Reasoning about Actions and
Causality
Θεωρίες Δράσης και Αιτιότητας
Θοδωρής Πάτκος
Αναπαράσταση και Επεξεργασία Γνώσης ΗΥ-467
2009
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Commonsense Reasoning
• The process of inferring aspects of our world based on our current
information and commonsense knowledge.
• Related issues
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Representation
Effects of Events
Indirect Effects of Events
Context-dependent Effects
Non-deterministic Effects
Concurrent Events
Preconditions
Inertia
Actions with duration
Delayed Effects and Continuous Change
Default Reasoning
…
Cognitive Robotics
• Agent = Action Theory + High-Level Program
Action Theories
• Action Theories are formal tools used to automate the
process of commonsense reasoning in dynamically-changing
worlds, such as
– predicting the outcome of a given action sequence
– explaining observations
– finding a situation in which certain goal conditions are met
• Informal definition
An action theory consists of a formal language that allows adequate
specifications of action domains and scenarios, and it tells us precisely what
conclusions can be drawn from these specifications.
Michael Thielscher – 2000
• The objective is to express the dynamics of the world,
therefore always include a more or less implicit general
notions of time, change and causality.
Action Theories and Logic
• Action theories have much in common with general purpose logics.
• They are based on a formal language and they include an
entailment relation to determine the way conclusions are drawn
from specifications.
• Yet entailment is somewhat different, because it reflects the special
notions of change and causality inherent in action theories.
• Automated reasoning using action theories exploits the naturalness
of the latter when it comes to formalizing domains and scenarios,
but also the progress in automated deduction (in FOL for instance).
Formalisms for Reasoning about
Action
• Situation Calculus
• Fluent Calculus
• Event Calculus
• Action Languages
• Other similar approaches
– STRIPS - PDDL
– Temporal Action Logic
– Dynamic Logic
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Event Calculus – A Multi-sorted FOL
• The EC distinguishes three kinds of objects – events, fluents
and timepoints. To do this, it appeals to a sorted first-order
language.
• Fluents are relations, whose truth value may vary from
timepoint to timepoint.
• Semantically, the universe is partitioned into disjoint subuniverses, one for each sort. A variable will range over its
own sub-universe, and a term will denote an element in its
corresponding sub-universe.
• Still, multi-sorted logics are no more powerful than ordinary
unsorted FOL.
Event Calculus – Basic Notions
• There are many different dialects; we
will use the Discrete Event Calculus
• Commonsense Law of Inertia: things
tend to persist unless affected by
some event.
• Circumscription: The Event Calculus
assumes by default that no
unexpected events occur and there
are no unexpected effects of events.
Fluent Predicates
• HoldsAt(f,t)
• ReleasedAt(f,t)
• Happens(e,t)
• Initiates(e,f,t)
• Terminates(e,f,t)
• Releases(e,f,t)
Event Calculus – Simple Example
• Initiates(WakeUp(a), Awake(a), t)
(1.1)
• Terminates(Sleep(a), Awake(a), t)
(1.2)
• HoldsAt(Awake(Nathan),0)
ReleasedAt(Awake(Nathan),0)
(1.3)
(1.4)
• Happens(WakeUp(Nathan),0)
(1.5)
• We have a domain axiomatization, observations and narrative of actions.
• We can prove that Nathan will be awake at timepoint 3 (or at any t>0).
• Why?? We need an Event Calculus Axiomatization (domain independent
axioms)
Event Calculus – Axiomatization
Fluent Predicates
• HoldsAt(f,t)
• ReleasedAt(f,t)
• Happens(e,t)
• Initiates(e,f,t)
• Terminates(e,f,t)
• Releases(e,f,t)
Event Calculus – Fluent and Action
Preconditions
• HoldsAt(On(o),t) 
Initiates(Flip(o), On(o), t)
HoldsAt(On(o),t) 
Terminates(Flip(o), On(o), t)
(2.1)
(2.2)
• If the preconditions do not hold the action may still occur, but will not have the
intended effect.
• Happens(WalkThroughDoor(a,d),t)  HoldsAt(Near(a,d),t)
(3.1)
• If the preconditions do not hold the action cannot occur (otherwise we get
inconsistency)
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Frame Problem
• Happens(TurnOn(l),t)  HoldsAt(On(l),t+1)
(4.1)
• HoldsAt(On(Light1),0)
• HoldsAt(On(Light2),0)
• Happens(TurnOn(Light2),0)
(4.2)
(4.3)
(4.4)
• Ok about Light2, but what can we say about Light1??
• The problem of representing which fluents do not change when an event
occurs is known as the frame problem.
Frame Problem
• The frame problem is understood as finding a way to succinctly specify the
effects and noneffects of actions, given the situations in which they are
performed.
Javier Pinto 1999
• A straightforward solution is to add frame axioms to specify when fluents
remain unchanged.
• The problem arises because the number of these axioms in the general case is
2xAxF (where A – number of actions, F – number of fluents)
• The solution within Event Calculus: inertia + circumscription
• We will see other solutions later on.
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Ramification Problem
• The problem of representing and reasoning about the indirect effects of events
is known as the ramification problem.
• Consider for example an agent moving around while carrying an object; the
object’s location changes due to the agent’s movement.
• The simplest solution is to divide fluents into primitive and derived and apply
state constraints.
• Better, we can use context-dependent inertia with state constraints, i.e.,
HoldsAt(Holding(a,o),t)  HoldsAt(InRoom(a,r),t) 
HoldsAt(InRoom(o,r),t)
(5.1)
Ramification Problem
• A multitude of benchmark problems have been proposed to
reveal weaknesses and counterintuitive inferences, such as the
Yale shooting scenario, Thielscher’s circuit etc
• A multitude of solutions have been proposed, but still this is an
open and very interesting issue.
Ramification Problem – Shanahan’s
circuit with vicious cycles
S1
S1
S2
S1
S2
S2
R
R

L
S3

L
Close(S1)
S3
R
Start(L)
Activate(R)
L
S3
L  S1  S2  Start(L)
Open(S2)
Close(S2)
R  S1  S2  S3  Activate(R)
S2  R  Open(S2)
S1
L  (S1  S2)  Stop(L)
R  (S1  S2  S3) 
Deactivate(R)
S2  R  Close(S2)
S1
 S2
S2
R
R

L
S3
L
Stop(L)
Deactivate(R)
S3
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Qualification Problem
• Whenever we intend to execute some plan we know that many things may go
wrong, i.e., in order to drive to the university the car must have gas, its engine
must not be broken, the tires must … … … …
• Still, if we lack any evidence to the contrary, commonsense instructs to
proceed assuming that none of them holds.
• If, on the other hand, we obtain information that some of the abnormal
situations is actually the case, then we must react appropriately and reconsider
our plan.
• It is impossible to list all contingencies! This is the so-called qualification
problem.
Qualification Problem
• We must find a way to represent qualifications without exploding the domain
axiomatization with vast number of axioms.
• Not only this solution does not scale well, but also any change in our
axiomatization will require tremendous effort.
• The Event Calculus assumes by default that no unexpected events occur and
there are no unexpected effects of events.
• It uses a technique introduced by J. McCarthy called circumscription to
minimize the extension of the language’s predicates.
• Circumscription allows us to perform default (nonmonotonic) reasoning.
Qualification Problem – Example
• Apples are usually red:
Apple(x)  Ab(x)  Red(x)
• If we write
Ab(x)  GrannySmith(x)  Rotten(x)
and wish to add other conditions, we have to modify the formula.
• Alternatively, we can use a more elaboration tolerant approach:
GrannySmith(x)  Ab(x)
Rotten(x)  Ab(x)
If we circumscribe Ab in these formulas, we get a formula equivalent to the
one above.
• Circumscription provides the necessary and sufficient conditions given the
specifications.
Other Issues – that we will not see
• Non-deterministic effects
• Continuous change
• Concurrent actions with cumulative/canceling effects
• Triggered actions
• Non-atomic actions (duration)
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Situation Calculus
Situation Calculus – Fluents
Situation Calculus – Preconditions and
Effects
Situation Calculus – Limitations
Situation Calculus – The Projection
Problem
Situation Calculus – The Legality Task
Situation Calculus – The Frame
Problem
Situation Calculus – The Frame
Problem
Situation Calculus – Successor State
Axioms
Situation Calculus – Successor State
Axioms
Situation Calculus – Axiomatization
•
A domain theory for the Situation Calculus consists of
– Successor state axioms for each relational and functional fluent. They provide solution
to the frame problem
– Actions precondition axioms for each action. They characterize the conditions required
for the action to be applied in a particular situation.
– Unique names axioms for actions
– A set of initial situation axioms. The initial values of fluents
– Foundational axioms for situations. They are domain independent
•
Instead of the principle of inertia, Situation Calculus adopts Reiter’s “causal
completeness assumption”.
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Comparing Action Theories
Comparing Action Theories
Comparing Action Theories
Comparing Action Theories
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Golog
Golog
Planning – Using the Situation Calculus
Outline
• Introduction to Action Theories
• Basic Issues using the Event Calculus
– Frame Problem
– Ramification Problem
– Qualification Problem
• Familiarization with the Situation Calculus
• Comparison of Action Theories
• Golog & Planning
• Knowledge
Knowledge
• As we saw, causality plays an essential role in commonsense
reasoning.
• In real-world implementations, software agents or robots
acquire information about their environment and extend their
knowledge as they operate.
• A good plan for such agents is one that not only achieves the
goal, but is also executable.
• Even if it can be shown that a plan must achieve the goal (and
terminate), the agent may not have enough knowledge to
execute it.
Knowledge – Example
GATE 1
GATE 2
GATE 3
Knowledge – Related Aspects
GATE 1
The agent must be endowed with
information about the world it
inhabits and the ability to reason
with and about this information.
??
GATE 2
??
GATE 3
??
??
The end