Planning and Optimization A7. Invariants, Mutexes and Task Reformulation Malte Helmert and Thomas Keller

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A7. Invariants, Mutexes and Task Reformulation

Malte Helmert and Thomas Keller

Universit¨at Basel

September 30, 2019

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Invariants Mutexes Reformulation Summary

Content of this Course

Planning

Classical

Foundations Logic Heuristics Constraints

Probabilistic

Explicit MDPs Factored MDPs

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Invariants

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Invariants

When we as humans reason about planning tasks,

we implicitly make use of “obvious” properties of these tasks.

Example: we are never in two places at the same time We can represent such properties as formulasϕ that are true in all reachable states.

Example: ϕ=¬(at-uni∧at-home)

Such formulas are called invariants of the task.

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Invariants: Definition

Definition (Invariant)

Aninvariant of a planning task Π with state variables V

is a formulaϕ overV with s |=ϕfor all reachable statess of Π.

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Computing Invariants

How does anautomated planner come up with invariants?

Theoretically, testing if a formula ϕis an invariant is as hard as planning itself.

proof idea: a planning task is unsolvable iff the negation of its goal is an invariant

Still, many practical invariant synthesis algorithms exist.

To remain efficient (= polynomial-time), these algorithms only compute a subsetof all useful invariants.

sound, but notcomplete

Empirically, they tend to at least find the “obvious”

invariants of a planning task.

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Invariant Synthesis Algorithms

Most algorithms for generating invariants are based on thegenerate-test-repairapproach:

Generate: Suggest some invariant candidates, e.g., by enumerating all possible formulas ϕof a certain size.

Test: Try to prove that ϕis indeed an invariant.

Usually done inductively:

1 Test thatinitial statesatisfiesϕ.

2 Test that ifϕis true in the current state, it remains true after applying a single operator.

Repair: If invariant test fails, replace candidateϕ

by a weakerformula, ideally exploitingwhy the proof failed.

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Invariant Synthesis: References

We will not cover invariant synthesis algorithms in this course.

Literature on invariant synthesis:

DISCOPLAN (Gerevini & Schubert, 1998) TIM (Fox & Long, 1998)

Edelkamp & Helmert’s algorithm (1999) Bonet & Geffner’s algorithm (2001) Rintanen’s algorithm (2008)

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Exploiting Invariants

Invariants have many uses in planning:

Regression search:

Prune subgoals that violate (are inconsistent with) invariants.

Planning as satisfiability:

Add invariantsto a SAT encoding of a planning task to get tighter constraints.

Proving unsolvability:

Ifϕis an invariant such that ϕ∧γ is unsatisfiable, the planning task with goalγ is unsolvable.

Finite-Domain Reformulation:

Derive amore compact FDR representation (equivalent, but with fewer states) from a given propositional planning task.

We now discuss the last point because it connects

to our discussion of propositional vs. FDR planning tasks.

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Mutexes

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Reminder: Blocks World (Propositional Variables)

Example

s(A-on-B) =F s(A-on-C) =F s(A-on-table) =T s(B-on-A) =T s(B-on-C) =F s(B-on-table) =F s(C-on-A) =F s(C-on-B) =F s(C-on-table) =T

A B

C

2⁹ = 512 states

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Reminder: Blocks World (Finite-Domain Variables)

Example

Use three finite-domain state variables:

below-a: {b,c,table}

below-b: {a,c,table}

below-c: {a,b,table}

s(below-a) = table s(below-b) = a s(below-c) = table

3³ = 27 states

A B

C

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Task Reformulation

Common modeling languages (like PDDL) often give us propositional tasks.

More compact FDR tasks are often desirable.

Can we do an automatic reformulation?

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Mutexes

Invariants that take the form ofbinary clauses are called mutexes because they express that certain variable assignments

cannot be simultaneously true (aremutually exclusive).

Example (Blocks World)

The invariant¬A-on-B∨ ¬A-on-Cstates that A-on-Band A-on-Care mutex.

We say that aset of literalsis a mutex group if every subset of two literals is a mutex.

Example (Blocks World)

{A-on-B,A-on-C,A-on-table} is a mutex group.

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Encoding Mutex Groups as Finite-Domain Variables

LetG ={`₁, . . . , `_n} be a mutex group over n different propositional state variablesV_G ={v₁, . . . ,v_n}.

Then a singlefinite-domainstate variablevG with

dom(v_G) ={`₁, . . . , `_n,none} can replace then variables V_G: s(v_G) =ì represents situations where (exactly)ì is true s(vG) = none represents situations where all ì are false Note: We can omit the “none” value if `₁∨ · · · ∨`_nis an invariant.

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Mutex Covers

Definition (Mutex Cover)

Amutex coverfor a propositional planning task Π

is a set of mutex groups{G₁, . . . ,G_n} where each variable of Π occurs in exactly one groupGi.

A mutex cover ispositive if all literals in all groups are positive.

Note: always exists (use trivial group{v} ifv otherwise uncovered)

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Positive Mutex Covers

In the following, we stick topositive mutex covers for simplicity.

If we have¬v in G for some group G in the cover, we can reformulate the task to use an “opposite” variable ˆv instead, as in the conversion to positive normal form(Chapter A6).

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Reformulation

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Mutex-Based Reformulation of Propositional Tasks

Given aconflict-free propositional planning task Π with positive mutex cover{G₁, . . . ,G_n}:

In all conditionswhere variable v ∈G_i occurs, replacev with v_G_i =v.

In all effects e where variablev ∈Gi occurs, Replace allatomic add effects v withvG_i :=v Replace allatomic delete effects ¬v with (v_G_i =v∧ ¬W

v⁰∈G_i\{v}effcond(v⁰,e))Bv_G_i := none This results in an FDR planning task Π⁰ that is equivalent to Π (without proof).

Note: the conditional effects can often be simplified away to an unconditional or empty effect.

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And Back?

It can also be useful to reformulate an FDR task into a propositional task.

For example, we might want positive normal form, which requires a propositional task.

Key idea: each variable/value combinationv =d becomes a separate propositional state variable hv,di

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Converting FDR Tasks into Propositional Tasks

Definition (Induced Propositional Planning Task)

Let Π =hV,I,O, γibe a conflict-free FDR planning task.

Theinduced propositional planning taskΠ⁰

is the propositional planning task Π⁰ =hV⁰,I⁰,O⁰, γ⁰i, where V⁰ ={hv,di |v ∈V,d ∈dom(v)}

I⁰(hv,di) =Tiff I(v) =d

O⁰ andγ⁰ are obtained from O andγ by

replacing each atomic formulav =d by the propositionhv,di replacing each atomic effectv :=d by the effect

hv,di ∧V

d⁰∈dom(v)\{d}¬hv,d⁰i.

Notes:

Again, simplifications are often possible to avoid introducing so many delete effects.

SAS⁺ tasks induce STRIPS tasks

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Summary

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Summary

Invariants are common properties of all reachable states, expressed as formulas.

Mutexes are invariants that express that certain literals are mutually exclusive.

Mutex coversprovide a way to express the information in a set of propositional state variables in a (potentially much smaller) set of finite-domain state variables.

Using mutex covers, we can reformulate propositional tasks as more compact FDR tasks.

Conversely, we can reformulate FDR tasksas propositional tasks by introducing propositions for each variable/value pair.