# Optimal Planning with State Constraints

## Date

2018

## Authors

Ivankovic, Franc

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## Abstract

In the classical planning model, state variables are assigned
values in the initial state and remain unchanged unless
explicitly affected by action effects. However, some properties
of states are more naturally modelled not as direct effects of
actions but instead as derived, in each state, from the primary
variables via a set of rules. We refer to those rules as state
constraints. The two types of state constraints that will be
discussed here are numeric state constraints and logical rules
that we will refer to as axioms.
When using state constraints we make a distinction between
primary variables, whose values are directly affected by action
effects, and secondary variables, whose values are determined by
state constraints. While primary variables have finite and
discrete domains, as in classical planning, there is no such
requirement for secondary variables. For example, using numeric
state constraints allows us to have secondary variables whose
values are real numbers. We show that state constraints are a
construct that lets us combine classical planning methods with
specialised solvers developed for other types of problems. For
example, introducing numeric state constraints enables us to
apply planning techniques in domains involving interconnected
physical systems, such as power networks.
To solve these types of problems optimally, we adapt commonly
used methods from optimal classical planning, namely state-space
search guided by admissible heuristics. In heuristics based on
monotonic relaxation, the idea is that in a relaxed state each
variable assumes a set of values instead of just a single value.
With state constraints, the challenge becomes to evaluate the
conditions, such as goals and action preconditions, that involve
secondary variables. We employ consistency checking tools to
evaluate whether these conditions are satisfied in the relaxed
state. In our work with numerical constraints we use linear
programming, while with axioms we use answer set programming and
three value semantics. This allows us to build a relaxed planning
graph and compute constraint-aware version of heuristics based on
monotonic relaxation.
We also adapt pattern database heuristics. We notice that an
abstract state can be thought of as a state in the monotonic
relaxation in which the variables in the pattern hold only one
value, while the variables not in the pattern simultaneously hold
all the values in their domains. This means that we can apply the
same technique for evaluating conditions on secondary variables
as we did for the monotonic relaxation and build pattern
databases similarly as it is done in classical planning.
To make better use of our heuristics, we modify the A* algorithm
by combining two techniques that were previously used
independently – partial expansion and preferred operators. Our
modified algorithm, which we call PrefPEA, is most beneficial in
cases where heuristic is expensive to compute, but accurate, and
states have many successors.

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## Keywords

Optimal Planning, State Constraints

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## Type

Thesis (PhD)

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