A7.1Invariants A7.1InvariantsA7.2MutexesA7.3ReformulationA7.4Summary PlanningandOptimization PlanningandOptimization ContentofthisCourse

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Planning and Optimization

A7. Invariants, Mutexes and Task Reformulation

Malte Helmert and Thomas Keller

Universit¨at Basel

September 30, 2019

M. Helmert, T. Keller (Universit¨at Basel) Planning and Optimization September 30, 2019 1 / 24

Planning and Optimization

September 30, 2019 — A7. Invariants, Mutexes and Task Reformulation

A7.1 Invariants A7.2 Mutexes A7.3 Reformulation A7.4 Summary

Content of this Course

Planning

Classical

Foundations Logic Heuristics Constraints

Probabilistic

Explicit MDPs Factored MDPs

A7. Invariants, Mutexes and Task Reformulation Invariants

A7.1 Invariants

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Invariants

I When we as humans reason about planning tasks,

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

I Example: we are never in two places at the same time I We can represent such properties as formulas ϕ

that are true in all reachable states.

I Example: ϕ=¬(at-uni∧at-home)

I Such formulas are calledinvariantsof the task.

Invariants: Definition

Definition (Invariant)

Aninvariant of a planning task Π with state variablesV

is a formulaϕover V with s |=ϕfor all reachable states s of Π.

Computing Invariants

How does an automated planner come up with invariants?

I Theoretically, testing if a formulaϕis an invariant isas hard as planning itself.

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

I Still, many practical invariant synthesis algorithms exist.

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

sound, but notcomplete

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

invariants of a planning task.

Invariant Synthesis Algorithms

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

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

I Test: Try to prove thatϕis 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.

I Repair: If invariant test fails, replace candidate ϕ

by aweakerformula, ideally exploiting whythe proof failed.

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Invariant Synthesis: References

We will not cover invariant synthesis algorithms in this course.

Literature on invariant synthesis:

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

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

Exploiting Invariants

Invariants have many uses in planning:

I Regression search:

Prune subgoals that violate (are inconsistent with) invariants.

I Planning as satisfiability:

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

I Proving unsolvability:

If ϕis an invariant such that ϕ∧γ is unsatisfiable, the planning task with goal γ is unsolvable.

I Finite-Domain Reformulation:

Derive a more compactFDR 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.

A7. Invariants, Mutexes and Task Reformulation Mutexes

A7.2 Mutexes

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

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Reminder: Blocks World (Finite-Domain Variables)

Example

Use three finite-domain state variables:

I below-a: {b,c,table}

I below-b: {a,c,table}

I below-c: {a,b,table}

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

3³= 27 states

A B

C

Task Reformulation

I Common modeling languages (like PDDL) often give uspropositional tasks.

I More compact FDR tasks are often desirable.

I Can we do an automatic reformulation?

Mutexes

Invariants that take the form of binary clausesare called mutexes because they express that certain variable assignments

cannot be simultaneously true (are mutually exclusive).

Example (Blocks World)

The invariant ¬A-on-B∨ ¬A-on-C states that A-on-B andA-on-Care mutex.

We say that a set 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.

Encoding Mutex Groups as Finite-Domain Variables

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

Then a singlefinite-domain state variablev_G with

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

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Mutex Covers

Definition (Mutex Cover)

A mutex coverfor a propositional planning task Π

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

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

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

Positive Mutex Covers

In the following, we stick to positivemutex covers for simplicity.

If we have¬v inG 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).

A7. Invariants, Mutexes and Task Reformulation Reformulation

A7.3 Reformulation

Mutex-Based Reformulation of Propositional Tasks

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

I In allconditions where variable v ∈G_i occurs, replace v with v_G_i =v.

I In all effects e where variablev ∈G_i occurs, I Replace allatomic add effects v withvG_i :=v I Replace allatomic delete effects¬v with

(v_G_i =v∧ ¬W

v⁰∈Gi\{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?

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

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

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

Converting FDR Tasks into Propositional Tasks

Definition (Induced Propositional Planning Task)

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

Theinduced propositional planning taskΠ⁰

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

I I⁰(hv,di) =Tiff I(v) =d

I O⁰ andγ⁰ are obtained fromO andγ by

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

hv,di ∧V

d⁰∈dom(v)\{d}¬hv,d⁰i.

Notes:

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

I SAS⁺ tasks induce STRIPS tasks

A7. Invariants, Mutexes and Task Reformulation Summary

A7.4 Summary

A7. Invariants, Mutexes and Task Reformulation Summary

Summary

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

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

I 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.

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

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