Slide 1

Propositional Definite Clause Logic: Syntax,

Semantics and Bottom-up Proofs

Jim LittleUBC CS 322 – CSP October 20, 2014

Slide 2

Lecture Overview

• Recap: Logic intro• Propositional Definite Clause

Logic: Semantics• PDCL: Bottom-up Proof

Logic: A general framework for reasoning

General problem: Query answering• tell the computer how the world works• tell the computer some facts about the world• ask a yes/no question about whether other facts must be

true

Solving it with Logic1. Begin with a task domain.2. Distinguish those things you want to talk about (the

ontology)3. Choose symbols in the computer to denote elements of your

ontology4. Tell the system knowledge about the domain5. Ask the system whether new statements about the domain

are true or false

live_w4? lit_l2?

Slide 3

Example: Electrical Circuit

/ up

/down

Slide 4

/ up

/down

Syntax: are these sentences that a reasoning procedure can process?

Semantics: what do these statements say about the world I need to represent?

Slide 5

To Define a Logic We Need

• Syntax: specifies the symbols used, and how they can be combined to form legal sentences • Knowledge base is a set of sentences in the language

• Semantics: specifies the meaning of symbols and sentences

• Reasoning theory or proof procedure: a specification of how an answer can be produced.• Sound: only generates correct answers with respect to

the semantics

• Complete: Guaranteed to find an answer if it exists

Slide 6

Propositional Definite Clauses: Syntax

Definition (atom)

An atom is a symbol starting with a lower case letter

Definition (body)

A body is an atom or is of the form b1 ∧ b2 where b1 and b2 are bodies.

Definition (definite clause)

A definite clause is

- an atom or

- a rule of the form h ← b where h is an

atom (“head”) and b is a body. (Read this as “h if b”.)

Definition (KB)

A knowledge base (KB) is a set of definite clauses

Examples: p1; live_l1

Examples: p1 ∧ p2; ok_w1 ∧ live_w0

Examples: p1 ← p2; live_w0 ← live_w1 ∧ up_s2

Slide 7

Propositional Definite Clauses: Semantics

Definition (model)A model of a knowledge base KB is an interpretation in

which KB is true.

Similar to CSPs: a model of a set of clauses is an interpretation that makes all of the clauses true

Definition (interpretation)

An interpretation I assigns a truth value to each atom.

Definition (truth values of statements)• A body b1 ∧ b2 is true in I if and only if b1 is true in

I and b2 is true in I.• A rule h ← b is false in I if and only if b is true in I

and h is false in I.• A knowledge base KB is true in I if and only if

every clause in KB is true in I.

Slide 8

To Obtain This We Need One More Definition

Definition (logical consequence)If KB is a set of clauses and G is a conjunction of atoms, G is a logical consequence of KB, written KB ⊧ G, if G is true in every model of KB.

• we also say that G logically follows from KB, or that KB entails G.• In other words, KB ⊧ G if there is no interpretation in which KB is true and G is false.

• when KB is TRUE, then G must be TRUE

• We want a reasoning procedure that can find all and only the logical consequences of a knowledge base

Slide 9

User’s View of Semantics

• Choose a task domain: intended interpretation.

• For each proposition you want to represent, associate a proposition symbol in the language.

• Tell the system clauses that are true in the intended interpretation: axiomatize the domain.

• Ask questions about the intended interpretation.

• If KB |= g , then g must be true in the intended interpretation.

Slide 10

Computer’s View of Semantics

• The computer doesn’t have access to the intended interpretation.

• All it knows is the knowledge base.

• The computer can determine if a formula is a logical consequence of KB.

• If KB |= g then g must be true in the intended interpretation.

• Otherwise, there is a model of KB in which g is false. This could be the intended interpretation.

The computer wouldn't know!

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Computer’s View of Semantics

• Otherwise, there is a model of KB in which g is false. This could be the intended interpretation.

The computer wouldn't know

.

.

.

sr

q

qp

KBp q r s

I1 true true true true

I2 true true true false

Slide 12

To Define a Logic We Need

• Syntax: specifies the symbols used, and how they can be combined to form legal sentences • Knowledge base is a set of sentences in the language

• Semantics: specifies the meaning of symbols and sentences

• Reasoning theory or proof procedure: a specification of how an answer can be produced (sound and complete)• Bottom-up Proof Procedure for Finding Logical Consequence

Slide 13

Proof Procedures• A proof procedure is a mechanically derivable

demonstration that a formula logically follows from a knowledge base.

• Given a proof procedure P, KB ⊦ P g means g can be derived from knowledge base KB with the proof procedure.

• If I tell you I have a proof procedure for PDCL• What do I need to show you in order for you to trust my

procedure?

That is sound and complete

Slide 14

Soundness and Completeness

• Completeness of proof procedure P: need to prove that

If g is true in all models of KB (KB g)⊧ then g is derived by the procedure (KB ⊦P g)

Definition (completeness)

A proof procedure P is complete if KB g implies KB ⊧ ⊦P g.

complete: every atom that logically follows from KB is derived by P

• Soundness of proof procedure P: need to prove that

Definition (soundness)

A proof procedure P is sound if KB ⊦P g implies KB g.⊧

If g can be derived by the procedure (KB ⊦P g) then g is true in all models of KB (KB g)⊧

sound: every atom derived by P follows logically from KB (i.e. is true in every model)

Slide 15

Simple Proof Procedure

problem with this approach?

• If there are n propositions in the KB, must check all the interpretations!

Goal of proof theory • find sound and complete proof procedures that allow us to

prove that a logical formula follows from a KB avoiding to do the above

Simple proof procedure S• Enumerate all interpretations• For each interpretation I, check whether it is a model of KB

i.e., check whether all clauses in KB are true in I

• KB ⊦S g if g holds in all such models

Slide 16

2𝑛

Lecture Overview

• Recap Lecture 7• Logical Consequences and Proof

Procedures

• Bottom-Up Proof Procedure• Soundness

• Completeness

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Bottom-up proof procedure

• One rule of derivation, a generalized form of modus ponens:

• If “h ← b1 … bm" is a clause in the knowledge base, and

each bi has been derived,

then h can be derived.

• This rule also covers the case when m = 0.

Slide 18

Bottom-up proof procedureC :={};repeat

select clause “h ← b1 ∧ … ∧ bm” in KB such that bi ∈ C for all i, and h ∉ C;C := C ∪ { h }

until no more clauses can be selected.KB ⊦ G if G ⊆ C at the end of this procedure

The C at the end of BU procedure is a fixed point:• Further applications of our rule of derivation will not

change C!Slide 19

C := {};repeat

select clause h ← b1 … bm in KB such that bi C for all i, and h C;

C := C {h}until no more clauses can be selected.

Bottom-up proof procedure: example

a ← b c

a ← e f

b ← f k

c ← e

d ← k

e.

f ← j e

f ← c

j ← c

{}

{e}

{c,e}

{c,e,f}

{c,e,f,j}

{c,e,f,j,a}

Done.Slide 20

Lecture Overview

• Recap of Lecture 6

• Planning as CSP

• Logic

• Intro

• Propositional Definite Clause Logic (PDCL)

• Sintax and Semantics

• Logical Consequences and Proof Procedures

• Bottom Up Proof Procedure

• Soundness and completenessSlide 21

Soundness of bottom-up proof procedure BU

What do we need to prove to show that BU is sound ?

sound: every atom g that P derives follows logically from KB

Definition (soundness)

A proof procedure P is sound if KB ⊦P g implies KB g⊧ .

C := {};repeat

select clause h ← b1 … bm in KB such that bi C for all i, and h C; C := C {h}until no more clauses can be selected.

Slide 22

Soundness of bottom-up proof procedure BU

What do we need to prove to show that BU is sound ?

If g C at the end of BU procedure, then g is true in all models of KB (KB g)⊧

sound: every atom g that P derives follows logically from KB

Definition (soundness)

A proof procedure P is sound if KB ⊦P g implies KB g⊧ .

C := {};repeat

select clause h ← b1 … bm in KB such that bi C for all i, and h C; C := C {h}until no more clauses can be selected.

Slide 23

Soundness of bottom-up proof procedure BU

By contradiction: Suppose there is a g such that KB ⊦ BU g but not KB ⊧ g.

If g C at the end of BU procedure, then g is true in all models of KB (KB g)⊧

What do we need to prove to show that BU is sound ?

• Let h be the first atom added to C that is not true in every model of KB. • In particular, suppose I is a model of KB in which h isn’t true

• There must be a clause in KB of form

h b1 ... bn

where each bi is true in I.

• Because h is false in I, this clause is false in I.

• Therefore I is not a model of KB => ContradictionSlide 24

Completeness of BU: general idea

• Generic completeness of proof procedure: If G is logically entailed by the KB (KB ⊧ G) then G can be proved by the BU procedure (KB

⊦BU G)

Sketch of our proof:1. Suppose KB ⊧ G. Then G is true in all models of KB.

2. Thus G is true in any particular model of KB

3. We will define a model (called minimal model) so that if G is true in that model, G is proved by the bottom up algorithm.

4. Thus KB ⊦ G.

Slide 25

We define a specific model of our KB, in which• every atom in C at the end of BU is true • every other atom is false

This is called minimal model

All atoms = {a, b, c, d,e, f, g}C = {e,d, c, f,}Minimal Model =a=F, b=F, c=T, d = T, e=T, f=T, g=F

• Using this model, we’ll then show that, if KB ⊧ G, then G must be in C, that is

Completeness of BU: general idea

If g is true in all models of KB (KB ⊧ g) then g C at the end of BU procedure (KB ⊦BU g)

KB a ← e ∧

g.b ← f ∧ g. c ← e.

f ← c e. d.

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Claim: MM is a model of KBProof by contradiction: assume that MM is not a model

of KB. • Then there must exist some clause in KB which is

false in MMLike every clause in KB, it is of the form h ← b1 … bm

(with m 0).

• h ← b1 … bm can only be false in MM if each bi is true in MM and h is false in MM.Since each bi is true in MM, each bi must be in C as well.BU would add h to C, so h would be true in MMContradiction! Thus, MM is a model of KB

DefinitionThe minimal model MM is the interpretation in which- every element of BU’s fixed point C is true - every other atom is false.

Slide 27

Completeness of bottom-up procedure

Direct proof based on minimal model:• Suppose KB ⊧ g. Then g is true in all models of

KB.• Thus g is true in the minimal model.• Thus g C at the end of BU procedure.• Thus KB ⊦BU g. Done. KB g implies KB ⊧ ⊦BU g

If g is true in all models of KB (KB ⊧ g) then g C at the end of BU procedure (KB ⊦BU g)

Slide 28

Summary for bottom-up proof procedure BU

• BU is sound: it derives only atoms that logically follow from KB

• BU is complete:it derives all atoms that logically follow from KB

• Together: it derives exactly the atoms that logically follow from KB

• And, it is efficient!• Linear in the number of clauses in KB

Each clause is used maximally once by BUSlide 29

Learning Goals Up To Here

• PDCL syntax & semantics- Verify whether a logical statement belongs to the

language of propositional definite clauses- Verify whether an interpretation is a model of a

PDCL KB. ‾ Verify when a conjunction of atoms is a logical consequence

of a knowledge base

• Bottom-up proof procedure• Define/read/write/trace/debug the Bottom Up (BU) proof

procedure• Prove that the BU proof procedure is sound and complete

Slide 30

Slide 31

Lecture Overview

• Recap: Logic intro• Propositional Definite Clause

Logic: Semantics• PDCL: Bottom-up Proof

Slide 32

Next class

(still section 5.2)• Soundness and Completeness of Bottom-up

Proof Procedure• Using PDC Logic to model the electrical

domain• Reasoning in the electrical domain