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...Gómez 2007), and express memoization (Michie 1968), which underlies dynamic programming. Since Rabin and Scott’s Turing-award paper (1959), nondeterminism has been applied to model checking, testing (=-=Claessen and Hughes 2000-=-), probabilistic inference, and search. These features are each available in mainstream functional languages. A call-by-value language can typically model nonstrict evaluation with thunks and observe ...

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...trict evaluation, sharing, and nondeterminism are all valuable features in functional programming. Nonstrict evaluation lets us express infinite data structures and their operations in a modular way (=-=Hughes, 1989-=-). Sharing lets us represent graphs with cycles, such as circuits (surveyed by Acosta-Gómez, 2007), and express memoization (Michie, 1968), which underlies dynamic programming. Since Rabin & Scott’s T...

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...on with thunks and observe sharing using reference cells, physical identity comparison, or a generative feature such as Scheme’s gensym or SML’s exceptions. Non-determinism can be achieved using amb (=-=McCarthy 1963-=-), threads, or first-class continuations (Felleisen 1985; Haynes 1987). In a non-strict language like Haskell, non-determinism can be expressed using a list monad Permission to make digital or hard co...

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...lculi The combination of sharing and non-strictness—known as call-byneed or lazy evaluation—has been extensively investigated theoretically. The first “natural” semantics for call-by-need evaluation (=-=Launchbury 1993-=-; Seaman 1993) both rely on heaps, which store either evaluated or unevaluated bindings of variables. Later, Ariola et al. (1995), Ariola and Felleisen (1997), and Maraist et al. (1998) proposed call-...

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...f (and is often same as) the meanings of eager non-deterministic programs. 1.1 Contributions We embed lazy non-determinism with call-time choice into mainstream functional languages in a shallow way (=-=Hudak 1996-=-), rather than, say, building a Curry interpreter in Haskell (Tolmach and Antoy 2003). This new approach is especially practical because these languages already have mature implementations, because fu...

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...oing so would require cooperation with garbage collection. 8.3 Hardware description languages The problem of explicit sharing has also been addressed in the context of hardware description languages (=-=Bjesse et al., 1998-=-; Acosta-Gómez, 2007). In order to model a circuit as an algebraic data type in a purely functional language, one needs to be able to identify shared nodes. The survey by Acosta-Gómez (2007, Section 2...

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...e the equational laws with observations of lazy nondeterministic programs with explicit sharing. Early choice. The motivation for our (Choice) law comes from Constructor-Based Rewriting Logic (CRWL) (=-=González-Moreno et al., 1999-=-), a standard formalization of FLP. To every term e (which we assume is closed in this informal explanation), CRWL assigns a denotation �e�, the set of partial values that e can reduce to. A432 S. Fi...

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...ed by the test predicate. This pattern applies to propertybased test-case generation (Christiansen and Fischer 2008; Fischer and Kuchen 2007; Runciman et al. 2008) as well as probabilistic inference (=-=Goodman et al. 2008-=-; Koller et al. 1997). Given the appeal of these applications, it is unfortunate that combining the three features naively leads to unexpected and undesired results, even crashes. For example, lazy in...

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..., constraint solving (Nordin and Tolmach 2001), and weighted results. For example, we have applied our approach to express and sample from probability distributions as OCaml programs in direct style (=-=Filinski 1999-=-). With less development effort than state-of-theart systems, we achieved comparable concision and performance (Kiselyov and Shan 2009). The implementation of our monad transformer is available as a H...

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... nondeterministic programming 443 We use the IO type to account for the unspecified order and multiplicities of computed results (which is similar to the role IO plays in imprecise exceptions (Peyton =-=Jones et al., 1999-=-)). As the result list is computed lazily, this function can be used to implement an interactive eval-print loop for querying results (see the Appendix). 6 Generalized, efficient implementation In thi...

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...ate. This pattern applies to propertybased test-case generation (Christiansen and Fischer 2008; Fischer and Kuchen 2007; Runciman et al. 2008) as well as probabilistic inference (Goodman et al. 2008; =-=Koller et al. 1997-=-). Given the appeal of these applications, it is unfortunate that combining the three features naively leads to unexpected and undesired results, even crashes. For example, lazy in OCaml is not thread...

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... shown, the expressive power of non-determinism simplifies programs because different non-deterministic results can be viewed individually rather than as members of a (multi-)set of possible results (=-=Antoy and Hanus 2002-=-). In Haskell, we can express non-deterministic computations using lists (Wadler 1985) or, more generally, monads that are instances of the type class MonadPlus. A monad m is a type constructor that p...

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...logic variables—perhaps the most conspicuous feature of FLP—and the associated solution techniques of narrowing and residuation, but logic variables can be emulated using nondeterministic generators (=-=Antoy & Hanus, 2006-=-). We present our concrete code in Haskell, but we have also implemented our approach in OCaml. Our monadic computations perform competitively against corresponding computations in Curry that use nond...

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...tity comparison, or a generative feature such as Scheme’s gensym or SML’s exceptions. Non-determinism can be achieved using amb (McCarthy 1963), threads, or first-class continuations (Felleisen 1985; =-=Haynes 1987-=-). In a non-strict language like Haskell, non-determinism can be expressed using a list monad Permission to make digital or hard copies of all or part of this work for personal or classroom use is gra...

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...e achieved using amb (McCarthy, 1963), threads, or first-class continuations (Felleisen, 1985; Haynes, 1987). In a nonstrict language like Haskell, nondeterminism can be expressed using a list monad (=-=Wadler, 1985-=-) or another MonadPlus instance, and sharing can be represented using a state monad (Acosta-Gómez, 2007, Section 2.4.1). These features are particularly useful together. For instance, sharing the resu...

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... to generate candidate solutions only to the extent demanded by the test predicate. This pattern applies to propertybased test-case generation (Christiansen and Fischer 2008; Fischer and Kuchen 2007; =-=Runciman et al. 2008-=-) as well as probabilistic inference (Goodman et al. 2008; Koller et al. 1997). Given the appeal of these applications, it is unfortunate that combining the three features naively leads to unexpected ...

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...s, physical identity comparison, or a generative feature such as Scheme’s gensym or SML’s exceptions. Non-determinism can be achieved using amb (McCarthy 1963), threads, or first-class continuations (=-=Felleisen 1985-=-; Haynes 1987). In a non-strict language like Haskell, non-determinism can be expressed using a list monad Permission to make digital or hard copies of all or part of this work for personal or classro...

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...nerate strategy, which is to generate candidate solutions only to the extent demanded by the test predicate. This pattern applies to propertybased test-case generation (Christiansen and Fischer 2008; =-=Fischer and Kuchen 2007-=-; Runciman et al. 2008) as well as probabilistic inference (Goodman et al. 2008; Koller et al. 1997). Given the appeal of these applications, it is unfortunate that combining the three features naivel...

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...ict evaluation, sharing, and non-determinism are all valuable features in functional programming. Non-strict evaluation lets us express infinite data structures and their operations in a modular way (=-=Hughes 1989-=-). Sharing lets us represent graphs with cycles, such as circuits (surveyed by Acosta-Gómez 2007), and express memoization (Michie 1968), which underlies dynamic programming. Since Rabin and Scott’s T...

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... infinite data structures and their operations in a modular way (Hughes 1989). Sharing lets us represent graphs with cycles, such as circuits (surveyed by Acosta-Gómez 2007), and express memoization (=-=Michie 1968-=-), which underlies dynamic programming. Since Rabin and Scott’s Turing-award paper (1959), nondeterminism has been applied to model checking, testing (Claessen and Hughes 2000), probabilistic inferenc...

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...d sample from probability distributions as OCaml programs in direct style (Filinski 1999). With less development effort than state-of-theart systems, we achieved comparable concision and performance (=-=Kiselyov and Shan 2009-=-). The implementation of our monad transformer is available as a Hackage package at: http://hackage.haskell.org/cgi-bin/ hackage-scripts/package/explicit-sharing-0.1 5.1 Non-deterministic data We have...

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...yov et al. 2005), the usual monadic encoding of nondeterminism in Haskell loses non-strictness (see §2.2). The triple combination has also been challenging for theoreticians and practitioners of FLP (=-=López-Fraguas et al. 2007-=-, 2008). After all, Algol has made us wary of combining non-strictness with any effect. The FLP community has developed a sound combination of laziness and non-determinism, call-time choice, embodied ...

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...n for non-deterministic data, our approach is typed. This tight integration with Haskell lets us be much more efficient than is possible using an interpreter. The KiCS compiler from Curry to Haskell (=-=Braßel and Huch 2009-=-) also aims to exploit the fact that many functional logic programs contain large deterministic parts. Unlike our approach, KiCS does not use monads to implement sharing but generates unique identifie...

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... the more efficient test-and-generate strategy, which is to generate candidate solutions only to the extent demanded by the test predicate. This pattern applies to propertybased test-case generation (=-=Christiansen and Fischer 2008-=-; Fischer and Kuchen 2007; Runciman et al. 2008) as well as probabilistic inference (Goodman et al. 2008; Koller et al. 1997). Given the appeal of these applications, it is unfortunate that combining ...

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...ds the result False if n>0. It does not terminate if n<=0 because an infinite list cannot be identified as being sorted without considering each of its elements. In this sense, isSorted is not total (=-=Escardó, 2007-=-). In a nonstrict language, variables can be bound to not yet evaluated expressions rather than only fully evaluated values like in a call-by-value language. This allows to evaluate arguments of funct...

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... the delayed computation raises an exception, let alone backtracks. Although sharing and nondeterminism can be combined in Haskell by building a state monad that is a MonadPlus instance (Hinze, 2000; =-=Kiselyov et al., 2005-=-), the usual monadic encoding of nondeterminism in Haskell loses nonstrictness (see Section 2.2). The triple combination has also been challenging for theoreticians and practitioners of FLP (López-Fra...

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...o (\ts -> unMemo a ts ++ unMemo b ts) It is crucial that the thunks are passed to both alternatives separately in the implementation of mplus. The list of thunks thus constitutes a first-class store (=-=Morrisett 1993-=-)—using mutable global state to store the thunks would not suffice because thunks are created and evaluated differently in different non-deterministic branches. We can implement a very specific versio...

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...s. 1.1 Contributions We embed lazy non-determinism with call-time choice into mainstream functional languages in a shallow way (Hudak 1996), rather than, say, building a Curry interpreter in Haskell (=-=Tolmach and Antoy 2003-=-). This new approach is especially practical because these languages already have mature implementations, because functional programmers are already knowledgeable about laziness, and because different...

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...ation. That is, they only say that replacing certain expressions by others does not affect the (non)termination of well-typed programs3 when the monad type constructors are held abstract (Hinze 2000; =-=Lin 2006-=-). Observing a computation in our monad requires applying run then observing the result in the target monad. Each of the two steps may identify many computations. For example, the order of choices may...

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...xisting functional languages. 2. The ability to plug in different underlying monads makes it possible to express techniques such as breadth-first search (Spivey 2000), heuristics, constraint solving (=-=Nordin and Tolmach 2001-=-), and weighted results. For example, we have applied our approach to express and sample from probability distributions as OCaml programs in direct style (Filinski 1999). With less development effort ...

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...tion of sharing and non-strictness—known as call-byneed or lazy evaluation—has been extensively investigated theoretically. The first “natural” semantics for call-by-need evaluation (Launchbury 1993; =-=Seaman 1993-=-) both rely on heaps, which store either evaluated or unevaluated bindings of variables. Later, Ariola et al. (1995), Ariola and Felleisen (1997), and Maraist et al. (1998) proposed call-by-need calcu...

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...nspecified if the delayed computation raises an exception, let alone backtracks. Although sharing and nondeterminism can be combined in Haskell by building a state monad that is a MonadPlus instance (=-=Hinze, 2000-=-; Kiselyov et al., 2005), the usual monadic encoding of nondeterminism in Haskell loses nonstrictness (see Section 2.2). The triple combination has also been challenging for theoreticians and practiti...

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...amming. Non-strict evaluation lets us express infinite data structures and their operations in a modular way (Hughes 1989). Sharing lets us represent graphs with cycles, such as circuits (surveyed by =-=Acosta-Gómez 2007-=-), and express memoization (Michie 1968), which underlies dynamic programming. Since Rabin and Scott’s Turing-award paper (1959), nondeterminism has been applied to model checking, testing (Claessen a...