knowledge representation & reasoning i · 2019-07-25 · knowledge representation and reasoning •...

Alexander Felfernig and Gerald Steinbauer

Institute for Software Technology

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Inffeldgasse 16b/2

A-8010 Graz

Austria

Wissensverarbeitung - Knowledge Representation & Reasoning I -



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Motivation



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Artificial Intelligence and Knowledge

Representation & Reasoning • Artificial Intelligence (AI) can be described as:

• the study of intelligent behavior achieved trough computational

means

• Knowledge Representation & Reasoning can be

viewed as: • the study of how to reason (compute) with knowledge in order to

decide what to do

Knowledge

Acquisition

Knowledge

Representation Reasoning

Decision on

Actions



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Clear Need for proper KR & R



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Agents

• perceive the environment through sensors (→ percepts)

• act upon the environment through actuators (→ actions)

• examples: humans and animals, robots and software

agents (softbots), temperature control, ABS, . . .



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Wumpus World

• performance measure • gold +1000

• death -1000

• step -1

• using arrow -10

• environment • squares adjacent to wumpus are smelly

• squares adjacent to pit are breezy

• glitter iff gold in the same square

• shooting kills wumpus if facing it

• grabbing picks up gold if in the same square

• releasing drops the gold in the same square

• actuators • left turn, right turn, grab, release, shoot

• sensing • breeze, glitter, smell, bump, hear



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Rational Agent

• … does the “right thing”!

• rational behavior is dependent on • performance measures (goals)

• percept sequences

• knowledge of the environment

• possible actions

• for each possible percept sequence, a rational agent

should select an action that is expected to maximize

its performance measure, given the evidence

provided by the percept sequence and whatever

built-in knowledge the agent has.



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Environment of Rational Agents I

• accessible vs. inaccessible (fully observable vs.

partially observable) • are the relevant aspects of the environment accessible to the

sensors?

• deterministic vs. stochastic • is the next state of the environment completely determined by the

current state and the selected action? if only actions of other agents

are nondeterministic, the environment is called strategic.

• episodic vs. sequential • can the quality of an action be evaluated within an episode

(perception + action), or are future developments decisive for the

evaluation of quality?



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Environment of Rational Agents II

• static vs. dynamic • can the environment change while the agent is deliberating? if the

environment does not change but if the agent’s performance score

changes as time passes by the environment is denoted as semi-

dynamic.

• discrete vs. continuous • is the environment discrete (chess) or continuous (a robot moving

in a room)?

• single agent vs. multi-agent • which entities have to be regarded as agents? there are

competitive and cooperative scenarios.



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Examples of Environments

task observable deterministic episodic static discrete agents

crossword

puzzle fully deterministic sequential static discrete single

chess with

clock fully strategic sequential semi discrete multi

poker partially stochastic sequential static discrete multi

backgammon fully stochastic sequential static discrete multi

taxi driving partially stochastic sequential dynamic continuous multi

medical

diagnosis partially stochastic sequential dynamic continuous single

image analysis fully deterministic episodic semi continuous single

part-picking

robot partially stochastic episodic dynamic continuous single

refinery

controller partially stochastic sequential dynamic continuous single

interactive

English tutor partially stochastic sequential dynamic discrete multi



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What’s about the Wumpus World ?



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Simple Reflex Agent

• direct use of perceptions is often not possible due to the large space

required to store them (e.g., video images)

• input therefore is often interpreted before decisions are made



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Interpretive Reflex Agent

• since storage space required for perceptions is too

large, direct interpretation of perceptions

function SIMPLE-REFLEX-AGENT(percept) returns an action

persistent: rules, a set of condition–action rules

state ← INTERPRET-INPUT(percept)

rule ← RULE-MATCH(state, rules)

action ← rule.ACTION

return action



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Model-based Reflex Agent

• in case the agent’s history in addition to the actual

percept is required to decide on the next action, it

must be represented in a suitable form (model)



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Model-based Reflex Agent

function MODEL-BASED-REFLEX-AGENT(percept) returns an action

persistent: state, the agent’s current conception of the world state

model, a description of how the next state depends on

current state and action

rules, a set of condition–action rules

action, the most recent action, initially none

state ← UPDATE-STATE(state, action, percept, model)

rule ← RULE-MATCH(state, rules)

action ← rule.ACTION

return action



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Goal-based Agent

• often, percepts alone are insufficient to decide what to do

• model-based, goal-based agents use an explicit representation

of goals and consider them for the choice of actions



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Utility-based Agents

• usually, there are several possible actions that can be taken in a

given situation

• in such cases, the utility of the next achieved state can come

into consideration to arrive at a decision



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Knowledge Representation and Reasoning

• often, our agents need knowledge before they can

start to act intelligently

• they then also need some reasoning component to

exploit the knowledge they have

• examples: • knowledge about the important concepts in a domain

• knowledge about actions one can perform in a domain

• knowledge about temporal relationships between events

• knowledge about the world and how properties are related to

actions



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Knowledge Bases

• knowledge base (KB) = set of sentences in a formal language

• declarative approach to build an agent

TELL it what it needs to know

• then it can ASK itself what to do – answers should follow from

the KB

• agents can be viewed at the knowledge level

i.e. what they know, regardless of the implementation

• or on the implementation level

i.e. data structures in KB and algorithms to manipulate them

inference engine domain-independent algorithms

knowledge base domain-specific content



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Knowledge-Based Agents

function KB-AGENT(percept) returns an action

persistent: KB, a knowledge base

t, a counter, initially 0, indicating time

TELL(KB, MAKE-PERCEPT-SENTENCE(percept,t))

action ← ASK(KB, MAKE-ACTION-QUERY(t))

TELL(KB, MAKE-ACTION-SENTENCE(action,t))

t ← t + 1

return action



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What’s about first order logic?



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Wumpus Models



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Logic-Based Reflex Agent

• … only reacts to percepts

• example for a percept statement (at time 5):

𝑝𝑒𝑟𝑐𝑒𝑝𝑡 𝑠𝑡𝑒𝑛𝑐ℎ, 𝑏𝑟𝑒𝑒𝑧𝑒, 𝑔𝑙𝑖𝑡𝑡𝑒𝑟, 𝑛𝑜𝑛𝑒, 𝑛𝑜𝑛𝑒, 5

1. ∀𝑏, 𝑔, 𝑢, 𝑐, 𝑡. 𝑝𝑒𝑟𝑐𝑒𝑝𝑡 𝑠𝑡𝑒𝑛𝑐ℎ, 𝑏, 𝑔, 𝑢, 𝑐 ⇒ 𝑆𝑡𝑒𝑛𝑐ℎ 𝑡 ∀𝑠, 𝑔, 𝑢, 𝑐, 𝑡. 𝑝𝑒𝑟𝑐𝑒𝑝𝑡 𝑠, 𝑏𝑟𝑒𝑒𝑧𝑒, 𝑔, 𝑢, 𝑐 ⇒ 𝐵𝑟𝑒𝑒𝑧𝑒 𝑡 ∀𝑠, 𝑏, 𝑢, 𝑐, 𝑡. 𝑝𝑒𝑟𝑐𝑒𝑝𝑡 𝑠, 𝑏, 𝑔𝑙𝑖𝑡𝑡𝑒𝑟, 𝑢, 𝑐 ⇒ 𝐴𝑡𝐺𝑜𝑙𝑑 𝑡

2. choice of action

∀𝑡. [𝐴𝑡𝐺𝑜𝑙𝑑(𝑡) ⇒ 𝐴𝑐𝑡𝑖𝑜𝑛(𝑔𝑟𝑎𝑏, 𝑡)]

• query: ∃𝑥. 𝐴𝑐𝑡𝑖𝑜𝑛(𝑥, 𝑡)



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Encyclopedic Knowledge

• knowledge about structure and properties of a

domain

• categories: allow to organize objects

• taxonomy: description using subclass relations

• inheritance: objects inherit properties based on

relations

• physical composition: description of larger structures

• instance relationship: obtain categories for an object

• common sense: general knowlegde shared by a large

group of individuals



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A General Category Hierarchy



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Knowledge about Dynamics

• agents have an internal model • of all basic aspects of their environment,

• of the executability and effects of their actions,

• of further basic laws of the world, and

• of their own goals

• important aspect: how does the world change?



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Situation Calculus

• a way to describe dynamic worlds with FOL

• states are represented by terms

• the world is in state 𝑠 and can only be altered through the execution of an action: 𝑑𝑜(𝑎, 𝑠) is the resulting situation, if 𝑎 is executed

• actions have preconditions and are described by their effects

• relations whose truth value changes over time are called fluents.

represented through a predicate with two arguments: the fluent

and a state term. for example, 𝑎𝑡(𝑥, 𝑦, 𝑠) means, that in situation 𝑠, the agent is at position 𝑥, 𝑦. ℎ𝑜𝑙𝑑𝑖𝑛𝑔(𝑦, 𝑠) means that in situation 𝑠, the agent holds object 𝑦.

• Atemporal or eternal predicates, e.g., 𝑝𝑜𝑟𝑡𝑎𝑏𝑙𝑒(𝑔𝑜𝑙𝑑)



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Example Wumpus World

let 𝑠0 be the initial situation and

𝑠1 = 𝑑𝑜(𝑓𝑜𝑟𝑤𝑎𝑟𝑑, 𝑠0) 𝑠2 = 𝑑𝑜(𝑡𝑢𝑟𝑛(𝑟𝑖𝑔ℎ𝑡), 𝑠1) 𝑠3 = 𝑑𝑜(𝑓𝑜𝑟𝑤𝑎𝑟𝑑, 𝑠2)



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Description of Actions

• preconditions: in order to pick something up, it must

be both present and portable:

∀𝑥, 𝑠[𝑃𝑜𝑠𝑠 𝑔𝑟𝑎𝑏 𝑥 , 𝑠 ⇔ 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑥, 𝑠 ∧ 𝑝𝑜𝑟𝑡𝑎𝑏𝑙𝑒(𝑥)]

• in the wumpus world:

𝑝𝑜𝑟𝑡𝑎𝑏𝑙𝑒 𝑔𝑜𝑙𝑑 , ∀𝑠[𝑎𝑡𝑔𝑜𝑙𝑑(𝑠) ⇒ 𝑝𝑟𝑒𝑠𝑒𝑛𝑡(𝑔𝑜𝑙𝑑, 𝑠)]

• positive effect axioms:

∀𝑥, 𝑠 𝑃𝑜𝑠𝑠 𝑔𝑟𝑎𝑏 𝑥 , 𝑠 ⇒ ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑥, 𝑑𝑜 𝑔𝑟𝑎𝑏 𝑥 , 𝑠

• negative effect axioms:

∀𝑥, 𝑠[𝑃𝑜𝑠𝑠(𝑟𝑒𝑙𝑒𝑎𝑠𝑒 𝑥 , 𝑠) ⇒ ¬ℎ𝑜𝑙𝑑𝑖𝑛𝑔(𝑥, 𝑑𝑜(𝑟𝑒𝑙𝑒𝑎𝑠𝑒 𝑥 , 𝑠))]



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Frame Problem

• we had: ℎ𝑜𝑙𝑑𝑖𝑛𝑔(𝑔𝑜𝑙𝑑, 𝑠0)

• following situation:

¬ℎ𝑜𝑙𝑑𝑖𝑛𝑔(𝑔𝑜𝑙𝑑, 𝑑𝑜(𝑟𝑒𝑙𝑒𝑎𝑠𝑒(𝑔𝑜𝑙𝑑); 𝑠0))?

• we had: ¬ℎ𝑜𝑙𝑑𝑖𝑛𝑔(𝑔𝑜𝑙𝑑, 𝑠0)

• following situation:

¬ℎ𝑜𝑙𝑑𝑖𝑛𝑔(𝑔𝑜𝑙𝑑, 𝑑𝑜(𝑡𝑢𝑟𝑛 𝑟𝑖𝑔ℎ𝑡 , 𝑠0))? • we must also specify which fluents remain unchanged!

• the frame problem: specification of the properties that do not

change as a result of an action.

• successor state axioms must also be specified



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Successor State Axioms

• an elegant way to solve the frame problem is to fully

describe the successor situation:

• true after action [ action made it true or,

already true and the action did not falsify it ]

• example for grab:

∀𝑎, 𝑥, 𝑠[ℎ𝑜𝑙𝑑𝑖𝑛𝑔(𝑥, 𝑑𝑜(𝑎, 𝑠)) ⇔ { 𝑎 = 𝑔𝑟𝑎𝑏 𝑥 ∧ 𝑃𝑜𝑠𝑠 𝑎, 𝑠 ∨ (ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑥, 𝑠 ∧ 𝑎 ≠ 𝑟𝑒𝑙𝑒𝑎𝑠𝑒(𝑥))}]

• can also be automatically compiled by only giving the

effect axioms (and then applying explanation

closure). Here we suppose that only certain effects

can appear



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Knowledge and Belief

• so far the agent has no notation about knowlegde

• its related to the mental state

• belief: the agent believes that a proposition holds

until proven otherwise, also related to defaults

𝑏𝑒𝑙𝑖𝑒𝑣𝑒𝑠(𝑎, 𝑝)

• knowlegde: is the justified true belief, there are no

doubts

𝑘𝑛𝑜𝑤𝑠 𝑎, 𝑝 𝑘𝑛𝑜𝑤𝑠𝑤ℎ𝑒𝑡𝑒𝑟 𝑎, 𝑝 ⇔ 𝑘𝑛𝑜𝑤𝑠 𝑎, 𝑝 ∨ 𝑘𝑛𝑜𝑤𝑠(𝑎, ¬𝑝)

• incomplete knowlegde: there is not enough

information to answer a query, closely related to

sensing



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Tradeoffs

• we like to build agents that react timely (tractability)

• all concepts seen so far can be mapped to FOL

• FOL is quite handy for knowlegde representation

• … but it is a problem for the reasoning

• there is always the tradeoff between expressiveness

and tractability

• solution: limit the expressive power, e.g. Horn

clauses

• use different representations for different aspects

(hybrid reasoning)



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Our Goals

• what are the goals we will tackle in the course • get to know example knowledge representations related to

intelligent agents

• use this representations for a concrete problem

• be aware of the modelling issues of different representations

• be able to automatically reason with different representations

(manually and using tools)

• understand the performance issues of different representations

• be able to manage specific aspects of KR & R



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Content

• next units

• foundations of different representations

• representing knowledge with different representations

• automated reasoning with different representations

• special issues with different representations

• we focus on • First Order Logic (FOL)

• Answer Set Programming (ASP)

• Description Logic (DL)

• Planning



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Tool Support

• we will use state-of-the-art tools

• please install them for the lecture and the practical

work

• FOL • Automated Theorem Prover – Prover9 (GUI)

• https://www.cs.unm.edu/~mccune/mace4/

• ASP • Potsdam Answer Set Solving Collection – clingo (command line)

• http://potassco.sourceforge.net/

• DL • ontology editor - Protégé (GUI)

• http://protege.stanford.edu/

https://www.cs.unm.edu/~mccune/mace4/https://www.cs.unm.edu/~mccune/mace4/http://potassco.sourceforge.net/http://potassco.sourceforge.net/http://potassco.sourceforge.net/http://protege.stanford.edu/http://protege.stanford.edu/



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Practical Work

• support Gargamel in setting the table



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Logic-based Knowledge Representation

• logic: properties of the world (domain) are

represented in the form of propositions and

sentences.

• syntax: expressions (formulae) ;

admissible/syntactically correct sentences

• semantics: – meaning of the sentences

– true sentences basis for logical consequences, for example,

from 1, 2, …, n we can derive .

– inference systems (calculi): allow „calculations“

– natural representation of facts and rules, for example:

from_Crete(Epimenides).

x(from_Crete(x) lies(x)).

logical conclusion: lies(Epimenides).



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Logic-based Knowledge Representation

• reference language – first order (predicate) logic

– most important formalism of logic

– expressive: all computable functions specifiable

(Church‘s thesis)

– simple, natural syntax and intuitive semantics

• disadvantages – non-decidable but decidable fragments

– high computational complexity further restrictions

of expressivity needed



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Logical KR

• use of FOL

• domain theory – a general description of the problem

domain

– e.g. groups in mathematics, geometry

• axioms – general valid sentences

• facts – specific knowledge

– literals, e.g., child(Bart,Homer), ¬likes(Moe,Bart)



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Logical KR

• general rules

– universal quantified sentences

– x1x2 …xn(1˄…˄n)→

– i: condition, premise

– : conclusion

– similar to if-then statement

• universal quantified Horn Clauses

– all i and are atoms

– subset of FOL, reduced expressiveness

– e.g., x(human(x)→mortal(x))



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Logical KR

• universe of Discourse

– in FOL in general no special domain

– in KR a particular domain D and interpretation

– e.g., arithmetic: {0,1,2,3} (constants), {+,*} (functions )



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Methodology of Modeling

• what is a good way to model in logic

• no formal approach

• clarity of names, concepts

– names of constants, predicates etc. are meaningful, e.g. partent(x,y)

• disclose relations

– x(senior(x)→discount(x))

– fact: senior(Peter)

– problem: gender? why discount ?

– xy[(y=gender(x)˄age(x)>age_limit(y))→ discount(x)]



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Methodology of Modeling

• validation

– why a sentence gets valid ?

• generality

– can one express a sentence more

generally ?

• requirement of predicates

– are new predicates necessary ?

– relation to other predicates

– description of super/sub classes



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Approach

1. conceptualization

– decide what to represent

– abstract concept

2. choice of vocabulary

– translation of the abstract concept to FOL

– the resulting vocabulary is an ontology of the problem domain

3. coding of the domain theory

– specify all relations and rules

4. coding of the specific knowledge



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Example – Alpine Club

• Tony, Mike and John belong to the Alpine Club. Every member

of the Alpine Club who is not a skier is a mountain climber.

Mountain climbers do not like rain, and anyone who does not

like snow is not a skier. Mike dislikes whatever Tony likes, and

likes whatever Tony dislikes. Tony likes rain and snow.

• Prove that the given sentences logically entail that there is a

member of Alpine Club who is a mountain climber but not a

skier.

• Suppose we had been told that Mike likes whatever Tony

dislikes, but we had not been told that Mike dislikes whatever

Tony likes. Prove that the resulting set of sentences no longer

logically entails that there is a member of Alpine Club who is a

mountain climber but not a skier.



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Thank You!

knowledge representation & reasoning i · 2019-07-25 · knowledge representation and reasoning •...

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