E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986. 53  Home/Search   Document Not in Database   Summary   Related Articles   Check

....way,byconfrontation: if action A k threatens link A i; j , plan for the occurrence of some consequence of A k that does not make p false. A final difference between classical planners and buridan concerns the relationship between buridan s subgoals and a classical planner s goal agenda [45]. In a classical planner, every entry on the agenda must be made true before the plan can be considered a solution, but in the case of a probabilistic planner this is no longer the case. Thus buridan need not consider all subgoals to devise a plan that achieves its goal with sufficient ....

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986. 53

....Abstract To build effective planning systems, it is crucial to find the right level of representation: too impoverished, and important actions and goals are impossible to express; too expressive, and planning becomes intractable. Within the classical framework, Pednault s adl [ 24 ] provided a happy compromise between the impoverished strips representation and the expensive situation calculus. Among languages handling sensing actions and information goals, there is a similar spectrum of expressiveness. uwl, an extension of strips, can t express goals like Rename ....

....in a directory, or rm , which deletes all writable files. Information Expressiveness 0 Complete strips adl Situation Calculus Incomplete uwl SADL Moore et al. In this paper, we define a new action representation language, sadl, 1 that combines ideas from with those from Pednault s adl [ 26, 24 ] . Just as adl marked the middle ground on the tractability spectrum between strips and the situation calculus, sadl offers an advantageous combination of expressiveness and efficiency. Since sadl supports universally quantified information goals and universally quantified, conditional, ....

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E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

....see section 3.4) 5. Allows the specification of control rules. 4. 2 UCPOP UCPOP is described by its authors as a clean Common Lisp implementation of an elegant algorithm for partial order planning with an expressive action representation [Barrett et al..95] UCPOP handles a large subset of ADL [Pednault86] including actions with conditional effects, universally quantified preconditions and effects, and universally quanitified goals. In addition UCPOP is sound and complete. The authors claim the simplicity and efficient implementation of UCPOP make it an excellent vehicle for further research on ....

Pednault, E. P. D., "Towards a mathematical theory of plan synthesis", PhD thesis, Stanford University, December 1986.

....do. Therefore, intuitively, the final result of regressing W should be a formula whose only situation term is S 0 . This is precisely the notion of goal regression in artificial intelligence planning procedures, although there, functional fluents are not normally considered (but see Pednault [26] for an exception) That the regression operator is well defined is by no means obvious; the following theorem confirms that it is, and also that regressing W produces an equivalent formula about the initial situation S 0 only. Theorem 2 Suppose W is a regressable formula of L sitcalc and D is a ....

E.P.D. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Department of Electrical Engineering, Stanford University, 1986.

....true in the initial state for an action or action sequence to achieve a given goal. We will use R a (#) to represent the result of regressing # through action a,andR a n 1 (#) to represent the result of regressing # through the action sequence a 1 ; a 2 ; a n . Regression closely follows (Pednault 1986). Regression of the empty plan succeeds i#, in the initial state, the LHS implies the RHS. R (# # #) # # #) Regression of a plan consists of successively regressing each action, starting with the last. R a n 1 (#) R a1 (R a2 ( R an (#) Conjunction, disjunction, quantification ....

Pednault, E. 1986. Toward a Mathematical Theory of Plan Synthesis. Ph.D. Dissertation, Stanford University.

....2 In xii, the world model consists of two databases: D W and D C . D W is a database of ground literals, and D C is a database of LCW sentences. The action language used by xii is essentially the union of uwl [ 4 ] and the ucpop action language, which is a subset of Pednault s adl [ 14 ] uwl is used to describe goals and effects involving sensing and action. Like ucpop, xii allows conjunction, disjunctive goals, conditional effects and universal and existential quantification. Much of this expressiveness comes for free, by borrowing code directly from ucpop with little or no ....

.... is based on the ucpop algorithm [ 15 ] and in fact was written by modifying the ucpop implementation written by Tony Barrett [ 2 ] The algorithm we used for interleaving planning and execution closely follows IPEM, by Ambros Ingerson and Steel [ 1 ] Our action language borrows both from adl [ 14 ] and uwl [ 4 ] xii was inspired by socrates [ 5, 9 ] which dealt with the problem of LCW, though in a more ad hoc manner. socrates also operated in the UNIX domain, and used uwl in its planning language. Like xii, socrates used LCW to avoid redundant sensing. However, socrates didn t deal ....

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986. 16

....planner the information needed to determine when obtaining information, such as the combination of a safe, would be useful, and then leave it to the planner to decide how and when to acquire that information. We do so quite simply by the use of conditional effects (see Figures 2 and 3) Following (Pednault 1986), we call the precondition of a conditional effect a secondary precondition (the precondition of the action itself is known as a primary precondition) If the agent wants to know whether a conditional effect will occur, it needs to know whether the corresponding secondary precondition is true. ....

Pednault, E. 1986. Toward a Mathematical Theory of Plan Synthesis. Ph.D. Dissertation, Stanford University.

....easier than writing declarative XII rules, mainly due to the familiar programming language interface to the task presented by Simon. May 6, 1996 A Study of Procedural Search Control in Simon 2 2 Background The operator and goal language used by both systems is the same, and is derived from ADL[4] and UWL[8] The semantics of this language are defined in [6] 10] so we present only a brief review and refer the reader to those papers for the details. The planner represents entities in the world as objects . Information about objects and relations between them are represented by ....

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

....domains in the Boyer Moore logic and verify plans using the Boyer Moore theorem prover, points 2 and 3 given above are fairly settled. Thus, we are left with point 1. Fortunately, a rigorous state transition model of problem domains is already available and, by now, well understood in AI 7 [37, 101]. It is also well known that a fairly large number of problems fit this model [100, 37, 99] A problem domain is given by a set of possible states of a physical world and a set of actions that can be executed sequentially to change the state of the world. Actions are deterministic and terminating. ....

E. P. D. Pednault. Toward a mathematical theory of plan synthesis. PhD thesis, Stanford University, Department of Electrical Engineering, 1986.

....in order for the induction to be carried through. A related area of study is robot planning in AI [9] Situational logic was first used in qa3 [11] to synthesize robot plans, which are straight line programs of basic actions. Hoare logic was used in plan synthesis methods based on goal reduction [30], which has the difficulty in generating plans with control structures more powerful than sequential composition. Manna and Waldinger recently adapted program synthesis techniques to the automated generation of recursive plans [21] They noticed however that fully rigorous theorem proving might ....

Pednault, E., "Toward a Mathematical Theory of Plan Synthesis"; PhD Dissertation, Computer Science Department, Stanford University, 1987.

....weldg cs.washington.edu Abstract To build effective planning systems, it is crucial to find the right level of representation: too impoverished, and important actions or goals are impossible to express; too expressive, and planning becomes intractable. Under the classical framework, Pednault [17] provided a happy compromise between the impoverished strips representation, and the expensive situation calculus. Among languages handling sensing actions and information goals, there is a similar spectrum of expressiveness. uwl, which is based on strips, can t express goals like Rename the file ....

....or rm , which deletes all writable files. Expressiveness 0 Complete Information strips adl Situation Calculus Incomplete Information uwl sadl Levesque s R In this paper, we define a new action representation language, sadl, 1 that combines ideas from uwl with those from Pednault s adl [18, 17]. Just as adl marked the middle ground on the tractability spectrum between strips and the situation calculus, sadl offers an advantageous combination of expressiveness and efficiency. Since sadl supports universally quantified information goals and universally quantified, conditional, ....

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

.... (f) F) Note the difference between this information goal and the (corresponding) achievement goal of making all files in bin read protected: 8 (file (f) 2 (parent.directory f bin) satisfy (readable (f) F) Thanks to the formalization of the Modal Truth Criterion [5] and Causality Theorem [36], the semantics of satisfy goals are well understood. In essence, a sequence of actions achieves the goal, satisfy( if two conditions are met: 1. one of the actions in the sequence has an effect that causes to be true or observes to be true or if is true in the initial state, and 2. all ....

....The second says to use the function ls sense to translate the output of the shell into a set of bindings for the run time variables f, n and p. primitive find out and satisfy goals. The semantics of complex goal satisfaction is defined in terms of the primitives in the same manner as for adl [36]. 3 Planning in the Face of Incomplete Information We have described the legal goals and action schemata of xiil, but we have not yet discussed how to generate plans for this language or how to relax the assumptions made by classical planners. Here we discuss how xii represents plans, and how it ....

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E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

....bound. With LCW, the softbot completes all the problems. The mean size of D C (the agent s store of LCW information) is 155 sentences. The maximum size is 167. becomes intractable (e.g. does introducing disjunction into LCW sentences make it so ) A ACTION SEMANTICS We extend adl, Pednault s (Pednault 1989; 1988; 1986) state transition model of conditional and universally quantified actions, to handle incomplete information. Formally, each action is modeled as a pair, hC; Oi, denoting its causational and observational aspects. Following adl (Pednault 1988, p. 357) we define the causational effects, C, of an ....

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

....x to c, then S j= P (x) because M j= P (c) and M is correct. If does not map x to c then, by Equation 5 S j= P (x) Either maps x to c or not, so LCW(P (x) 2 Proving the next two theorems requires commitment to a formal action semantics. Here we sketch an extension to adl, Pednault s [39, 38, 37] statetransition model of conditional and universally quantified actions, to handle incomplete information. Formally, each action is modeled as a pair, hC; Oi, denoting its causational and observational aspects. Following adl [38, p. 357] we define the causational effects, C, of an action as a ....

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

....some world, represented by the STRIPS database D 0 . To find out, simply perform the operator ff on D 0 (by applying ff s elementary add and delete revision operators to D 0 ) We describe this process as progressing the database D 0 in response to the action ff (cf. Rosenschein [24] and Pednault [16]) The resulting database describes the effects of the action on the world represented by the initial database. 1 However, it may not always be convenient or even possible to describe the effects of actions as a simple process of progressing an initial world description. As we shall see in this ....

....be a progression of the initial database to S ff . For any sentence 2 L 2 S ff , D S ff j= iff D j= Proof: If D j= then by Lemma 4.1, we have D S ff j= If D S ff j= then D j= by Proposition 4.1. This theorem informs us that D S ff is a strongest post condition (cf. Pednault [16], Dijkstra and Scholten [3] and others) of the pre condition D S 0 with respect to the action ff. Pednault [16] shows that his definition of progression cannot in general be a finite set of first order sentences in L S ff . By Theorem 1, this result applies to our definition as well. In the next ....

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E. P. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Department of Electrical Engineering, Stanford University, Stanford, CA, 1986.

....to logically incomplete theories. In this paper we provide a semantics for STRIPS like systems in terms of a purely declarative situation calculus axiomatization for actions and their effects. On our view, STRIPS is a mechanism for computing the progression (Lin and Reiter [ 6 ] Pednault [ 8 ] of an initial situation calculus database under the effects of an action. We illustrate this idea by describing two different STRIPS mechanisms, and proving their correctness with respect to their situation calculus specifications. 1 Preliminaries The language L of the situation calculus is ....

....X 6= Y: D: clear(Y ) ontable(X) A: clear(X) on(X; Y ) movetotable(X; Y ) P: clear(X) on(X; Y ) X 6= Y . D: on(X; Y ) A: clear(X) clear(Y ) ontable(X) 4 Pednault s ADL The only prior literature similar to our progression semantics for STRIPS like systems is by Pednault ( 9 ] 8 ] Like us, Pednault relates a STRIPS database to the initial state of a situation calculus axiomatization. But 9 See Section 2 for the relevant definitions. our interpretation of such a database, namely as a statesuppressed situation calculus theory, distinguishes our approach from ....

E.P.D. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Department of Electrical Engineering, Stanford University, 1986.

....under uncertainty in the third section. The example systems I begin this section with a description of the language I use to discuss the various planning systems. A probabilistic extension to ADL Buridan and Weaver both have action representations that are simplifications of Pednault s ADL (Pednault 1986). This language has a syntax similar to STRIPS (Fikes Nilsson 1971) but is equivalent in representational power to the situation calculus (McCarthy Hayes 1969) In Pednault s formalism, a state is represented as an algebraic structure, specifying a domain, a set of relations and functions over ....

Pednault, E. P. D. 1986. Toward a MathematicalTheory of Plan Synthesis. Ph.D. Dissertation, Stanford University.

....Airplane routing 151:56 6 27:56 sipe example 1:42 6 0:24 Figure 4: Execution times for the zeno planner. Related Work Because of our interest in formal properties, zeno is closest in spirit to the work of Allen(Allen 1991) Chapman(Chapman 1987) McAllester(McAllester Rosenblitt 1991) and Pednault(Pednault 1986). Allen and Pelavin (Allen et al. 1991) describe an elegant theory of temporal planning based on first order logic and an interval model of time. In contrast, we model time using the real numbers; this allows metric duration and continuous change. Numerous systems with some of zeno s features have ....

Pednault, E. 1986. Toward a Mathematical Theory of Plan Synthesis. Ph.D. Dissertation, Stanford University.

....be informal due to the use of different time structures. We will, however, let Lin s work guide the development of the results presented in this paper. 2 Goal Regression in SitCalc and Lin s Proposal Initially, goal regression was developed as a plan synthesis method ( Waldinger, 1977 ] Pednault, 1986 ] Reiter [ 1991 ] provided a situation calculus account for goal regression with a completeness proof. Reiter relies on his solution to the frame problem, in particular the successor state axioms, which are biconditional relations between the values of a fluent after the execution of an action, ....

E. P. D. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Dept. of Electrical Engineering, Stanford University, December 1986.

....Abstract To build effective planning systems, it is crucial to find the right level of representation: too impoverished, and important actions and goals are impossible to express; too expressive, and planning becomes intractable. Within the classical framework, Pednault s adl [ 24 ] provided a happy compromise between the impoverished strips representation and the expensive situation calculus. Among languages handling sensing actions and information goals, there is a similar spectrum of expressiveness. uwl, an extension of strips, can t express goals like Rename the file ....

....a directory, or rm , which deletes all writable files. Information Expressiveness 0 Complete strips adl Situation Calculus Incomplete uwl SADL Moore et al. In this paper, we define a new action representation language, sadl, 1 that combines ideas from uwl with those from Pednault s adl [ 26, 24 ] Just as adl marked the middle ground on the tractability spectrum between strips and the situation calculus, sadl offers an advantageous combination of expressiveness and efficiency. Since sadl supports universally quantified information goals and universally quantified, conditional, ....

[Article contains additional citation context not shown here]

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

....implemented in Allegro Common Lisp and runs on a variety of platforms. 4 To maintain completeness we use A or IDA search to implement the exploration of nondeterministic choices. Figure 1 shows the actual run times for the briefcase example and others found in the literature [McDermott 1991,Pednault 1986,Pednault 1988,Sussman 1975] The number of unifications and the total number of partial plans that ucpop considered when solving a particular planning problem are also shown. The problems in figure 1 are summarized as follows: ffl Briefcase. Pednault s conditionalized briefcase domain. The U ....

....proofs of soundness and completeness. The paper concludes with a brief discussion of related work. 2 Representing Actions Frustrated with the restrictive strips representation but frightened by the prospect of implementing a planner for the full situation calculus, we gravitated naturally toward Pednault s [1986,1989] Action Description Language (ADL) ADL is essentially a reformulation of the situation calculus into action schemata, akin to the add and delete lists of STRIPS [Fikes and Nilsson 1971] ADL is more expressive than STRIPS yet less expressive than full, first order logic. 2.1 Action ....

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E.P.D. Pednault. Toward a mathematical theory of plan synthesis. PhD thesis, Stanford University, December 1986.

....way, by confrontation: if action A k threatens link A i; p A j , plan for the occurrence of some consequence of A k that does not make p false. A final difference between classical planners and buridan concerns the relationship between buridan s subgoals and a classical planner s goal agenda [45]. In a classical planner, every entry on the agenda must be made true before the plan can be considered a solution, but in the case of a probabilistic planner this is no longer the case. Thus buridan need not consider all subgoals to devise a plan that achieves its goal with sufficient ....

E. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Stanford University, December 1986.

.... out, simply perform the operator ff on D 0 (by applying ff s elementary add and delete revision operators to D 0 ) We describe this process as Fellow of the Canadian Institute for Advanced Research progressing the database D 0 in response to the action ff (cf. Rosenschein [15] and Pednault [9]) The resulting database describes the effects of the action on the world represented by the initial database. 1 However, it may not always be convenient or even possible to describe the effects of actions as a simple process of progressing an initial world description. As we shall see in this ....

....database to S ff . For any sentence 2 L 2 S ff , D S ff j= iff D j= Proof: If D j= then by Lemma 4.1, we have DS ff j= If D S ff j= then D j= by Proposition 4.1. From this theorem, we see that if D S ff is a progression, then it is a strongest post condition (cf. Pednault [9], Dijkstra and Scholten [3] and others) of the precondition DS0 wrt the action ff. A result by Pednault [9] shows that D S ff cannot in general be a finite set of first order sentences in L S ff . In the following, we shall extend this result, and show that D S ff cannot in general be a set of ....

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E. P. Pednault. Toward a Mathematical Theory of Plan Synthesis. PhD thesis, Department of Electrical Engineering, Stanford University, Stanford, CA, 1986.

....axioms. AMT provides a relational algebra based on the ternary predicate symbol 2. However, an algebra refers to a formal systems with free variables only (i.e. the variables denoting the individuals of the system) whereas the Strips environment description language deals with bound variables [7, 9, 11]. We now need a calculus. We thus here introduce a simple predicate calculus (with only one predicate symbol) built upon the algebra that AMT defines. 3 Algebraic uncertainty and classical planning We have previously extended the notion of membership. We are to introduce symbolic probabilities ....

E. Pednault, Towards a Mathematical Theory of Plan Synthesis, Ph.D. Thesis, Electrical Engineering Department, Stanford University (December 1986), 245 pages.

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