative framework for solving constraint satisfaction problems, an important subclass of combinatorial search problems. These problems are both practically significant and computationally hard. Finding solutions involves combining good general-purpose search algorithms with problem-specific heuristics. Conventional imperative algorithms are usually implemented and presented monolithically, which makes them hard to understand and reuse, even though new algorithms often are combinations of simpler ones. Lazy functional languages, such as Haskell, encourage modular structuring of search algorithms by separating the generation and testing of potential solutions into distinct functions communicating through an explicit, lazy intermediate data structure. But only relatively simple search algorithms have been treated this way in the past. Our framework uses a generic generation and pruning algorithm parameterized by a labeling function that annotates search trees with conflict sets. We show that many advanced imperative search algorithms, including conflict-directed backjumping, backmarking, minimal forward checking, and fail-first dynamic variable ordering, can be obtained by suitable instantiation of the labeling function. More importantly, arbitrary combinations of these algorithms can be built by simply composing their labeling functions. Our modular algorithms are as efficient as the monolithic imperative algorithms in the sense that they make the same number of consistency checks, and most of our algorithms are within a constant factor of their imperative counterparts in runtime and space usage. We believe our framework is especially well-suited for experimenting to find good combinations of algorithms for specific problems.

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“…Like the earlier development in [9], the approach presented here uses a constructive type theory as the formal framework for specifying and developing programs. There, the proofs were mechanically checked in the Nuprl theorem prover [12], here the development is formal but proofs have not been mechanically checked.…”

Section: Introductionmentioning

confidence: 99%

Generalized support and formal development of constraint propagators

Caldwell

Gent

Nightingale

2017

AIC

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Constraint programming is a family of techniques for solving combinatorial problems, where the problem is modelled as a set of decision variables (typically with finite domains) and a set of constraints that express relations among the decision variables. One key concept in constraint programming is propagation: reasoning on a constraint or set of constraints to derive new facts, typically to remove values from the domains of decision variables. Specialized propagation algorithms (propagators) exist for many classes of constraints.The concept of support is pervasive in the design of propagators. Traditionally, when a domain value ceases to have support, it may be removed because it takes part in no solutions. Arc-consistency algorithms such as AC2001 [8] make use of support in the form of a single domain value. GAC algorithms such as GAC-Schema [7] use a tuple of values to support each literal. We generalize these notions of support in two ways. First, we allow a set of tuples to act as support. Second, the supported object is generalized from a set of literals (GAC-Schema) to an entire constraint or any part of it.We design a methodology for developing correct propagators using generalized support. A constraint is expressed as a family of support properties, which may be proven correct against the formal semantics of the constraint. We show how to derive correct propagators from the constructive proofs of the support properties. The framework is carefully designed to allow efficient algorithms to be produced. Derived algorithms may make use of dynamic literal triggers or watched literals [15] for efficiency. Finally, three case studies of deriving efficient algorithms are given.

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Search algorithms in type theory

Cited by 7 publications

References 31 publications

A call-by-value lambda-calculus with lists and control

A call-by-value lambda-calculus with lists and control

Modular lazy search for Constraint Satisfaction Problems

Generalized support and formal development of constraint propagators

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