The Calculation of Invariants for Ordered Sets

Bouchitté, Vincent; Habib, Michel

doi:10.1007/978-94-009-2639-4_6

Cited by 22 publications

(5 citation statements)

References 63 publications

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“…Given a plan-library Π, one potential approach to that end is to obtain the plan-library Π ′ consisting of all linear extensions of plan-rules in Π, and then use Π ′ as the input into Algorithm 1 (Summ). We could use existing, fast algorithms to generate linear extensions [17], or consider simpler plan-rules corresponding to restricted classes of partially ordered sets [4]. Finally, it would be interesting to formally characterise the restricted class of domains in which the presented algorithms are complete.…”

Section: Discussion and Future Workmentioning

confidence: 99%

“…Definition 1 (Ranking) A ranking for a plan-library Π is a function RΠ : EΠ → N0 from event-goal types mentioned in Π to natural numbers, such that for all event-goals e1, e2 ∈ EΠ where e2 is the same type as some e3 ∈ children(e1, Π), we have that RΠ(e1) > RΠ(e2). 4 In addition, we define the following two related notions: first, given an event-goal type e, RΠ(e) denotes the rank of e in Π; and second, given any event-goal e(t) mentioned in Π, we define RΠ(e(t)) = RΠ(e(x)) (where |x| = |t|), i.e., the rank of an event-goal is equivalent to the rank of its type. In order that these and other definitions also apply to event-goal programs, we sometimes blur the distinction between eventgoals e and event-goal programs !e.…”

Section: Assumptionsmentioning

confidence: 99%

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Addendum to: Summary Information for Reasoning About Hierarchical Plans

Silva¹,

Sardiña²,

Padgham³

2017

Preprint

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Hierarchically structured agent plans are important for efficient planning and acting, and they also serve (among other things) to produce "richer" classical plans, composed not just of a sequence of primitive actions, but also "abstract" ones representing the supplied hierarchies. A crucial step for this and other approaches is deriving precondition and effect "summaries" from a given plan hierarchy. This paper provides mechanisms to do this for more pragmatic and conventional hierarchies than in the past. To this end, we formally define the notion of a precondition and an effect for a hierarchical plan; we present data structures and algorithms for automatically deriving this information; and we analyse the properties of the presented algorithms. We conclude the paper by detailing how our algorithms may be used together with a classical planner in order to obtain abstract plans.

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Section: Discussion and Future Workmentioning

confidence: 99%

Section: Assumptionsmentioning

confidence: 99%

Addendum to: Summary Information for Reasoning About Hierarchical Plans

Silva¹,

Sardiña²,

Padgham³

2017

Preprint

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“…2.2. Indeed, the following theorem is an 9 Recall that a linear extension of a partial order (C, <) is a total order (C, < ) such that a~(b~a < b.…”

Section: The Hierarchy Of Distributed Computation Classesmentioning

confidence: 99%

Synchronous, asynchronous, and causally ordered communication

1996

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This article studies characteristic properties of synchronous and asynchronous message communications in distributed systems. Based on the causality relation between events in computations with asynchronous communications, we characterize computations which are realizable with synchronous communications, which respect causal order, or where messages between two processes are always received in the order sent. It is shown that the corresponding computation classes form a strict hierarchy. Furthermore, an axiomatic definition of distributed computations with synchronous communications is given, and it is shown that several informal characterizations of such computations are equivalent when they are formalized appropriately. As an application, we use our results to show that the distributed termination detection algorithm by Dijkstra et al. is correct under a weaker synchrony assumption than originally stated.

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“…If G is a graph then by G k we denote the graph with the same vertex set as G but which has an edge between every pair of vertices that are connected by a path of length at most k in G. In other words, if M is the incidence matrix of G, then M k is the incidence matrix of G k , where arithmetic is done mod 2. The cube of G is G 3 and the square of G is G 2 . A poset P has a delay k ordering if and only if G 0 (P) k is Hamiltonian.…”

Section: Strategy and De Nitions A Popular Strategy For E Ciently Gementioning

confidence: 99%

Generating Linear Extensions Fast

Pruesse¹,

Ruskey²

1994

SIAM J. Comput.

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One of the most important sets associated with a poset P is its set of linear extensions, E(P). In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)), where e(P) = jE(P)j. The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)), where n is the number of elements of the poset. Our algorithm is the rst constant amortized time algorithm for generating a \naturally de ned" class of combinatorial objects for which the corresponding counting problem is #P-complete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension di ers from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modi ed to e ciently count linear extensions, and to compute P (x < y), for all pairs x; y, in time O(n 2 + e(P)).

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The Calculation of Invariants for Ordered Sets

Cited by 22 publications

References 63 publications

Addendum to: Summary Information for Reasoning About Hierarchical Plans

Addendum to: Summary Information for Reasoning About Hierarchical Plans

Synchronous, asynchronous, and causally ordered communication

Generating Linear Extensions Fast

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