Universal Quantification in a Constraint-Based Planner
Keith Golden and Jeremy Frank NASA Ames Research Center Mail Stop 269-1 Moffett Field, CA 94035 {kgolden,frank}@ptolemy.arc.nasa.gov
Abstract Incomplete information: It is common for softbots to • have only incomplete information about their environ- We present a general approach to planning with a restricted class of universally quantified constraints. These constraints ment. For example, a softbot is unlikely to know about stem from expressive action descriptions, coupled with large all the files on the local file system, much less all the files or infinite universes and incomplete information. The ap- available over the Internet. proach essentially consists of checking that the quantified Large or infinite universes: The size of the universe is constraint is satisfied for all members of the universe. We • generally very large or infinite. For example, there are present a general algorithm for proving that quantified con- straints are satisfied when the domains of all of the variables hundreds of thousands of files accessible on a typical file are finite. We then describe a class of quantified constraints system and billions of web pages publicly available over for which we can efficiently prove satisfiability even when the Internet. The number of possible files, file path names, the domains are infinite. These form the basis of constraint etc., is effectively infinite. Given the presence of incom- reasoning systems that can be used by a variety of planners. plete information and the ability to create new files, it is necessary to reason about these infinite sets. 1 Introduction Constraints: As noted in (Chien et al. 1997; Lansky & • Softbots (software robots) are intelligent software agents Philpot 1993), data processing domains typically involve that sense and act in an environment, such as a com- a rich set of constraints. By constraints, we mean any puter operating system. Since software environments are relations whose truth values can be computed. so rich, there is almost no limit to the kinds of tasks that The intersection of these features poses some interesting softbots can perform, including on-line comparison shop- challenges. For example, the intersection of universal quan- ping, managing email, scheduling meetings, and process- tification and incomplete information means that standard ing data. Planner-based softbots (Etzioni & Weld 1994; approaches to dealing with universal quantification in plan- Golden 1997) accept goals from users and invoke a planner ning (Penberthy & Weld 1992) don’t work, and other ap- to find a sequence of actions (e.g., commands or program proaches are needed (Golden 1998; Etzioni, Golden, & Weld invocations) that will achieve the goal. 1997; Babaian & Schmolze 2000). This paper discusses the We are working on softbots for data processing, includ- effect of universal quantification and large or infinite uni- ing image processing, managing file archives, and running verses on constraint reasoning and proposes a way to accom- scientific models. Due to the richness of softbot problem modate universally quantified constraints into a constraint- domains in general, and data processing domains in particu- based planner. lar, the planner must handle a rich action representation. In particular, it must support: 1.1 Universally quantified constraints Universal quantification: Many commands and pro- • grams operate on sets of things, where membership in the Universally quantified constraints can be exceedingly useful set can be defined in terms of necessary and sufficient con- when representing image processing domains. For example, ditions. For example, to represent an image-processing command that performs a horizontal flip of the pixels in a rectangular region of an im- – The Unix ls (or DOS dir) command lists all files in a age between (MINX, MINY) and (MAXX, MAXY), we might given directory. write something like: – The “tar x” (or unzip) command extracts all files in a given archive. x,y when(MINX x MAXX && MINY y MAXY) output∀ .value(x,y)≤ := input≤ .value(MAXX+MINX≤ ≤-x,y) – The grep command returns all lines of text in a file matching a given regular expression. where output.value(x,y) is the pixel value of the image out- – Most image processing commands operate on all pixels put at coordinates x,y, and similarly for input.value. The in an image or in a given region of an image. keyword when indicates a conditional effect. We might also
AIPS 2002 233 want to specify spatial transforms of an image, such as scal- Replacing goals with their universal base depends on the ing or projections, or changes to color values. All of these Closed World Assumption (all objects must be known) and are convenient to represent using numeric constraints, quan- on the number of objects in the universe being relatively tified over the pixels in the image or in the specified region. small. In softbot domains, neither assumption is likely to In describing commands that act on text files, it is useful be valid. For example, not all files accessible to the softbot to quantify over lines or characters of text. For example, will be known, and the number of available files can eas- the grep command outputs all lines of text contained in the ily be thousands or millions. To address the problem that input that match a given regular expression: not all files are known, the softbot can first achieve a sub- goal of knowing all the relevant files and then proceed as line when (input.containsLine(line) ∀ above (Etzioni, Golden, & Weld 1997), but that still leaves && input.matches(regexp)) the problem that the number of files may be large. For ex- output.containsLine(line) ample, suppose the softbot has the goal of making all of the Similarly, many commands operate on sets of files, which files in the user’s home directory group readable. This goal can often be expressed in terms of a regular expression sat- could be achieved by identifying all the files (recursively) isfied by their path names. For example, the files recur- contained in the home directory “~user” and then ensuring sively contained in directory “/foo/bar” all have a path name that each one is group readable, but it would take some time matching “/foo/bar/.+”, where “.+” means “any string at just to identify all the files. It is much simpler and faster to least one character long.” handle them all at once with a single Unix command, which In these examples, we see that it is necessary to reason recursively makes all files in the directory group readable: about constraints on variables with either infinite or very chmod -R g+r ~user large domains. Such an approach is supported in the PUCCINI planner 1.2 Road map (Golden 1998) by directly linking from universally quanti- fied goals to universally quantified effects. The approach In the remainder of the paper, we discuss how universally used by PUCCINI presupposes that the goals and effects are quantified constraints arise in the planning process and how all expressed in terms of predicates, like group-readable, for they are solved. In Section 2 we describe how universally which entailment can be determined using simple unifica- quantified constraints arise as subgoals in the planning pro- tion. When conditions include constraints as well as pred- cess. In Section 3 we present a general approach to solv- icates, determining entailment requires additional mecha- ing universally quantified constraints in a constraint network nisms, as we discuss in Section 2.2. and an algorithm for implementing this approach, and we prove that the algorithm is both sound and complete. The 2.1 Restrictions on universally quantified general approach is not always possible to instantiate when expressions there are infinite domains. In Section 4 we describe how to efficiently handle constraints with infinite domains un- Given the requirement to support universally quantified der certain restrictions. In Section 5, we discuss how these goals directly with universally quantified effects, it is im- techniques apply to an Earth Science domain that we are portant to specify exactly what kinds of expressions the lan- working on, and in Section 6 we present a detailed example guage will allow, since the unrestricted case would require covering both planning and constraint reasoning. In Section first-order theorem proving, which is undecidable. In a goal, 7 we describe related work, and in Section 8 we conclude the use of the keyword when indicates that the antecedent and describe future work. and consequent refer to different times. For example, the goal when(Φ(~x)) Ψ(~x) means that for all~x that satisfy Φ(~x) 2 Planning with universal quantification when the goal is given (i.e., in the initial state), we want Ψ(~x) to be true when the goal is achieved (i.e., in the final The traditional approach to planning with universal quantifi- state). Thus, we can specify goals like “paint all the blue cation, used by UCPOP (Penberthy & Weld 1992) and other chairs green” without contradiction: planners works as follows: c: chair when (c.color = blue) c.color = green 1. Universally quantified goals are replaced with the equiva- ∀ The planner has no control of what is true in the initial state, lent universally ground conjunctive goal, which is called so it will never try to achieve the goal by falsifying the the universal base. antecedent. To borrow a term from contingency planning, 2. Universally quantified effects are peeled as needed; that the antecedent specifies the context in which the consequent is, given an effect should be achieved. x when(P(x)) Q(x) Effects All universally quantified effects are conditional ∀ and a goal, Q(a), a new ground effect is “peeled off” effects, in which the antecedent specifies restrictions on the the forall effect to satisfy the goal: universe(s) of the quantified variable(s) and the consequent specifies what will become true for members of the specified when(P(a)) Q(a) universes. These effects are of the form The result is the subgoal P(a). ~x,~y (when(Φ(~x,~y,~w)) Ψ(~x,~w)). ∀
234 AIPS 2002 where Φ and Ψ are conjunctive expressions and variables {when(Φg)ψg}, ψg is matched against each of the literals in ~w are action parameters, variables in action schemas that ψe Ψe, using the following procedure. ∈ need to be instantiated in order to obtain concrete actions. 1. regress({when(Φ )ψ }, {when(Φ )ψ }) Limiting Φ to a conjunction is not a real limitation, since an e e g g 2. β =MGU(ψ ,ψ ) expression of the form e g 3. C = Φ {} Φ when (Φ1 Φ2) Ψ 4. n := copy( e) ∨ 5. if β = then return failure can be rewritten as the conjunction of “when(Φ1) Ψ” and ⊥ 6. for each ve,vg β “when(Φ2) Ψ.” h i ∈ 1 7. if ve is quantified Effects cannot contain existential quantifiers, or any- 8. then replace v in∀ Φ with v . thing equivalent to existentials, such as universal quantifiers e n g 9. else if vg is quantified nested within an antecedent or negation. Allowing existen- 10. then return failure.∀ tials or disjunctive consequents in effects would make them 11. else C := C (ve = vg). non-deterministic. Given the lack of nesting and existentials, 12. end for ∧ all universals can be treated as free variables. All quantified 13. for each ve β variables appearing in Ψ must also appear in Φ. This is just 6∈ Φ 14. replace ve in n with ve0 a sanity check, since the domain of any quantified variable 15. end for ∀ ∃ that does not appear in Φ is completely unrestricted. Φ may 15. return {when(Φg)Φn} C contain additional quantified variables, ~y, that don’t appear ∧ in Ψ. The reason that unmatched universally quantified variables can be replaced with existentials (line 14) is as follows: Goals and preconditions The syntax of universally quan- since the effect occurs for all v that satisfy Φ, and v isn’t tified goals and action preconditions is the same as that of mentioned in the goal, it is only necessary to find some value effects, except that existential quantifiers nested within the of v that satisfies Φ. Any new variables are written inside Ψ universal quantifiers are allowed in : the scope of all variables from∃ the goal.2 ∀ ~x,~y, ~z (when(Φ(~x,~y,~w)) Ψ(~x,~z,~w)). ∀ ∃ Examples of Goal Regression We will now present some All universal quantifiers precede all existential quantifiers; examples of goal regression. Suppose that we have an action this is simply the negation of Skolem Normal Form. Goals to give a Mothers’ Day card to all new mothers: can also explicitly refer to time. For example, we can ask p1, p2:person when(p1 =parent(p2) && ∀ for data on last Tuesday’s rainfall. Whereas effects are not sex(p1) = F && age(p2) < 1) really restricted compared to the commonly supported sub- has-card(p1) set of ADL (Pednault 1989), the limitations on universally quantified goals are more restrictive. This particular set of and our goal is to give a card to Mary (i.e., has-card(Mary)). restrictions was chosen to support the class of goals required Applying this action to satisfy the goal will result in the sub- for the softbot domains that interest us, while simplifying the goal inference procedures. p :person (Mary = parent(p ) && ∃ 20 20 sex(Mary) = F && age(p0 ) < 1) 2.2 Goal regression with quantified variables 2 That is, the action will achieve the goal if Mary is female The subgoaling, or goal regression, procedure we use is sim- and has a child less than one year old. ilar to that used by PUCCINI. We use the peeling technique Now suppose our goal is to give a card to all mothers of outlined above, with the addition that quantified variables in newborn boys: the effect can be replaced by quantified variables in the goal. : ( ( ) Suppose we have a goal when(Φg)Ψg that we want to satisfy m,s person when m =parent s ∀ ( ) F using an effect when(Φe)Ψe. If the right-hand side (RHS) && sex m = ( ) M ( ) 0) of a goal Ψg contains multiple conjuncts, they are solved in- && sex s = && age s = ( ) dependently, so subgoals are all of the form when(Φg)ψg, has-card m ψ where g is a single literal. We rely on a unification func- If we use the action to give a card to all new mothers, the ψ ψ tion MGU( e, g), which returns the most general unifier subgoal then becomes between the effect literal ψe and the goal literal ψg. If the literals don’t unify, MGU returns . Otherwise, it returns a m,s:person when(m =parent(s) ⊥ ∀ ( ) F set of pairs ve,vg , whose interpretation is that ψe unifies && sex m = {h i} ( ) M ( ) ) with ψg if all the constraints ve = vg are satisfied. && sex s = && age s = 0 {m =parent(s); sex(m) = F; age(s) < 1} The Goal Regression Algorithm To determine the con- 2 ditions required for {when(Φe)Ψe} to satisfy the goal For completeness, it is also necessary to determine whether two or more effects combine to achieve a universally quantified 1Effects can introduce the creation of new objects, through the goal. A technique called goal partitioning (Golden 1997), provides new keyword, which is similar in some respects to an existential this ability, but at a high computational cost. We are investigating quantifier, but that is outside the scope this paper. a way to lower this cost, but that is outside the scope of this paper.
AIPS 2002 235 Note that the left hand side of this expression is just the We assume that the planner produces candidate plans that left-hand side of the original goal, and the right hand side are complete except for the instantiation of some action pa- is the “peeled” left hand side (LHS) of the effect. All sub- rameters and are correct subject to a list of subgoals being goals from conditional effects are generated the same way, “trivially” satisfied (i.e., no more actions need to be inserted so the same LHS expression is carried back through succes- into the plan). The planner sends the constraint reasoner this sive goal regressions. list of subgoals, which are of the form The right-hand side (RHS) literals m =parent(s) and ~x,~y, ~z (Φ(~x,~y,~w) Ψ(~x,~z,~w)) sex(m) = F are clearly entailed by the LHS, which we can ∀ ∃ ⇒ determine by unification, using a slight variation on the re- along with some additional constraints on the parameters. gression procedure above. When the LHS entails a literal on The job of the constraint network is to either return an as- the RHS, we say that the goal literal is trivially satisfied, and signment to all of the unspecified parameters (~w) such that remove it without further subgoaling. all of the subgoals are trivially satisfied, or return failure The remaining goal condition, a constraint, is not so in case there is no such assignment. If the constraint net- straightforward. Although age(s) = 0 clearly entails age(s) work returns failure then the candidate plan is invalid, so the < 1, the two do not unify. As we discuss below, the pur- planner should continue searching. Otherwise, the candi- pose of reasoning about universally quantified constraints is date plan, instantiated with the values for ~w returned by the to answer the entailment question for constraints. constraint network, is a valid plan. The Form of Subgoals Subgoals are just goals, and obey 3 Solving Quantified Constraints the same restrictions. However, since subgoals are generated through a specific process, outlined above, it is worth show- In order to determine whether the subgoals are trivially sat- isfied, it is necessary to reason about the solutions to the ing that the process maintains the restriction on the form of Φ Ψ subgoals. CSPs induced by and . Before proceeding, we review some standard CSP notation. Let X be a set of variables. Since the subgoaling process always copies the LHS of Denote the domain of x X as d(x). Let D be the set of do- • ∈ the goal to the LHS of the subgoal, all restrictions obeyed mains. Let k = (x1 ...xi ...xn;R) be a constraint; xi X and ∈ by the former are obeyed by the latter. In particular, the R d(x1) ... d(xn) is a relation defining the permitted LHS is conjunctive and it must not contain existentials. assignments⊆ × to the× variables. Let K be the set of constraints. The RHS of the subgoal comes from the (peeled) LHS of Then C(X) = (X,D,K) is a CSP. A solution to the CSP is • the effect. Since the latter is conjunctive, so is the former. an assignment of values to the variables such that all con- straints are satisfied. Let S(C) be the set of solutions to C. Quantified variables appearing in the RHS but not in the Let L be a relation on a set of variables U, and let π (L) • V LHS are existential. To see why, consider that every quan- be the projection of the relation L onto the set V U .A tified variable that appears in the RHS either originated in CSP is k-consistent if any consistent assignment to⊆ k-1 vari- the goal or is a copy of a variable from the effect. ables can be extended to an assignment to k variables (k=2 1. If the variable appeared in the goal, then it cannot have is arc consistency.) A CSP is strongly k-consistent if it is j-consistent for all j k. been in the LHS of goal, since otherwise it would be in ≤ the LHS of the subgoal, contradicting our assumption. Having reviewed these definitions, we now formally de- Since it was not in the LHS of the goal, it must be an fine quantified constraints: existential. Definition 1 Let Φ,Ψ be CSPs. We then refer to a subgoal 2. If the variable came from the effect, then it must be an ~x,~y ~z(Φ(~x,~y,~w) Ψ(~x,~z,~w)) as a quantified constraint, ∀ ∃ ⇒ existential, since, as indicated in line 14 of the regres- and refer to the constraints comprising Φ,Ψ as primitive sion algorithm, all universals in the effect that aren’t constraints. A quantified constraint is satisfied for ~w =~θ iff ~ ~ replaced by variables from the goal are replaced by ex- π ~x S(Φ(~x,~y,θ)) π ~x S(Ψ(~x,~z,θ)). istentials. { } ⊆ { } The general approach to solving quantified implications is 2.3 From planning to constraints straightforward. Given an expression of the form “all things that satisfy Φ also satisfy Ψ,” we identify the set of things In the remainder of the paper, we discuss how to tell if the that satisfy Φ and check whether they also satisfy Ψ. We can LHS of a universally quantified subgoal entails the RHS think of this as an empirical proof technique: we’re doing when both sides contain constraints. We will not concern nothing more than checking the validity of the expression ourselves further with the details of the planning algorithm. for all members of the universe. We can convert the whole planning problem into a constraint Given a quantified constraint problem, but it would also be possible to use a causal-link ~x,~y ~z(Φ(~x,~y,~w) Ψ(~x,~z,~w)), planner like PUCCINI (Golden 1998), and perform constraint ∀ ∃ ⇒ reasoning to answer questions about whether certain sub- the variables in ~w must be assigned values by a search pro- goals are trivially satisfied (i.e., the LHS entails the RHS). cedure. As mentioned in Section 2, these variables repre- In either case, we can separate the problem of solving con- sent the parameters of actions; the search over these values straints to check subgoal satisfaction from the rest of the is a search over candidate plans. During this search, we can planning problem. propagate the domains of the variables in~x,~y based on Φ, but
236 AIPS 2002 do not assign these variables. We do not propagate based on Theorem 1 The algorithm for checking the satisfiability the constraints in Ψ, because these constraints do not hold of quantified constraints is sound: it will not return suc- if the domains of the variables in Φ are empty. Once all of cess if, for any quantified constraint, ~x,~y, ~z.Φ(x,~y,~w) these variables are assigned, we are left with the constraint Ψ(~x,~z,~w), there is some assignment∀ ~α to∃ ~x such that⇒ ~y, ~z.Φ(~α,~y,~w) Ψ(~α,~z,~w). ~x,~y ~z(Φ(~x,~y) Ψ(~x,~z)), ∃ ∀ ∧ ¬ ∀ ∃ ⇒ where ~x represents one or more universally quantified vari- Proof: Suppose otherwise. Then there is some some ~α Φ Ψ ables common to and . Again, as described above, the such that ~y, ~z.Φ(~α,~y,~w) Ψ(~α,~z,~w). The algorithm will desired semantics of this implication is that everything sat- only return∃ success∀ if each∧¬w ~w is singleton, and line 7 is Φ Ψ i isfying also satisfies . Thus, we must identify the set not reached. This happens if ∈ of tuples corresponding to the assignments to ~x that satisfy Φ(~x,~y), and check that each tuple also satisfies Ψ(~x,~z). To 1. There are no quantified constraints (line 3). This contra- do this, we solve both Φ(~x,~y) and Ψ(~x,~z) for ~x. We then dicts the assumption that there is such a constraint. π (Φ(~,~)) π (Ψ(~,~)) check to see if ~x S x y ~x S x z . Because 2. S(Φ(~x,~y,~w)) = 0/ (line 4). This is equivalent to saying Φ the quantified constraint{ } takes the⊆ form{ } of an implication, Φ is false for all ~x, contradicting our assumption that there if the set of solutions to is empty, then the implication was some ~α for which Φ was true. is satisfied vacuously, and there are no constraints on the values of the variables in ~x . If there are solutions to Φ 3. S(Φ(~x,~y,~w)) = 0/ and there is no ~α such that ~α π Φ 6 α π Ψ ∈ but π ~x S(Φ(~x,~y)) π ~x S(Ψ(~x,~z)), then the quantified con- ~x S( (~x,~y,~w)) and ~ ~x S( (~x,~y,~w)) (lines 5,6). { } 6∈ { } straint{ is} not satisfied,6⊆ and{ } some other assignment to the vari- That is, there is no ~α such that ~y.Φ(~α,~y,~w) and ables in ~w must be generated. Otherwise, the constraint is ~z.( Ψ(~α,~z,~w)), contradicting the∃ assumption that satisfied, and the domains of~x are defined by the the restric- ∀~y, ¬~z.Φ(~α,~y,~w) Ψ(~α,~z,~w). tions imposed by Φ. ∃ ∀ ∧ ¬ If the set of tuples satisfying Φ is finite, then enumerat- Theorem 2 The algorithm for checking the satisfiability of ing them and checking that each one of them satisfies Ψ is quantified constraints is complete: If, for all quantified con- straints, ~x,~y, ~z.Φ(x,~y,~w) Ψ(~x,~z,~w), then the algorithm relatively straightforward, though possibly time consuming. ∀ ∃ ⇒ But what if the set is infinite? In the general case, there is returns success. nothing that can be done. However, as we will see, there are some useful classes of problems where it is possible to iden- Proof: Suppose the algorithm returns failure, but for Φ tify the infinite set of tuples satisfying (~x,~y) and check that all quantified constraints, ~x,~y, ~z.Φ(x,~y,~w) Ψ(~x,~z,~w). Ψ they all satisfy (~x,~z) using efficient constraint propagation The algorithm will return∀ failure∃ if there is⇒ some quan- techniques. tified constraint for which S(Φ(~x,~y,~w)) = 0/ and ~α It should be noted that the steps presented above can be 6 ∈ π ~x S(Φ(~x,~y,~w)) but ~α π ~x S(Ψ(~x,~z,~w)) (line 6). But done in a variety of ways. There is no need to assign all { } 6∈ { } then π ~x S(Φ(~x,~y,~w)) π ~x S(Ψ(~x,~z,~w)), which in turn variables in ~w before beginning the process of identifying violates{ } the assumption6⊆ that{ } for all quantified constraints, the domain of ~x. It is also possible to fix the domains of ~x ~x,~y, ~z.Φ(x,~y,~w) Ψ(~x,~z,~w). after solving Φ before solving Ψ and only check to see if any ∀ ∃ ⇒ elements of these domains are eliminated during the solving Complexity Let nΦ be the number of variables in Φ and of Ψ. These refinements are left as future work. let dΦ be the size of the largest domain of any variable in Φ We present an algorithm for proving that quantified con- . Denote nΨ and dΨ similarly. The complexity of the algo- nΦ nΨ straints are satisfied. The only assumptions are that there is rithm is O((dΦ) + (dΨ) ), because checking the satisfia- a way of enumerating the variables in ~w, and that there is bility of the constraints potentially requires enumerating the Φ Ψ some way of representing the values satisfying Φ(~x,~y) and solution space for both CSPs , . Ψ(~x,~y). In the following sections, we discuss specific tech- niques for performing these operations. 4 Handling infinite universes 1. isSatisfied(γ) The general approach discussed in Section 3 works for rel- 2. choose assignments for all variables ~w. atively small, finite domains. To handle large or infinite do- 3. for (each( ~x,~y, ~z.Φ(~x,~y) Ψ(~x,~z)) γ) mains efficiently, we need to employ special-case constraint 4. if(S(Φ(~x,~∀y)) =∃0/) ⇒ ∈ propagation techniques. We describe one such technique in 6 5. for(each ~α π ~x S(Φ(~x,~y))) detail in this section. The technique depends on being able ∈ { } to represent infinite domains concisely. In Sections 4.1 and 6. if(~α π ~x S(Ψ(~x,~y))) 7. return6∈ { } failure. 4.2, we discuss concise representations of infinite domains 8. end for for numbers and strings, and describe classes of constraints 9. end for for which these concise representations can store the valid 10. return success. domains exactly. In Section 4.3, we describe further restric- tions on the form of the quantified constraints that allow us We now prove that the algorithm is both sound and com- to check the satisfiability of these quantified constraints effi- plete: ciently, even if the variable domains are infinite.
AIPS 2002 237 the relation is true for all points inside the region and false for all points outside it, then the projection of any interval on y will be an interval on x (or vice versa). Examples of convex regions are: x < 10, y > 2x + 1, x2 + y2 r2. y dom y dom ≤ Continuous to discrete A function from a continuous x dom x dom (real) variable to a discrete (integer) variable is by defini- tion not a continuous function. However, it may be regarded (a) (c) as a continuous function whose range is projected onto the integer number line. If such a description is valid, then the projection of any continuous interval on x will be a discrete interval on y. Going the other direction, intervals on y will map to intervals on x under the same circumstances as in
y dom the fully continuous case: non-decreasing functions, non- y dom increasing functions, and relations defining convex regions.
x dom x dom Discrete A function whose domain is discrete will not, in general, project an interval onto another interval. For ex- ample, consider the simple case of y = 2x, where x and y (b) (d) are integers. The domain of y is the set of even numbers, Figure 1: Reasoning about numeric functions and relations which cannot be represented as an interval. However, when we consider relations defining convex regions, we again find that the projection of an interval is an interval. So although y = 2x does not give an interval, y 2x does. 4.1 Numeric domains ≤ Large or infinite sets of numbers can be represented con- Other domain representations The decision to represent cisely using intervals. Additionally, we can determine a numeric domain using a single interval has had a profound whether one interval contains another efficiently. If we as- impact on the class of constraints that we can “solve” for sume that all infinite numeric domains are represented as particular variables. Another representation, such as a fi- single intervals, the question of whether the domain of a nite set of intervals, would allow additional constraints to numeric variable can represent exactly the possible values be handled, though at the cost of additional complexity in allowed by a constraint reduces to the question of whether constraint execution. the values for that variable allowed by the constraint can be represented as an interval. Assuming that the domains of 4.2 String domains the other variables in the constraint are also represented as Just as infinite sets of numbers can be represented by inter- intervals, the question then becomes whether the projection vals, infinite sets of strings can be represented by regular of an interval on one variable is an interval on another. We expressions. Regular expressions are a much more flexible will consider both continuous (real) and discrete (integer) representation than intervals, in that the set of regular ex- domains. pressions is closed under intersection, union and negation, Continuous If the domain of x is continuous, then for ev- whereas the set of intervals is only closed under intersec- ery continuous function y = f (x), if the domain of x is an in- tion. Regular expressions (regexps) are equivalent to finite terval, the domain of y will also be an interval. The converse automata (FAs) in expressive power, and in fact we repre- is not necessarily true. However, the converse is true if f sent regexps as FAs, since the latter are easier to compute is either non-decreasing or non-increasing. If f (x) increases with. For example, deciding whether two FAs accept the and decreases in x, then there will be some y interval that same language can be done efficiently. corresponds to multiple x intervals (Figure 1a). However, if Concatenation The concatenation of two strings, x and y, the y interval obeys certain restrictions, then the domain of yields another string, z. This constraint is represented as x will still be an interval. In particular, z = x + y. If the domains of x and y are regexps, the domain neither of the horizontal lines representing the bounds of of z will simply be the regexp resulting from concatenating • the y interval may cross f more than twice. Crossing the regexps for x and y. twice corresponds to passing through one peak or trough Less obviously, if the domains of x and z are regexps, the in f . domain of y is a regexp. To construct an FA for y given FAs if one of the lines passes through a peak, the other line for x and z, we in effect traverse the FAs for z and x in par- • must be above the peak (Figure 1b), and if one line passes allel, exploring the cross-product of the nodes from the two through a trough, then the other line must be below the FAs, starting with the pair of initial states and adding a tran- lab trough. sition sn,tm sp,tq from every node sn,tm and every { } → { } lab { lab} We can apply the same sort of reasoning to relations (Figure label lab such that the transitions sn sp and tm tq appear 1c); however a special class of relations is worth noting. If in the original FAs (see Figure 2). This→ is simply→ the opera- any relation defines a convex region (Figure 1d), such that tion that is performed when intersecting two FAs. Whenever
238 AIPS 2002 a b {I,1} {II,3} tices of GC are the variables of C and the hyperedges are 2 {III,4} a the constraints. Assume we have imposed a total order o on b b the variables X. Freuder (Freuder 1982) defines the width of a variable x X induced by ordering o as the number 1 3 4 {I,2} a {II,4} a b of variables earlier∈ in the ordering that are in the scope of RE1 = ab | ba a constraint on x. The width of an ordering o is the maxi- b mum width of any variable induced by the ordering o, and the width of a CSP is the minimum width over all orderings. I II III IV a b a We restate the following theorem from (Freuder 1982) without proof: II III IV RE3 = b*aba b a Theorem 3 Let C be a CSP. If C is strongly k-consistent and RE2 = ba | a the width of C is < k, then there is a variable order that will result in a backtrack-free search for a solution to C. Figure 2: Given FAs for RE1 and RE3, find an FA for RE2 such that RE3 is concatenation of RE1 and RE2. First, tra- We can now prove the following: verse FAs for RE3 and RE1 in parallel, constructing cross- Corollary 1 Let C be a CSP and assume C is strongly k- product FA (upper right). Then, identify states that are ac- consistent and the width of C is w < k. Let x be the first cept states for RE1 and mark the corresponding states in the variable in a search order inducing a width of w < k. Then FA for RE3 (shaded circles). Construct a new NFA (bottom) d(x) = πx(S(C)). for RE2 by copying FA for RE3 and making marked nodes Proof: We will show that each element of d(x) can be start nodes. extended to a solution to C. For each α d(x), make the assignment x = α. Consider the assignment∈ of any variable we reach a node s,t , such that node s is an accept state in y. Now, since the width of C is w < k, we know that when the FA for x, we mark{ } node t. After the traversal is complete, we use a variable ordering that induces a width w < k, fewer the marked nodes in the FA for z represent all of the states than k variables sharing constraints with y are assigned be- that can be reached by reading a string accepted by x. fore assigning y. Further, since we also know that C is A new nondeterministic FA (NFA) for y is constructed by strongly k-consistent, any consistent assignment of fewer copying the FA for z, making the start node a non-start node than k variables can always be extended by one assignment. and making all the marked nodes new start nodes. The com- Thus, we can continue assigning variables without failure plexity of the whole operation is dominated by generating until all variables are assigned, regardless of the initial as- the cross-product FA (O(mn), where m and n are the number signment to x. of nodes in the FAs for x and z, respectively). A similar pro- Theorem 4 Let ~x,~y, ~z.Φ(x,~y,~w) Ψ(x,~z,~w) be a quan- cedure can be used to construct an NFA for x, given FAs for tified constraint such∀ that:∃ ⇒ y and z. 1. Φ and Ψ share one universally quantified variable x Note that, in Figure 2, the FA for RE3 does not yet reflect whose domain is infinite, and x and any other infinite do- the concatenation constraint. That is, RE3 accepts strings, main variables are only involved in constraints for which such as bbaba, for which RE1 is not a prefix. When the con- strong k-consistency can be enforced. straint is enforced for all three variables, RE3 = aba | baba. Φ Ψ It doesn’t matter what order the variables are considered. 2. and are strongly k-consistent. 3. There exists an ordering o such that Φ has width w < k Containment The relation contains(a, b) means that 1 induced by o1 and x is the first variable in the order. string b is a substring of a. If the domain of b is a regexp Ψ r, then the domain of a is simply the regexp “.*r.*”, where 4. There exists an ordering o2 such that has width w < k “.” means “accept any character,” so “.*” means “accept any induced by o2 and x is the first variable in the order. string of zero or more characters.” Less obviously, if the do- Then the quantified constraint is satisfied if and only if main of a is a regexp, then so is the domain of b. Given an dΦ(x) dΨ(x). FA for a, we can construct an NFA for b by eliminating any ⊆ Proof: Since x is the only universally quantified vari- dead-end nodes from a (that is, nodes from which it is im- able shared between Φ and Ψ, we only need to check that possible to reach an accept node), and then making all nodes π S(Φ(x,~y,~w)) π S(Ψ(x,~z,~w)). Since we have as- in a both start and accept nodes. x x sumed{ } Φ and Ψ are⊆ k-consistent,{ } and that each has an or- dering that induces width less than k, the previous theorem 4.3 Tractable Reasoning allows us to conclude that all values of the first variable in In the previous sections we established that we can enforce the ordering are part of the solution space. But we have also consistency on a variety of constraints, even when the do- assumed that, for both orderings, that variable is x. Thus, mains are infinite. We now show how to use these results π x S(Φ(x,~y,~w)) = dΦ(x) and π x S(Ψ(x,~z,~w)) = dΨ(x), to demonstrate that a quantified constraint is satisfied. In and{ } we are done. { } order to do this, we need some additional definitions. Let We are now confronted with the problem of establishing C(X) be a CSP. Consider the hypergraph GC, where the ver- strong k-consistency. For CSPs with variables with infinite
AIPS 2002 239 domains, arc-consistency can be enforced on tree-structured ing entailment through unification and computing entail- (width 1) CSPs in polynomial time, but no stronger result ment for universally quantified constraints with infinite do- is known. In the case of finite domains, Freuder (Freuder mains. 1990) has shown that, for certain families of CSPs called Suppose we have a grayscale image corresponding to the k-trees, strong k-consistency can be established in polyno- elevation over some region: mial time in the number of variables. Our current imple- plot. mentation maintains strong k-consistency for primitive k-ary xSize = XMAX; plot. constraints over infinite numeric or string domains but only ySize = YMAX; x,y: unsigned, el: real. maintains arc consistency globally. Thus, we limit our atten- ∀ tion to tree-structured CSPs. when(x < XMAX && y < YMAX && Universally quantified constraints with infinite domains el=elevation(xProj(x),yProj(y))) plot.value , hProj can be solved in time polynomial in the number of vari- (x y) = (el) ables, but it is also necessary to consider the cost of com- where words in ALL CAPS are constants, xProj and yProj puting the domain for each variable. In the case of numeric are linear functions mapping the x, y coordinates of the im- constraints, this cost is generally trivial, consisting of a few age to the corresponding longitude, latitude that they rep- arithmetic operations. In the case of string domains, the cost resent, hProj is a linear function mapping elevation to pixel depends on the size of the regular expressions representing values in the image, with lower (blacker) values correspond the domains. Given two domains represented by FAs of size to lower elevations, and elevation(x, y) is the elevation at lon- m and n, intersection of the two domains is O(mn), union gitude x, latitude y. The notation plot.xSize denotes the hor- is O(m + n), negation is O(m), and enforcement of the con- izontal size of the image plot, and plot.value(x, y) means straints discussed in Section 4.2 is at worst O(mn). However, the pixel value at the coordinates x,y in the image plot. some of these operations produce NFAs as outputs, and oth- Say we would like to produce a color image showing the ers require deterministic FAs (DFAs) as inputs. Converting same elevations, but highlighting particular ranges of eleva- from an NFA to a DFA can result in an exponential increase tion using different colors. For example, pixels correspond- in the size of the FA. ing to points below sea level should be blue and points above the snow line should be shades of gray. 5 Applicability One way to accomplish this would be by creating bitmaps We have implemented this approach in a constraint-based or monochrome images corresponding to the the pixels of planner and are applying it to an Earth Science data process- interest (i.e., pixels above or below a particular value), and ing domain that involves a mixture of image processing, text using these bitmaps to select the pixels on which particular processing and other operations. Preliminary results indicate operations, like coloring the pixels blue, will be performed. that the assumptions we make in this paper are valid for this Suppose we have a threshold command, which takes an domain. There are two main assumptions that potentially image, in, as input and has an argument specifying a thresh- limit the applicability of our approach. old value, and outputs an image, out, the same size as the input, with a value of 255 for every pixel in the input whose 1. Constraints can be fully captured by the domain represen- value is above the threshold and a value of zero for every tation. This is really only a limitation for numeric con- pixel below the threshold: straints, since every string constraint in the domain can be captured fully using regexps. Most numeric constraints x,y: unsigned, v: pixelValue that appear in universally quantified expressions repre- ∀ when ( x < in.xSize && y < in.ySize && sent either convex regions of images or functions from v = in.value(x, y)) real-valued measurements to integral pixel values. These when(v thresh) out.value(x,y) := 0; constraints all obey this restriction. when (v≤>thresh) out.value(x,y) := 255; 2. The width of the constraint network defined by quanti- where thresh is an action parameter of type pixelValue (i.e., fied constraints must less than the level of consistency a variable from ~w) denoting the threshold value, and a pix- enforced, and the left and right hand sides must share at elValue is an integer in the range [0,255]. The use of nested most one quantified variable. This is a more serious lim- when statements is merely a shorthand, where “when (Φ1) itation. Since the nature of the quantified constraints is {when (Φ2) Ψ}” is equivalent to “when (Φ1 Φ2) Ψ}.” dictated by quantified goals, it is possible to formulate Here, we focus on a single subgoal that arises during∧ plan- goals that violate this restriction. Since the set of goals ning: to generate a threshold map, sea, based on elevation at is open, we can’t draw any conclusions about which goals sea level: are common without extensive user tests. On the other x ,y :unsigned, elev: real. hand, in most goals we have looked at, quantified con- ∀ 0 0 straints result in tree-structured CSPs that trivially obey when(x0
240 AIPS 2002 x ,y :unsigned, elev: real, v :unsigned domain of hProj(e1) [43,255], resulting in a domain for e1 ∀ 0 0 ∃ 0 when(x0
AIPS 2002 241 serving an interval representation of variables involved in ternational Conference on the Principles and Practices of arithmetic constraints while eliminating infeasible values. Constraint Programming, 67–82. However, they explicitly assume that the interval representa- Chien, S.; Fisher, F.; Lo, E.; Mortensen, H.; and Greeley, tion is an unsound approximation to the domain of feasible R. 1997. Using artificial intelligence planning to automate values. Benhamou and Goualard (Benhamou & Goualard science data analysis for large image database. In Proc. 2000) describe a method of sound but incomplete approxi- 1997 Conference on Knowledge Discovery and Data Min- mate propagation of infinite domains. Since we require both ing. soundness and completeness in cases where that set may be Etzioni, O., and Weld, D. 1994. A softbot-based interface infinite, we have made stronger restrictions on the types of to the Internet. C. ACM 37(7):72–6. reasoning performed. Etzioni, O.; Golden, K.; and Weld, D. 1997. Sound and 8 Conclusions and Future Work efficient closed-world reasoning for planning. J. Artificial Intelligence 89(1–2):113–148. We have described a planning methodology for softbots Freuder, E. 1982. A sufficient condition for backtrack-free that supports universal quantification, incomplete informa- search. Journal of the Association for Computing Machin- tion, and constraints on variables with very large or infinite ery 29(1):24–32. domains. We restrict the form of both goals and effects, while preserving the ability to express conditional effects Freuder, E. 1990. Complexity of k-tree structured con- and reason about incomplete information. Our approach straint satisfaction problems. In Proceedings of the 8th Na- uses a combination of unification and constraint reasoning tional Conference on Artificial Intelligence, 4–9. to demonstrate entailment. We described an algorithm for Ginsberg, M., and Parkes, A. 2000. Satisfiability algo- proving or disproving entailment for constraints over finite rithms and finite quantification. In Proceedings of the 7th domains, and identified a subclass of constraints for which Conference on Knowledge Representation. the same algorithm can prove or disprove entailment for Golden, K.; Etzioni, O.; and Weld, D. 1994. Omnipotence variables with infinite domains. This class of constraints has without omniscience: Sensor management in planning. In proven useful in the domains of planning for image process- Proc. 12th Nat. Conf. AI, 1048–1054. ing and managing file archives. Golden, K. 1997. Planning and Knowledge Representation When describing the algorithm to validate quantified con- for Softbots. Ph.D. Dissertation, University of Washington. straints, we assumed that all parameters of the actions were Available as UW CSE Tech Report 97-11-05. assigned before validation occurs. As described in Section Golden, K. 1998. Leap before you look: Information gath- 6, there are times when it is worth deferring the decision ering in the PUCCINI planner. In Proc. 4th Intl. Conf. AI about parameters to actions, because propagation will limit Planning Systems. the possibilities. Exploiting these possibilities is the subject of future work. Lansky, A. L., and Philpot, A. G. 1993. AI-based plan- We can potentially weaken the conditions on quantified ning for data analysis tasks. In Proceedings of the Ninth constraints required to reason about variables with infinite IEEE Conference on Artificial Intelligence for Applications domains. The condition that Φ and Ψ share only one vari- (CAIA-93). able can be relaxed when there is a procedure for check- L’Homme, O. 1993. Consistency techniques for numeric ing the validity of the constraint without checking infinitely csps. In Proceedings of the 13th International Conference many values. One case is when all of the constraints de- on Artificial Intelligence. scribe linear equations or inequalities. In addition, it may Marriott, K., and Stuckey, P. 1998. Programming with be possible to generalize the conditions under which consis- Constraints: An Introduction. The MIT Press. tency enforcement allows us to conclude that all the values Pednault, E. 1989. ADL: Exploring the middle ground be- of a variable participate in solutions to a CSP. Finally, we tween STRIPS and the situation calculus. In Proc. 1st Int. can try to find more constraints on which we can enforce Conf. Principles of Knowledge Representation and Rea- consistency when domains are infinite. soning, 324–332. Acknowledgments We would like to thank Tania Bedrax- Penberthy, J., and Weld, D. 1992. UCPOP: A sound, com- Weisss, Ari Jónsson, Wanlin Pang, Robert Morris and Ellen plete, partial order planner for ADL. In Proc. 3rd Int. Conf. Spertus for their helpful comments and contributions to this Principles of Knowledge Representation and Reasoning, work. This work was supported by the NASA Intelligent 103–114. See also http://www.cs.washington.edu/ Systems program. research/projects/ai/www/ucpop.html. Stickel, M.; Waldinger, R.; Lowry, M.; Pressburger, T.; and References Underwood, I. 1994. Deductive composition of astronom- Babaian, T., and Schmolze, J. 2000. Psiplan: Open world ical software from subroutine libraries. In Proceedings of planning with ψ-forms. In Proceedings of the 5th Confer- the 12th Conference on Automated Deduction. ence on Artificial Intelligence Planning and Scheduling. Benhamou, F., and Goualard, F. 2000. Universally quan- tified interval constraints. In Proceedings of the 6th In-
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