Propositions and Inference - jbnu.ac.krnlp.jbnu.ac.kr/AI2018/slides_cbnu/nash_ai_ca_slides_ch5.pdf · Suppose that there are three atoms: ai is fun, happy, and light on Interpretation

Propositions and Inference

Seung-Hoon Na1

1Department of Computer ScienceChonbuk National University

2018.9.19

Seung-Hoon Na (Chonbuk National University) Propositions and Inference 2018.9.19 1 / 95

Why Propositions?

Extensional description for constraints: Tables as legal assignmentsto variables

Intensional description for constraints: In terms of formulas

Propositions provide a suitable language for providing intensionalconstraints

Propositions are:

More concise and readable than extensional representationMore efficient reasoning, from the form of knowledgeModular and easy to debug: small changes to the problem result insmall changes to the knowledge baseRicher types of queries to be answeredExtended to reason about individuals and relations (Predicate logic)


Syntax for propositional calculus

An atomic proposition (or an atom): a symbolE.g.) ai fun, lit l1, sunny , rainy , etc.Here, a symbol starts with a lower-case letter

Proposition (or logical formula)an atomic proposition ora compound proposition of form

¬p

p ∧ q

p ∨ q

p → q

p ← q

p ↔ q

Logical connectives: ¬, ∧, ∨, →, ←, ↔

E.g.) ¬a ∨ b ∧ c → d ∧ ¬e ∨ f


Semantics of the Propositional Calculus

Semantics: defines the meaning of the sentence of a languageWhen the sentences are about a (real or imagined) world, semanticsspecifies how to put symbols of the language into correspondence withthe world

Semantics for propositional calculusAtoms: have meaning to someone and are either true or false ininterpretationsPropositions: Decomposed into the semantics of atoms. The truth ofa proposition is compositionally obtained from the truth of its involvedatoms

Interpretation; a function π that maps atoms to {true, false}Suppose that there are three atoms: ai is fun, happy , and light onInterpretation I1 and I2 defined by π1 and π2, respectively:

atoms π1ai is fun true

happy falselight on true

atoms π2ai is fun false

happy truelight on false


Example: Interpretation



Knowledge base: a set of propositions that are stated to be true.

Axiom: an element of the knowledge base.

Model: Given a knowledge base KB, a model of KB is aninterpretation in which all the propositions in KB are true.

KB |= g : if g is true in every model of KB.

g is a logical consequence of KB.g logically follows from KBKB entails gNo interpretation exists in which KB is true and g is false.




Propositional constraints

Propositional satisfiability problem consists of:

Boolean variables: a variable with domain {true, false}if X is a Boolean variable, x means X = true and ¬x means X = false.E.g.) happy means Happy = true, and ¬happy means Happy = false.

Clausal constraintsClause: an expression of the form l1 ∨ l2 ∨ · · · ∨ lk , where each li is aliteral.Literal: an atom or the negation of an atom.Satisfiability of a clause: A clause l1 ∨ l2 ∨ · · · ∨ lk is satisfied in apossible world iff there is at least one literal li is true in the possibleworld.


Propositional constraints: An example

The clause ψ = happy ∨ sad ∨ ¬living is a constraint on Booleanvariables Happy , Sad , and Living .

What assignments make the clause ψ true, being satisfied?

What assignments make the clause ψ false, being unsatisfied?


CSP as Propositional Satisfiable Problem

Convert a CSP variable Y with domain {v1, · · · , vk} to k Booleanvariables {Y1, · · · ,Yk}

Yi is true when Y has value vi and is false otherwise.Yi : called an indicator variableThus, k atoms, y1, · · · , yk are used to represent the CSP variable Y .

Add basic clausal constraints to KB such as:

¬yi ∨ ¬yj for i < j : yi and yj cannot be both true when i 6= jy1, · · · , yk : one of the yi must be true.


Propositional definite clauses

Propositional definite clauses represent KB by definite clauses, sonot all compound propositions are allowed

An atomic proposition or atom: the same as in propositionalcalculus

A definite clause: h← a1 ∧ · · · anh: the head of the clausem > 0: the clause is called a rulem = 0: the clause is called an atomic clause or fact.

A knowledge base: a set of definite clauses


Propositional definite clauses: Example

Electrical environment:


Propositional definite clauses: Example

Electrical environment:


Propositional definite clauses

A query: Ask whether a formula is a logical consequence of theknowledge base, with the form:

ask b

that means KB |= b. Here, b is an atom or a conjunction of atoms(similar to the body of a rule)

The answer “yes”: if b is a logical consequence of KB (KB |= b)

The answer “no”: if b is not a logical consequence of KB (KB 6|= b)


Propositional definite clauses: Proofs

The definition of |=: specify which propositions are logicalconsequences of KB, but not specify how to compute them

Deduction: a specific form of inference to determine if someproposition is logical sequence of KB

A proof: a mechanically derivable demonstration that a propositionlogically follows from KB

A theorem: a provable proposition

A proof procedure: a algorithm for deriving consequences of aknowledge base

KB ` g : g can be proved or derived from KB

A proof procedure is sound: If KB ` g , then KB |= g

A proof procedure is complete: If KB |= g , then KB ` g


Bottom-up proof procedure

Based on one rule of derivation (inference rule), a generalized formof the inference rule called modus ponens:

If “h ← a1 ∧ · · · ∧ am” is a definite clause in KB, andeach ai has been derived, then h can be derived.

Compute the consequence set C of a set of KB of definite clauses.




Bottom-up proof procedure: An example



Soundness?

Completeness?

Complexity

Fixed point


Top-down proof procedure

Based on propositional definite clause resolution or SLDresolution

An answer clause is of the form:

yes ← a1 ∧ · · · ∧ am

Given query q1 ∧ · · · ∧ qm, an initial answer clause is given:(ψq) yes ← q1 ∧ · · · ∧ qm

A definite clause for the selected atom q1:(ψc) q1 ← b1 ∧ · · · ∧ bp

The resolvent of ψq with ψc is the answer clause:

yes ← b1 ∧ · · · ∧ bp ∧ q2 ∧ · · · ∧ qm


Top-down proof procedure

Select and choose: Select an atom and choose a clause


Top-down proof procedure: Example


Top-down proof procedure: Example (cont.)


Search graph in top-down proof procedure

A node in the search graph: represents an answer clause -yes ← a1 ∧ · · · ∧ am.

The neighbors of a node: represents all possible answer clausesobtained by resolving on a1 where a1 is the selected atom.

The goal nodes of the search graph: of the form yes ←.


Search graph in top-down proof procedure: Example

Knowledge base:


Search graph in top-down proof procedure: Example(cont.)


Infinite loop in proof procedure: Example

Knowledge base:

g ← a

a← b

b ← a

g ← c

c

Query:

ask g


Background knowledge and observations

ObservationInformation received online from users, sensors, or other knowledgesourcesIn propositional world, an observation is assumed to be a set of atomicpropositionsDoes not provide rules directly

Background knowledgeKB allows the agent to do something useful with observations

Reasoning types1) Observations are added to the background knowledge, or2) Observations are treated separately from background knowledge

Users’ knowledge: limitedCannot tell us everything that is trueDon’t know what is relevant and what vocabulary to use

OntologySpecifies the meaning of the symbolsDon’t know what is relevant and what vocabulary to use

Relevant knowledge is too big in many problems


Querying the User

ask-the-user mechanism: acquire information from a user intop-down proof procedure

askable atom: if the user may know its truth value at run time

Classes of atoms to be selected:

(non-askable) atoms for which the user is not expected to know: thesystem never asksaskable atoms for which the user has not already provided an answer:the user should be asked for the answer and the answer is recordedaskable atoms for which the user has already provided an answer: therecorded answer should be used, the user should not be asked again

ask-the-user mechanism in bottom-up proof procedure: shouldavoid asking about all askable atoms


Querying the User: Example


Knowledge-level Explanation

The explicit use of semantics allows explantation and debugging atthe knowledge level.

Three complementary means of interrogation to explain relevantknowledge

a how question: used to explain how an answer was proveda why question: used to ask the system why it is asking the user aquestiona whynot question: used to ask why an atom was not proven


How did the system prove an atom?

Suppose that there is a proof for g

Then, either g must be an atomic clause or three must be a rule:

g ← a1 ∧ · · · ∧ ak

Now, suppose that the system has proved g

Then the user asks how, and the system can display the clause thatwas used to prove g

But, when this clause is a rule (i.e. k ≥ 1), the user could further ask:

how i

Then, this will give the rule that was used to prove ai . In this way, theuser can continue using the how to explore how g was proved.


How did the system prove an atom?: Example


How did the system prove an atom?: Example (cont.)


Why did the system ask a question?


How and why are used together

how: moves from higher-level to lower-level subgoals

why: moves from lower-level to higher-level subgoals


Knowledge-level debugging

knowledge-level debugging: the process of finding errors in KB withreference only to what the symbols mean and what is true in theworld.

Four types of non-syntactic errors:

Incorrect answers: some atom that is false in the intendedinterpretation was derivedMissing answers: the proof failed when it should have succeeded(some particular true atom was not derived).Infinite loops.The system asks irrelevant questions


Incorrect answers

An incorrect answer: a false-positive error, an answer that hasbeen proved yet is false in the intended interpretation.

Suppose there is an atom g that was proved yet is false in theintended interpretation.

Then there must be a rule g ← a1 ∧ · · · ∧ ak in KB that was used toprove g .Either,

One of the ai is false in the intended interpretation, in which case itcan be debugged in the same way, orall of the ai are true in the intended interpretation. In this case, thedefinite clause g ← a1 ∧ · · · ∧ ak must be incorrect.


Incorrect answers: debug false positives


Incorrect answers: Example


Incorrect answers: Example (cont.)


Missing answers

a false-negative error: A goal g that is true in the domain, but isnot a consequence of the knowledge base.

Suppose g is an atom that should have a proof, but which fails.

Because the proof for g fails, the bodies of all of the definite clauseswith g in the head fail.

Suppose one of these definite clauses for g should have resulted in aproof: this means all of the atoms in the body must be true in theintended interpretation. Because the body failed, there must be anatom in the body that fails. This atom is then true in the intendedinterpretation, but fails. So we can recursively debug it.Otherwise, there is no definite clause applicable to proving g : the usermust add a definite clause for g .

A whynot question can be used to ask why some g was not proved


Missing answers


Missing answers: Example


Infinite loops

There can be an infinite loop only if the knowledge base is cyclicA knowledge base is cyclic: there is an atom a such that there is asequence of definite clauses of the form

a← · · · a1 · · ·a1 ← · · · a2 · · ·· · ·an ← · · · a · · ·

To detect a cyclic knowledge base, modify the top-down proofprocedure to maintain the set of all ancestors for each atom inthe proof

Initially, the set of ancestors of each atom is empty.When the rule a← a1 ∧ · · · ∧ ak is used to prove a, then the ancestorsof ai will be:

ancestors(ai ) = ancestors(a) ∪ {a}The proof can fail if an atom is in its set of ancestors: only occurs on acyclic KB.


Horn clauses

The definite clause language does not allow a contradiction to bestated

To allow proof by contradiction, a simple extension is necessary

An integrity constraint:

false ← a1 ∧ · · · ∧ ak

Horn clause: either 1) a definite clause or 2) an integrity constraint

That is, a Horn clause has either false or a normal atom as its head


Horn clauses

A Horn clause KB can imply negations of atoms


Horn clauses

The language of Horn clauses does not allow disjunctions andnegations to be input, however, it can derive disjunctions of negationsof atoms


Horn clauses

A set of clauses is unsatisfiable if it has no models

A set of clauses is provably inconsistent with respect to a proofprocedure if false can be derived from the clauses using that proofprocedure

A set of definite clauses: always satisfiablewe can always find its model

A set of Horn clauses: can be unsatisfiableE.g.) {a, false ← a} is unsatisfiable


Assumables and conflicts

An assumable: an atom that can be assumed in a proof bycontradiction

A proof by contradiction: derives a disjunction of the negation of theassumables.

Suppose that KB is a set of Horn clauses

A conflict of KB is a set of assumables that implies false given KB.

C = {c1, · · · , cr} is a conflict of KB if

KB ∪ {c1, · · · , cr} |= false

In this case, an answer is

KB |= ¬c1 ∨ · · · ¬cr

A minimal conflict: a conflict such that no strict subset is also aconflict.


Assumables and conflicts: Example

Consider KB2:

false ← a ∧ b

a← c

b ← d

b ← e

Suppose that {c , d , e, f , g , h} is the set of assumables

{c , d} and {c , e}: minimal conflicts of KB2

{c , d , e, h}: a conflict of KB2


Consistency-based diagnosis

Consistency-based diagnosisMake assumptions about what is working normally, and derive whatcomponents could be abnormal

E.g.) Electrical environment: Assume that all the devices are okay

A = {ok cb1, ok s1, ok s2, ok s3, ok l1, ok l2}

Derive what are a conflict C ⊆ A, which derives false?

KB ∪ C |= false

To determine the possible faults based on a model of the system andobservations of the systemConflicts can be used to prove what is wrong with the system


Consistency-based diagnosis: Example



The user is able to observe the switch positions and whether a light islit or dark.

Suppose the user observes that all three switches are up, and that l1and l2 are both dark.

up s1 up s2 up s3

dark l1 dark l2

Given KB, we have two minimal conflicts:

{ok cb1, ok s1, ok s2, ok l1}{ok cb1, ok s3, ok l2}

Thus, it follows:

KB |= ¬ok cb1 ∨ ¬ok s1 ∨ ¬ok s2 ∨ ¬ok l1

KB |= ¬ok cb1 ∨ ¬ok s3 ∨ ¬ok l2


Consistency-based diagnosis

a consistency-based diagnosis: a set of assumables that has at leastone element in each conflict

a minimal diagnosis is a diagnosis such that no subset is also adiagnosis

For one of the diagnoses, all of its elements must be false in the worldbeing modeled.




Reasoning in Horn clauses: Bottom-up proof procedure

Find all the conflicts Conflicts in KB:

Conflicts = {A : KB ∪ A |= false and A ⊆ Assumables}

Maintain the conclusion set C that consists of pairs 〈a,A〉:a: an atom, A: a set of assumables that imply a in KB

Initially, C : {〈a, {a}〉 : a is assumable}Apply the extended modus ponen rule to derive new conclusionsfrom KB:

If there is a clause h← b1 ∧ · · · ∧ bm such that for eachbi there exists Ai such that 〈bi ,Ai 〉 ∈ C , then 〈h,A1 ∪· · · ∪ Am〉 can be added to C .

When the pair 〈false,A〉 is generated, the assumptions A form aconflict

This bottom-up proof procedure is called assumption-based truthmaintenance system(ATMS)


Reasoning in Horn clauses: Bottom-up proof procedure


Reasoning in Horn clauses: Bottom-up - Example


Reasoning in Horn clauses: Bottom-up - Example (Cont.)

Thus, KB entails:

¬ok cb1 ∨ ¬ok s3 ∨ ¬ok l2

The other conflict can be found by continuing the algorithm


Reasoning in Horn clauses: Bottom-up - Example2

Consider KB2:

false ← a ∧ b

a← c

b ← d

b ← e

Suppose that {c , d , e, f } is the set of assumables

Initially, the conclusion set C is:

C = {〈c , {c}〉, 〈d , {d}〉, 〈e, {e}〉, 〈f , {f }〉}

Select the clause a← c : add 〈a, {c} to C

Select the clause b ← d : add 〈b, {d}〉 to C

Select the clause false ← a ∧ b: add 〈false, {c, d}〉 to C

· · ·Seung-Hoon Na (Chonbuk National University) Propositions and Inference 2018.9.19 61 / 95

Reasoning in Horn clauses: Top-down proof procedure

Different from the definite clause case, the top-level query is to provefalse, and the assumables encountered in a proof are not proven butcollected.


Reasoning in Horn clauses: Top-down - Example


Reasoning in Horn clauses: Top-down - Example2

Consider KB2:

false ← a ∧ b

a← c

b ← d

b ← e

Suppose that {c , d , e, f } is the set of assumables

The sequence of G is given as:

{false}{a, b}{c , b}{c, d}


Complete knowledge assumption

A database is often complete in the sense that anything not stated isfalse.


Complete knowledge assumption

Complete knowledge assumption is also called the closed-worldassumption.

Assumes that, for every atom, the clauses with the atom as the headcover all the cases when the atom is true.An agent can conclude that an atom is false if it cannot derive that theatom is true.Requires that everything relevant about the world be known to theagent

Open-world assumption: the agent does not know everything and socannot make any conclusions from a lack of knowledge


Clark’s completion

Suppose the clauses for atom a are:

a← b1

· · ·a← bn

where an atomic clause a is the rule a← trueThe complete knowledge assumption specifies:

a→ b1 ∨ · · · ∨ bn

The clauses defining a is equivalent to:

a← b1 ∨ · · · ∨ bn

Clark’s completion of the clauses for a:

a↔ b1 ∨ · · · ∨ bn

Clark’s completion for the case that there are no rules for an atom a:

a↔ falseSeung-Hoon Na (Chonbuk National University) Propositions and Inference 2018.9.19 67 / 95

Clark’s completion: Example


Clark’s completion: Example (Cont.)


Negation as failure

With the completion, the system can derive negations

Thus, it is useful to extend the language to allow negations in thebody of clauses.

An extended definite clause: allow literals in the body rather thanjust atoms

Negation as failure: ∼a:

The negation of atom a under the complete knowledge assumptionDistinguished from classical negation that does not assume thecompletion

A negation ∼a in the body of a clause or the query becomes ¬a inthe completion.


Definite clause with negation as failure: Example


Negation as failure: Example (Cont.)


Definite clause with negation as failure

With negation as failure, Clark’s completion of an acyclic knowledgebase: always consistent and always gives a truth value to each atom

However, non-acyclic knowledge bases: semantically problematic

E.g.) non-acyclic KB:

a← ∼b

b ← ∼a

Clark’s completion is a↔ ¬b: just specifies that a and b havedifferent truth values but not which one is true.

E.g.) non-acyclic KB:

a← ∼a

Clark’s completion is a↔ ¬a: logically inconsistent


Non-monotonic reasoning

Monotonic logic: if any proposition that can be derived from a KBcan also be derived when extra propositions are added to the KB

The definite clause logic: monotonicAdding knowledge does not reduce the set of propositions that canderived

Non-monotonic logic: if some conclusions can be invalidated byadding more knowledge.

The logic of definite clause with negation as failure:non-monotonicUseful for representing defaults

A default: a rule that can be used unless it overridden by anexception

To say that b is normally true if c is true, a KB designer can write arule of the form:

b ← c ∧ ∼ab a

where ab a is an atom that means abnormal with respect to someaspect a.Given c , the agent can infer b unless it is told ab a.


Non-monotonic reasoning: Example


Non-monotonic reasoning: Example (Cont.)


Negation as failure: Bottom-up proof procedure

∼p is added to C when it can determine that p must fail

The recursive definition of failure for p failswhen every body of a clause with p as the head fails.A body fails if one of the literals in the body failsAn atom bi in a body fails if ∼bi has been derivedA negation ∼bi in a body fails if bi has been derived.


Negation as failure: Bottom-up proof procedure


Bottom-up negation as failure procedure: Example


Negation as failure: Top-down proof procedure

When a negated atom ∼a is selected, a new proof for atom a isstarted.

If the proof for a fails, ∼a succeeds.

If the proof for a succeeds, the algorithm fails and must make otherchoices.


Negation as failure: Top-down proof procedure


Top-down negation as failure procedure: Example


Abduction

Abduction: a form of reasoning where assumptions are made toexplain observation

In abduction, an agent hypothesizes what may be true about anobserved case.An agent determines what implies its observations – what could be trueto make the observations true.

To formalize abduction, we use the language of Horn clauses andassumables:

a knowledge base, KB, which is a set of Horn clauses, anda set A of atoms, called the assumables; the assumables are thebuilding blocks of hypotheses.

A scenario of 〈KB,A〉 is a subset H of A such that KB ∪ H issatisfiable

KB ∪ H is satisfiable: if a model exists in which every element of KBand every element H is true.


Abduction

An explanation of proposition g from 〈KB,A〉 is a scenario that,together with KB, implies g .

An explanation of proposition g is a set H, H ⊂ A such that

KB ∪ H |= g

KB ∪ H 6|= false

A minimal explanation of g from 〈KB,A〉: an explanation H of gfrom 〈KB,A〉 such that no strict subset of H is also an explanation ofg from 〈KB,A〉.


Abduction: Example


Abduction: Example (Cont.)


Abductive diagnosis

The agent hypothesizes diseases and malfunctions, as well as thatsome parts are working normally, to explain the observed symptoms

Abductive diagnosis vs. consistency-based diagnosis (CBD):Behavior:

In CBD, only normal behavior needs to be represented, and thehypotheses are assumptions for normal behavior.In abductive diagnosis, faculty behavior as well as normal behaviorneeds to be represented, and the assumables need to be for normalbehavior and for each fault.

Observations:

In CBD, observations are added to KB, and false is provedIn abductive diagnosis, observations need to be explained.


Abductive diagnosis: Example


Abductive diagnosis: Example (Cont.)










Causal models

A causal model: a model of causality, is a representation of adomain that predicts the results of interventions

An intervention: an action that forces a variable to have a particularvalue

It changes the value in some way other than manipulating othervariables in the modelTo predict the effect of interventions, a causal model represents howthe cause implies its effect. When the cause is changed, its effectshould be changed

An evidential model: represents a domain in the other direction fromeffect to cause


Causal models: Example


Causal models: Example (Cont.)


Documents

Propositions and Inference - jbnu.ac.krnlp.jbnu.ac.kr/AI2018/slides_cbnu/nash_ai_ca_slides_ch5.pdf · Suppose that there are three atoms: ai is fun, happy, and light on Interpretation