slide 1 logic: td as search, datalog (variables) jim little ubc cs 322 – csp october 24, 2014...

Logic: TD as search, Datalog (variables)

Jim LittleUBC CS 322 – CSP

October 24, 2014Textbook §5.2 and 12

Lecture Overview

• Recap Top Down• TopDown Proofs as search• Datalog

Bottom-up vs. Top-down

• Key Idea of top-down: search backward from a query G to determine if it can be derived from KB.

KB C

G is proved if G C

When does BU look at the query G?

• At the end

Bottom-up Top-down

TD performs a backward search starting at G

KB answer

Query G

Slide 3

Top-down Ground Proof Procedure

Key Idea: search backward from a query G to determine if it can be derived from KB.

Top-down Proof Procedure: Basic elements

Notation: An answer clause is of the form: yes ← a1 ∧ a2 ∧ … ∧ am

Rule of inference (called SLD Resolution)Given an answer clause of the form:

yes ← a1 ∧ a2 ∧ … ∧ am

and the clause: ai ← b1 ∧ b2 ∧ … ∧ bp

You can generate the answer clauseyes ← a1 ∧ … ∧ ai-1 ∧ b1 ∧ b2 ∧ … ∧ bp ∧ ai+1 ∧ … ∧ am

i.e., resolving ai with ai ← b1 ∧ b2 ∧ … ∧ bp

Express query as an answer clause (e.g., query a1 ∧ a2 ∧ … ∧ am ) yes ← a1 ∧ a2 ∧ … ∧ am

Rule of inference: ExamplesRule of inference (called SLD Resolution)Given an answer clause of the form:

yes ← a1 ∧ a2 ∧ … ∧ am

and the KB clause: ai ← b1 ∧ b2 ∧ … ∧ bp

You can generate the answer clauseyes ← a1 ∧ … ∧ ai-1 ∧ b1 ∧ b2 ∧ … ∧ bp ∧ ai+1 ∧ … ∧ am

yes ← b ∧ c. b ← k ∧ f.

yes ← e ∧ f. e.

KB clause

-> yes ← k ∧ f ∧ c.

-> yes ← f.

(successful) Derivations• An answer is an answer clause with m = 0. That is,

it is the answer clause yes ← .

• A (successful) derivation of query “?q1 ∧ … ∧ qk “ from KB is a sequence of answer clauses γ0 , γ1 ,…,γn

such that• γ0 is the answer clause yes ← q1 ∧ … ∧ qk

• γi is obtained by resolving γi-1 with a clause in KB, and

• γn is an answer.• An unsuccessful derivation…..

yes ← .

yes ← a b∧ .

4: yes e3: yes c

1: yes e f

5: yes

Example: successful derivation

0: yes a

2: yes e c

1

2

435

a b c. a e f. b f k.

c e. d k e.

f j e. f c. j c.

Query: ?a

Done. The question was “Can we derive a?”

The answer is “Yes, we can”

8

a b c. a e f. b f k.

c e. d k e.

f j e. f c. j c.

Query: ?a

Example: failing derivation

4: yes e k c3: yes c k c

1: yes b c

5: yes k c

0: yes a

2: yes f k c

6: yes k e7: yes k

1 2

3

4 5

There is no rule with k as its head, thus … fail

6 7

9

Example: derivationsk ← e. a ← b ∧ c. b ← k ∧ f.

c ← e. d ← k. e. f ← j ∧ e. f ← c. j ← c.

KB

Sound and Complete?

• When you have derived an answer, you can read a bottom up proof in the opposite direction.

• Every top-down derivation corresponds to a bottom up proof and every bottom up proof has a top-down derivation.

• This equivalence can be used to prove the soundness and completeness of the derivation procedure.

11

Sound

• If KB ⊦TD G there is a top down derivation for G

• Therefore there is a bottom up derivation of G.

• Therefore G is in C and KB |=G (because BU is sound)

If KB ⊦TD G then KB |=G

12

Complete

• If KB |=G then we can find a bottom-up derivation for G.

• Therefore there is a top-down derivation as well, which we can find because the search space is finite.

If KB |=G then KB ⊦TD G

13

SLD resolution as search

• SLD resolution can be seen as a search• from a query stated as an answer clause

• to an answer

• Through the space of all possible answer clauses

15

Course Big PictureEnvironme

ntProblem

Inference

Planning

Deterministic

Stochastic

SearchArc Consistency

Search

Search Value Iteration

Var. Elimination

Constraint Satisfactio

n

Logics

STRIPS

Belief Nets

Vars + Constraint

s

Decision Nets

Markov Processes

Var. Elimination

Static

Sequential

RepresentationReasoningTechnique

SLS

• Constraint Satisfaction (Problems):• State: assignments of values to a subset of the variables• Successor function: assign values to a “free” variable• Goal test: set of constraints• Solution: possible world that satisfies the constraints• Heuristic function: none (all solutions at the same distance from start)

• Planning : • State: full assignment of values to features• Successor function: states reachable by applying valid actions• Goal test: partial assignment of values to features• Solution: a sequence of actions• Heuristic function: relaxed problem! E.g. “ignore delete lists”

• Query (Top-down/SLD resolution)• State: answer clause of the form yes a1 ... ak

• Successor function: state resulting from substituting firstatom a1 with b1 … bm if there is a clause a1 ← b1 … bm

• Goal test: is the answer clause empty (i.e. yes ) ?• Solution: the proof, i.e. the sequence of SLD resolutions• Heuristic function: ?????

Inference as Standard Search

17

Search Graph

Prove: ?← a ∧ d.

a ← b c. ∧ a ← g.a ← h. b ← j.b ← k. d ← m.d ← p. f ← m. f ← p. g ← m.g ← f. k ← m. h ←m. p.

KB

Why are these dead ends in the search space?Why are these dead ends in the search space?

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Search Graph



KB

Why are these dead ends in the search space?

• the successor function resolves the first atom in the body of the answer clause • But j and m cannot be resolved

Why are these dead ends in the search space?

can trace the example in the Deduction Applet at http://aispace.org/deduction/ using file kb-for-top-down-search available in course schedule

19

http://aispace.org/deduction/

Search Graph



KB

Heuristics?

Slide 20

Search Graph


KB

Slide 21

Representation and Reasoning in complex domains

• Expressing knowledge with propositions can be quite limiting

up_s2 up_s3

ok_cb1

ok_cb2

live_w1

connected_w1_w2

up( s2 ) up( s3 ) ok( cb1 ) ok( cb2 ) live( w1)connected( w1 , w2 )

• What we need is a more natural way to consider individuals and their properties

E.g. there is no notion that Now there is a notion that

up_s1 and up_s1 are about the same property

w1 is the same in live_w1

and in connected_w1_w2

w1 is the same in live(w1)and in connected(w1, w2)

up is the same in up(s1)and up(s3)

22

Lecture Overview

• Recap Top Down• TopDown Proofs as search• Datalog

Datalog

• An extension of propositional definite clause (PDC) logic• We now have variables• We now have relationships between variables• We can express knowledge that holds for a set of

individuals, writing more powerful clauses by introducing variables, such as:

• We can ask generic queries, E.g. “which wires are connected to w1?“

live(W) wire(W) connected_to(W,W∧ 1) wire(W∧ 1) live(W∧ 1).

? connected_to(W, w1)24

Datalog: a relational rule language

A variable is a symbol starting with an upper case letter

A constant is a symbol starting with lower-case letter or a sequence of digits.

A predicate symbol is a symbol starting with a lower-case letter.

A term is either a variable or a constant.

Datalog expands the syntax of PDCL….

Examples: X, Y

Examples: alan, w1

Examples: live, connected, part-of, in

Examples: X, Y, alan, w1

25

Datalog Syntax (cont’d)An atom is a symbol of the form p or p(t1 …. tn) where p is a predicate

symbol and ti are terms

A definite clause is either an atom (a fact) or of the form:

h ← b1 … ∧ ∧ bm

where h and the bi are atoms (Read this as ``h if b.'')

A knowledge base is a set of definite clauses

Examples: sunny, in(alan,X)

Example: in(X,Z) ← in(X,Y) part-of(Y,Z) ∧

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Summary of Datalog Syntax

Definite Clause

atom

p p(t1,….tn)

a <- b1 ^…^ bn

where a and b1.. bn are atoms

Datalog Expression

Term Query: ?body

constant variable

27

DataLog Sematics• Role of semantics is still to connect symbols and sentences in the

language with the target domain. Main difference:• need to create correspondence both between terms and

individuals, as well as between predicate symbols and relations

We won’t cover the formal definition of Datalog semantics, but if you are interested see 12.3.1 and 12.3.2 in textbook

28

Lecture Overview

• Recap Lecture 8

• TopDown-Up Proof Procedure

• DataLog• Syntax and semantics• SLD resolution

29

Datalog: Top Down Proof Procedure

• Extension of Top-Down procedure for PDCL. How do we deal with variables? Idea:

- Find a clause with head that matches the query- Substitute variables in the clause with their matching constants

• Example:

• We will not cover the formal details of this process, called unification. See textbook Section 12.4.2, p. 511 for the details.

in(alan, r123).

part_of(r123,cs_building).

in(X,Y) part_of(Z,Y) & in(X,Z).

Query: yes in(alan, cs_building).

yes part_of(Z,cs_building), in(alan, Z).

in(X,Y) part_of(Z,Y) & in(X,Z).with Y = cs_building X = alan

30

Example proof of a Datalog query

in(alan, r123).



Query: yes in(alan, cs_building).

yes part_of(Z,cs_building), in(alan, Z).

yes in(alan, r123).

yes part_of(Z, r123), in(alan, Z).

yes .

Using clause: in(X,Y) part_of(Z,Y) & in(X,Z), with Y = cs_building X = alan

Using clause: part_of(r123,cs_building) with Z = r123

Using clause: in(alan, r123).

Using clause: in(X,Y) part_of(Z,Y) & in(X,Z). With X = alan Y = r123

fail

No clause with matching head: part_of(Z,r123). 31

Tracing Datalog proofs in AIspace

• You can trace the example from the last slide in the AIspace Deduction Applet at http://aispace.org/deduction/ using file in-part-of available in course schedule

32

http://aispace.org/deduction/

Datalog: queries with variables

What would the answer(s) be?

Query: in(alan, X1).

in(alan, r123).



yes(X1) in(alan, X1).

33

Datalog: queries with variables

What would the answer(s) be?

yes(r123).

yes(cs_building).

Query: in(alan, X1).

in(alan, r123).



yes(X1) in(alan, X1).Again, you can trace the SLD derivation for this query in the AIspace Deduction Applet,

34

Learning Goals For Logic• 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 KB• 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 • Top-down proof procedure

- Define/read/write/trace/debug the Top-down (SLD) proof procedure

- Define it as a search problem- Prove that the TD proof procedure is sound and complete

• Datalog- Represent simple domains in Datalog- Apply the Top-down proof procedure in Datalog

35

Logics: Big picture

• We only covered rather simple logics• There are much more powerful representation

and reasoning systems based on logics e.g. full first order logic (with negation, disjunction and

function symbols)second-order logics (predicates over predicates)non-monotonic logics, modal logics, …

• There are many important applications of logic• For example, software agents roaming the web

on our behalfBased on a more structured representation: the

semantic webThis is just one example for how logics are used

36

Logics: Big Picture

Propositional Logics

First-Order Logics

Propositional Definite Clause

Logics

Semantics and Proof Theory

Satisfiability Testing (SAT)

Description Logics

Cognitive Architectures

Video Games

Hardware Verification

Product ConfigurationOntologies

Semantic Web

Information Extraction

Summarization

Production Systems

Tutoring Systems

37

• Examples for typical \Web queries• How much is a typical flight to Mexico for a

given date?• What’s the cheapest vacation package to some

place in the Caribbean in a given week?Plus, the hotel should have a white sandy beach and

scuba diving

• If webpages are based on basic HTML• Humans need to scout for the information and

integrate it• Computers are not reliable enough (yet)

Natural language processing (NLP) can be powerful (see Watson and Siri!)

But some information may be in pictures (beach), or implicit in the text, so existing NLP techniques still don’t get all the info.

Semantic Web: Extracting data

38

More structured representation:

the Semantic Web• Beyond HTML pages only made for humans• Languages and formalisms based on description logics

that allow websites to include rich, explicit information on • relevant concepts, individual and their relationships \

• Goal: software agents that can roam the web and carry out sophisticated tasks on our behalf, based on these richer representations

• Different than searching content for keywords and popularity. • Infer meaning from content based on metadata and assertions

that have already been made. • Automatically classify and integrate information

• For further material, P&M text, Chapter 13. Also• the Introduction to the Semantic Web tutorial given at 2011

Semantic TechnologyConference http://www.w3.org/People/Ivan/CorePresentations/SWTutorial/

39

http://www.w3.org/People/Ivan/CorePresentations/SWTutorial/

http://www.w3.org/People/Ivan/CorePresentations/SWTutorial/

Examples of ontologies for the Semantic Web

“Ontology”: logic-based representation of the world

• eClassOwl: eBusiness ontology • for products and services• 75,000 classes (types of individuals) and 5,500 properties

• National Cancer Institute’s ontology: 58,000 classes

• Open Biomedical Ontologies Foundry: several ontologies• including the Gene Ontology to describe

gene and gene product attributes in any organism or protein sequence

• OpenCyc project: a 150,000-concept ontology including• Top-level ontology

describes general concepts such as numbers, time, space, etc

• Many specific concepts such as “OLED display”, “iPhone”

40

A different example of applications of logic

Cognitive Tutors (http://pact.cs.cmu.edu/)

• computer tutors for a variety of domains (math, geometry, programming, etc.)• Provide individualized support to problem

solving exercises, as good human tutors do

• Rely on logic-based, detailed computational models (ACT-R) of skills and misconceptions underlying a learning domain.

• CanergieLearning (http://www.carnegielearning.com/ ): • a company that commercializes these tutors,

sold to hundreds of thousands of high schools in the USA

41

http://pact.cs.cmu.edu/

http://www.carnegielearning.com/

ACT-R Models for Cognitive Tutors• One of ACT-R main assumptions:

• Cognitive skills (procedural knowledge) are represented as production rules:

IF this situation is TRUE, THEN do X

• ACT-R model representing expertise in a given domain:• set of production rules mimicking how a human would reason

to perform tasks in that domain

• An ACT-R model for a Tutor encodes all the reasoning steps necessary to solve problems in the target domain• Example: rules describing how to solve

5x+3=30 42

ACT-R Models for Intelligent Tutoring Systems

Eq: 5x+3=30 ; Goals: [Solve for x]• Rule: To solve for x when there is only one occurrence, unwrap (isolate) x.

Eq:5x+3=30 ; Goals: [Unwrap x]• Rule: To unwrap ?V, find the outermost wrapper ?W of ?V and remove ?W

Eq: 5x+3=30; Goals: [Find wrapper ?W of x; Remove ?W]• Rule: To find wrapper ?W of ?V, find the top level expression ?E on side of

equation containing ?V, and set ?W to part of ?E that does not contain ?V

Eq: 5x+3=30; Goals: [Remove “+3”]• Rule: To remove “+?E”, subtract “+?E” from both sides

Eq: 5x+3=30; Goals: [Subtract “+3” from both sides]• Rule: To subtract “+?E” from both sides ….

Eq: 5x+3-3=30-3

43

Cognitive Tutors

• Computer tutors that use Act-R models of target domains (e.g. algebra, geometry), in order to• trace student performance by firing rules and do a stepwise

comparison of rule outcome with student action• Mismatches signal incorrect student knowledge that requires

tutoring• These models showed good fit with student performance,

indicating value of the ACT-R theory• Cognitive Tutors are great examples of AI success – used in

thousands of high schools in the USA (http://www.carnegielearning.com/success.cfm)

44

Where are we?Environment

Problem Type

Query

Planning

Deterministic Stochastic

Constraint Satisfaction Search

Arc Consistency

Search

Search

Logics

STRIPS

Vars + Constraints

Value Iteration

Variable

Elimination

Belief Nets

Decision Nets

Markov Processes

Static

Sequential

Representation

ReasoningTechnique

Variable

Elimination

Done with Deterministic Environments 45

Where are we?Environment

Problem Type

Query

Planning



Arc Consistency

Search

Search

Logics

STRIPS

Vars + Constraints

Value Iteration

Variable

Elimination

Belief Nets

Decision Nets

Markov Processes

Static

Sequential

Representation

ReasoningTechnique

Variable

Elimination

Second Part of the Course 46

Where Are We?Environment

Problem Type

Query

Planning



Arc Consistency

Search

Search

Logics

STRIPS

Vars + Constraints

Value Iteration

Variable

Elimination

Belief Nets

Decision Nets

Markov Processes

Static

Sequential

Representation

ReasoningTechnique

Variable

Elimination

We’ll focus on Belief Nets 47

48

Full First-Order Logics (FOLs)We have constant symbols, predicate symbols

and function symbolsSo interpretations are much more complex

(but the same basic idea – one possible configuration of the world)

INFERENCE:• Semidecidable: algorithms exists that says yes

for every entailed formulas, but no algorithm exists that also says no for every non-entailed sentence

• Resolution Procedure can be generalized to FOL

constant symbols => individuals, entities

predicate symbols => relationsfunction symbols => functions

slide 1 logic: td as search, datalog (variables) jim little ubc cs 322 – csp october 24, 2014...

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