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Map Coloring with Logic Two Formulations of Map Coloring

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Map Coloring

Map Coloring with Logic

Map Coloring

- Color the nodes of the graph so that
- no two adjacent nodes are the same color

Map Coloring with Logic

Map Coloring

- Color the nodes of the graph so that
- no two adjacent nodes are the same color

Map Coloring with Logic

r6

r4

r2

r5

r1

Student FormulationPremises:

color(X,C) => region(X)

color(X,C) => hue(C)

adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)

region(X) => Y.color(X,Y)

<ground atoms for adjacent, region, hue>

Query:

TUVXYZ | color(r1,T) ^ color(r2,U) ^ color(r3,V) ^ …

- - This is not the formulation used in the CSP/LP literature.
- - LP engines today do not understand this formulation.

Map Coloring with Logic

r6

r4

r2

r5

r1

Logic Programming Formulation- Axioms

next(red,blue)

next(red,green)

next(red,yellow)

next(blue,red)

next(blue,green)

…

- Query

R1 R2 R3 R4 R5 R6 | next(R1,R2) ^ next(R1,R3) ^ next(R1,R5) ^

next(R1,R6) ^ next(R2,R3) ^ next(R2,R4) ^

next(R2,R5) ^ next(R2,R6) ^ next(R3,R4) ^

next(R3,R6) ^ next(R5,R6)

This is the typical formulation used in the literature.

This form (deductive) is required by LP.

Map Coloring with Logic

Map Coloring with Logic

- Two formulations
- The one students produce in CS157 every year
- The one used in the literature [McCarthy82]
- (1) is more natural (and non-deductive)

(2) is more efficient (and deductive)

- Question for the day:
- How do we translate (1) into (2)?

Map Coloring with Logic

Agenda

- Consistency to Deduction: Reformulate non-deductive problem into deductive problem (a form of closing a theory)
- Database Reformulation: turn output of (1) into the Logic Programming formulation.
- Bilevel Reasoning: speeding up (1)

Map Coloring with Logic

Running Example

- Example: a very simple graph and two colors
- 3 nodes, 2 edges
- red and blue

Problem:

Solutions:

Map Coloring with Logic

Constraint Satisfaction

- Definition (Constraint Satisfaction Problem):

Input:

<V, DV, CV>

V: set of variables v1,v2,…

DV: domain for each variable -- dom(vi)

CV: constraints on values of variables

Output: Assignment of values to variables so that

1. val(vi) in dom(vi)

2. all the constraints are satisfied

Map Coloring with Logic

Map Coloring as a CSP

- Variables (V): nodes
- Domains (DV): same for each variable

for every i, dom(vi) = the set of colors

- Constraints (CV): mathematically, a set of permissible tables, i.e. for every pair of variables x and y, a table for all the allowed values of x and y.

Map Coloring with Logic

Example

V: {R1, R2, R3}

DV:

dom(R1) = {r, b}

dom(R2) = {r, b}

dom(R3) = {r, b}

r1 r2 r3

CV:

Map Coloring with Logic

Example Solution

V: {R1, R2, R3}

DV:

dom(R1) = {r, b}

dom(R2) = {r, b}

dom(R3) = {r, b}

r1 r2 r3

CV:

Map Coloring with Logic

All Solutions

V: {R1, R2, R3}

DV:

dom(R1) = {r, b}

dom(R2) = {r, b}

dom(R3) = {r, b}

r1 r2 r3

CV:

Map Coloring with Logic

All Solutions as a DB Query

- The set of all solutions can also be computed using the natural join operator.

r1 r2 r3

=

Map Coloring with Logic

Constraint Representations

- Representing constraints as permissible tables can be expensive and cumbersome.
- Often, constraints are more naturally and economically represented as logical sentences.

Map Coloring with Logic

Student Formulation

Premises:

color(X,C) => region(X)

color(X,C) => hue(C)

adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)

region(X) => Y.color(X,Y)

adjacent(r1,r2) ^ adjacent(r2,r3) ^ hue(r) ^ hue(b)

region(r1) ^ region(r2) ^ region(r3)

Entailment Query:

XYZ | color(r1,X) ^ color(r2,Y) ^ color(r3,Z)

Models:

Map Coloring with Logic

Student Formulation

Premises:

color(X,C) => region(X)

color(X,C) => hue(C)

adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)

region(X) => Y.color(X,Y)

adjacent(r1,r2) ^ adjacent(r2,r3) ^ hue(r) ^ hue(b)

region(r1) ^ region(r2) ^ region(r3)

Build a model:

XYZ | color(r1,X) ^ color(r2,Y) ^ color(r3,Z)

Models:

Map Coloring with Logic

Logic Programming Formulation

- Is different than the student formulation
- [McCarthy82] borrowed from Pereira and Porto [1980]
- McCarthy started with their formulation and showed various reformulations that improved efficiency.
- Served as an exploration of Kowalski’s doctrine:

Algorithm = Logic + Control

- Originally used to illustrate issues in Constraint Satisfaction/Logic Programming

Map Coloring with Logic

LP Formulation

- Axioms

next(r,b)

next(b,r)

- Entailment query

R1 R2 R3 | next(R1,R2) ^ next(R2,R3)

r1 r2 r3

Deductive

Map Coloring with Logic

Problem Statement

- Produce an algorithm that converts

Axioms:

color(X,C) => region(X)

color(X,C) => hue(C)

adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)

region(X) => Y.color(X,Y)

…

Build Model: XYZ |color(r1,X) ^ color(r2,Y) ^ color(r3,Z)

into

Axioms:

next(red,blue)

next(blue,red)

Entailment Query: XYZ |next(X,Y) ^ next(Y,Z)

Map Coloring with Logic

Herbrand Logic

- Assume no function constants
- The axioms include a
- DCA over all ground terms
- UNA over all ground terms
- Assume these hold implicitly
- Finite Herbrand Logic

Map Coloring with Logic

An Observation: Student

color(X,C) => region(X)

color(X,C) => hue(C)

adjacent(X,Y) ^ color(X,C) => -color(Y,C)

- Write the last with V instead of =>

-adjacent(X,Y) V -color(X,C) V -color(Y,C)

- Except for typing, these constraints are entirely negative
- Entail what colorings are invalid.
- Every other coloring is valid.

Map Coloring with Logic

Student Colorings

adjacent(X,Y) ^ color(X,C) => -color(Y,C)

- Invalid answers (entailed)

XYZ | -(color(r1,X) ^ color(r2,Y) ^ color(r3,Z))

- Remaining answers

…

Map Coloring with Logic

Idea

- To reformulate from Student to LP,
- Compute all invalid colorings (which are entailed).
- Compute the complement of that set.

=

…

Map Coloring with Logic

Implementation

: the set of Student constraints

Query: color(r1,X) ^ color(r2,Y) ^ color(r3,Z)

- Reify the notion of an invalid coloring.

(*) invalid(X,Y,Z) <=>

-(color(r1,X) ^ color(r2,Y) ^ color(r3,Z))

- Ask for all the invalid colorings that are logically entailed.

XYZ | ^ (*) |= invalid(X,Y,Z)

Map Coloring with Logic

Implementation

XYZ | invalid(X,Y,Z)

- The following expression is logically entailed and represents the set of all invalid colorings:

X=r1 v X=r2 v X=r3 v

Y=r1 v Y=r2 v Y=r3 v

Z=r1 v Z=r2 v Z=r3 v

X=Y v Y=Z

X Y Z

Map Coloring with Logic

Implementation

- To compute an expression that represents the complement of that set, it turns out that all we need to do is negate that expression.

X≠r1 ^ X≠r2 ^ X≠r3 ^

Y≠r1 ^ Y≠r2 ^ Y≠r3 ^

Z≠r1 ^ Z≠r2 ^ Z≠r3 ^

X≠Y ^ Y≠Z

- The above (with unique names axioms) entails all the valid colorings.

X Y Z

Map Coloring with Logic

Progress: Both are deductive

- Student after reformulation: all deductive answers to

XYZ | X≠r1 ^ X≠r2 ^ X≠r3 ^

Y≠r1 ^ Y≠r2 ^ Y≠r3 ^

Z≠r1 ^ Z≠r2 ^ Z≠r3 ^

X≠Z ^ Y≠Z

UNA[r,b,r1,r2,r3]

- LP formulation: all deductive answers to

XYZ | next(X,Y) ^ next(Y,Z)

next(r,b)

next(b,r)

X Y Z

Map Coloring with Logic

Because Student* is now a query over a complete theory (≠), it can be viewed as a DB query.

XYZ |

X≠r1 ^ X≠r2 ^ X≠r3 ^

Y≠r1 ^ Y≠r2 ^ Y≠r3 ^

Z≠r1 ^ Z≠r2 ^ Z≠r3 ^

X≠Z ^ Y≠Z

Student* as Database Query≠

Map Coloring with Logic

Database Reformulation

- Given one database query, find an equivalent query that is more efficient.
- Includes performing transformations on the data itself, with a space limit.
- Theorem [Chirkova]: There are infinitely many distinct transformations.
- Theorem [Chirkova]: There is an algorithm that given the query constructs a finite space of queries that is guaranteed to include the optimal one.

Map Coloring with Logic

Back to Map Coloring

- Given the query

X≠r1 ^ X≠r2 ^ X≠r3 ^ Y≠r1 ^ … Z≠r3 ^ X≠Y ^ Y≠Z

one of the transformations Chirkova’s work considers is

view(X,Y) <= X≠r1 ^ X≠r2 ^ X≠r3 ^

Y≠r1 ^ Y≠r2 ^ Y≠r3 ^

X≠Y

which can be used to rewrite the above query as

view(X,Y) ^ view(Y,Z)

Map Coloring with Logic

Comparison

- Student**

XYZ | view(X,Y) ^ view(Y,Z)

view(r,b)

view(b,r)

- Logic Programming

XYZ | next(X,Y) ^ next(Y,Z)

next(r,b)

next(b,r)

Map Coloring with Logic

Overview

- Starting with the Student formulation,

XYZ | color(r1,X) ^ color(r2,Y) ^ color(r3,Z)

color(X,C) => region(X)

color(X,C) => hue(C)

adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)

use logical entailment to find an expression for all the invalid colorings and negate it.

XYZ | X≠r1 ^ X≠r2 ^ X≠r3 ^ … ^ X≠Y ^ Y≠Z

- Then apply Chirkova’s work to produce the LP formulation, or one that is at least as good as LP.

XYZ | view(X,Y) ^ view(Y,Z)

Map Coloring with Logic

Another Example

- N-Queens: place N queens on an NxN chessboard s.t.
- No queen can be attacked by another queen.

That is

- No two queens share the same row
- No two queens share the same column
- No two queens share the same diagonal

Map Coloring with Logic

8 Queens Solution

Map Coloring with Logic

N-Queens Constraints

- All queens in different rows

X≠Y ^ row(X,R) => -row(Y,R)

- All queens in different columns

X≠Y ^ col(X,C) => -col(Y,C)

- All queens in different diagonals

X≠Y ^ diag(X,D1,D2)

=> -diag(Y,D1,D2)

Map Coloring with Logic

N-Queens Diagonals

- Row, Col is enough to compute the two diagonals

diag(X,pos,D) <=

row(X,R) ^ col(X,C) ^

sub(C, R, D)

diag(X,neg,D) <=

row(X,R) ^ col(X,C) ^

add(C, R, D)

- Diagonals represented by their slope and y-intercepts

4

3

2

1

0

Map Coloring with Logic

4 Queens Query

- Satisfiability Query:

(X1 Y1 X2 Y2 X3 Y3 X4 Y4) |

row(q1,X1) ^ col(q1,Y1) ^

row(q2,X2) ^ col(q2,Y2) ^

row(q3,X3) ^ col(q3,Y3) ^

row(q4,X4) ^ col(q4,Y4)

Map Coloring with Logic

Efficiency in N-Queens

- The expression that represents all invalid positions can become quite large and might take a long time to compute because of the add and sub facts.
- About 150 conjuncts
- Idea: produce an expression in terms of add and sub, as well as =.

Map Coloring with Logic

Bilevel Reasoning

- Separate constraints from data.

- X≠Y ^ row(X,R) => -row(Y,R)
- X≠Y ^ col(X,C) => -col(Y,C)
- X≠Y ^ diag(X,D1,D2) => -diag(Y,D1,D2)
- diag(X,pos,D) <= row(X,R) ^ col(X,C) ^ sub(C, R, D)
- diag(X,neg,D) <= row(X,R) ^ col(X,C) ^ add(C, R, D)
- add(1,1,2)
- add(1,1,3)
- ...
- add(4,4,8)
- sub(1,1,0)
- sub(1,2,-1)
- ...
- sub(4,4,0)

Map Coloring with Logic

Bilevel Reasoning

- Separate constraints from data.

- X≠Y ^ row(X,R) => -row(Y,R)
- X≠Y ^ col(X,C) => -col(Y,C)
- X≠Y ^ diag(X,D1,D2) => -diag(Y,D1,D2)
- diag(X,pos,D) <= row(X,R) ^ col(X,C) ^ sub(C, R, D)
- diag(X,neg,D) <= row(X,R) ^ col(X,C) ^ add(C, R, D)
- add(1,1,2)
- add(1,1,3)
- ...
- add(4,4,8)
- sub(1,1,0)
- sub(1,2,-1)
- ...
- sub(4,4,0)

Map Coloring with Logic

Bilevel Reasoning

- Separate constraints from data.

- X≠Y ^ row(X,R) => -row(Y,R)
- X≠Y ^ col(X,C) => -col(Y,C)
- X≠Y ^ diag(X,D1,D2) => -diag(Y,D1,D2)
- diag(X,pos,D) <= row(X,R) ^ col(X,C) ^ sub(C, R, D)
- diag(X,neg,D) <= row(X,R) ^ col(X,C) ^ add(C, R, D)

- add(1,1,2)
- add(1,1,3)
- ...
- add(4,4,8)
- sub(1,1,0)
- sub(1,2,-1)
- ...
- sub(4,4,0)

Map Coloring with Logic

Implementation

’: the set of n-queens constraints w/o add/sub

Query: row(q1,X1) ^ col(q2,Y1) ^ row(q3,X2) ^ ...

- Reify the notion of an invalid coloring.

(*) invalid(X1,Y1,X2,Y2,X3,Y3,X4,Y4) <=>

-(row(q1,X1) ^ col(q2,Y1) ^ row(q3,X2) ^ ...)

- Ask for all the invalid colorings where add/sub can be assumed, i.e. use residues (a form of abduction).

XYZ | ’ ^ (*) |= invalid(X,Y,Z)

Map Coloring with Logic

4-Queens Partial Reformulation

goal(X1,Y1,X2,Y2,X3,Y3,X4,Y4) <=

X1 ≠ X2

Y1 ≠ Y2

(-sub(Y2,X2,Z) v -sub(Y1,X1,Z))

(-add(Y2,X2,Z)) v -add(Y1,X1,Z))

X1 ≠ X3

Y1 ≠ Y3

X2 ≠ X3

Y2 ≠ Y3

(-sub(Y3,X3,Z) v -sub(Y1,X1,Z))

(-sub(Y3,X3,Z) v -sub(Y2,X2,Z))

(-add(Y3,X3,Z) v -add(Y1,X1,Z))

(-add(Y3,X3,Z) v -add(Y2,X2,Z))

...

Map Coloring with Logic

Consistency to Deduction

- Computing the sentences that are consistent with the premises and making them deductive consequences.
- Definition (query consistency to deduction):

qC2D[, (xbar)] =

{tbar | (tbar) is satisfiable}

- In map coloring,

= the set of map coloring axioms

(xbar) = color(r1,X) ^ color(r2,Y) ^ color(r3,Z)

qC2D = {<r,b,r>, <b,r,b>}

Map Coloring with Logic

Descriptions

- Definition (Description): e(xbar) is a description for the set of n-tuples S, where |xbar| = n if and only if

|= e(tbar) if tbar S

|= -e(tbar) if tbar S

- Example: X=Y V Y=Z describes {rrr,rrb,brr,bbr,rbb,bbb} in FHL

|=H r=r V r=r |=H r=r V r=b |=H b=r V r=r

|=H b=b V b=r |=H r=b V b=b |=H b=b V b=b

and

|=H -(r=b V b=r)

|=H -(b=r V r=b)

Map Coloring with Logic

Algorithm

- Input

: a finite set of FHL sentences

(xbar): a query in the vocabulary of Delta

- Using algorithm Alg, compute an expression e(xbar) in terms of equality such that for any model M in R,

|=M e(tbar) if and only if |= -(tbar)

- Output -e(xbar)

Map Coloring with Logic

Theorems

- Theorem (Soundness and Completeness):

Let Alg(, (xbar)) = -e(xbar).

|=M -e(tbar) if and only if (tbar) qC2D[, (xbar)]

Proof: Nontrivial

- Theorem (Decidability): qC2D[, (xbar)] for FHL is decidable.

Proof: Walk over Herbrand models and if |=M , collect all tbar s.t. |=M (tbar)

Map Coloring with Logic

Relationship to CWA

- CWA[] includes -p(tbar) whenever |# p(tbar)
- CWA[] is inconsistent whenever
- Delta entails some disjunction d1 v … v dn and
- Delta entails none of the di.
- Adding the map coloring answers to the Student formulation is inconsistent.

Map Coloring with Logic

Relationship to CWA

- Student axioms entail a disjunctive answer:

color(r1,b) ^ color(r2,r) ^ color(r3,b)

v

color(r1,r) ^ color(r2,b) ^ color(r3,r)

- but they entail neither disjunct alone.
- Adding both answers to the axioms produces an inconsistency

color(r1,b) (by the first answer) and

color(r2,b) (by the second answer)

Map Coloring with Logic

Relationship to CWA

- CWA: for every ground p, add -p if p is not entailed.
- C2D: for every ground p, add p if -p is not entailed.

Map Coloring with Logic

Map Coloring with Logic

[McCarthy82] McCarthy. Coloring Maps and the Kowalski Doctrine. Stanford Tech Report 1982.

[Pereira80] Pereira and Porto. Selective Backtracking for Logic Programs. CADE 1980.

[Chirkova2002] Chirkova. Automated Database Restructuring. Stanford thesis 2002.

Map Coloring with Logic

Scrap starts here

Map Coloring with Logic

Bilevel Reasoning

- Separate constraints from data.

- color(X,C) => region(X)
- color(X,C) => hue(C)
- adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)
- adjacent(r1,r2)
- adjacent(r2,r3)
- hue(r)
- hue(b)
- region(r1)
- region(r2)
- region(r3)

Map Coloring with Logic

Bilevel Reasoning

- Separate constraints from data.

- color(X,C) => region(X)
- color(X,C) => hue(C)
- adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)
- adjacent(r1,r2)
- adjacent(r2,r3)
- hue(r)
- hue(b)
- region(r1)
- region(r2)
- region(r3)

Map Coloring with Logic

- color(X,C) => hue(C)
- adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)

- adjacent(r1,r2)
- adjacent(r2,r3)
- hue(r)
- hue(b)
- region(r1)
- region(r2)
- region(r3)

- Separate constraints from data.

Map Coloring with Logic

- hue(b)

- It turns out that separating hue off from the rest of the axioms is an interesting case.

- color(X,C) => region(X)
- color(X,C) => hue(C)
- adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)
- adjacent(r1,r2)
- adjacent(r2,r3)
- region(r1)
- region(r2)
- region(r3)

Map Coloring with Logic

Partial Reformulation

- The algorithm presented earlier started with axioms talking about hues, regions, adjacency, and color

adjacent(X,Y) ^ color(X,C) => -color(Y,C)

to produce

XYZ | X≠r1 ^ X≠r2 ^ X≠r3 ^

Y≠r1 ^ Y≠r2 ^ Y≠r3 ^

Z≠r1 ^ Z≠r2 ^ Z≠r3 ^

X≠Z ^ Y≠Z

but if we separate hue we should get the following result

XYZ | hue(X) ^ hue(Y) ^ hue(Z) ^ X≠Y ^ Y≠Z

Map Coloring with Logic

Implementation

- Epilog (the Logic Group’s theorem prover) produces an expression that represents the set of all invalid colorings, in terms of hue:

XYZ | invalid(X,Y,Z), wrt hue

=> -hue(X) v -hue(Y) v -hue(Z) v X=Y v Y=Z

- Negating (just as before) gives us

hue(X) ^ hue(Y) ^ hue(Z) ^ X≠Y ^ Y≠Z

Map Coloring with Logic

Database Reformulation

- Given: a schema R and a set of queries Q={Q1,…,Qn}
- Compute a set of views on R that minimize the cost of evaluating all the Qi.

(Each relation in R is assigned a size.)

- Rada Chirkova [Chirkova2002]: conjunctive queries.
- Decidable.
- Triply exponential upper bound (in Q).
- Exponential lower bound (in Q).

Map Coloring with Logic

Database Reformulation

Find views V on R producing schema T such that

Eval(Q,T) < Eval(Q,R)

Viewset V

R

T

Map Coloring with Logic

Example

- Schema: parent(x,y)
- Query: XY | samefamily(X,Y)

samefamily(X,Y) <= anc(X,Z) ^ anc(Y,Z)

anc(X,Y) <= parent(X,Y)

anc(X,Y) <= anc(X,Z) ^ anc(Z,Y)

- Can be evaluated more efficiently as:

samefamily(X,Y) <=

foundingfather(X,F) ^ foundingfather(Y,F)

Map Coloring with Logic

Map Coloring in Math

- 1852: Guthrie makes Four Colour Conjecture:

Any planar graph can be colored using only four colors.

- Definition(Planar graph): Graph that can be drawn on the plane without edges crossing.
- Incorrect proofs given by Kempe (1879) and Tait (1880). Kempe’s was accepted for over a decade.
- 1977: Appel and Haken wrote a program to exhaustively check a large number of cases. They (debatably) proved the conjecture.

Map Coloring with Logic

Map Coloring in AI

- Used to illustrate Constraint Satisfaction techniques, e.g. Russell and Norvig
- Trying to color a particular graph
- Not trying to show every graph with particular characteristics can be colored.

Map Coloring with Logic

…

Student Formulation (2)Premises:

color(X,C) => region(X)

color(X,C) => hue(C)

adjacent(X1,X2) ^ color(X1,C) => -color(X2,C)

adjacent(r1,r2) ^ adjacent(r2,r3) ^ hue(r) ^ hue(b)

region(r1) ^ region(r2) ^ region(r3)

Build a model:

Map Coloring with Logic

DB Reformulation Again

- Given the query

hue(X) ^ hue(Y) ^ hue(Z) ^ X≠Y ^ Y≠Z

one of the views Rada’s work considers is

view(X,Y) <= hue(X) ^ hue(Y) ^ X≠Y

which can be used to rewrite the above query as

view(X,Y) ^ view(Y,Z)

Map Coloring with Logic

Back to Map coloring

- In Student*, the schema consists of a single relation: ≠.
- The query

XYZ | X≠r1 ^ X≠r2 ^ X≠r3 ^

Y≠r1 ^ Y≠r2 ^ Y≠r3 ^

Z≠r1 ^ Z≠r2 ^ Z≠r3 ^

X≠Z ^ Y≠Z

- Unfortunately, this query includes constants, which Rada’s work does not cover.
- Notice that this query is defined in terms of -hue.

X Y Z

Map Coloring with Logic

Efficiency

- It is unsettling and inefficient to have the answer defined in terms of -hue when it is more natural to define it in terms of hue.

hue(X) ^

hue(Y) ^

hue(Z) ^

X ≠ Y ^

Y ≠ Z

- Independent of which colors are available
- No constants

X Y Z

Map Coloring with Logic

A Technical Note about Descriptions

- e(X,Y,Z) <=> X=Y V Y=Z

describes S = {rrr,rrb,brr,bbr,rbb,bbb}

For every tuple tbar in S, |=H e(tbar)

For every tuple tbar not in S, |=H -e(tbar)

- In Herbrand logic, suppose we know that
- |=H e(tbar) implies tbar in S
- e(xbar) is built on a single relation constant: =.

Then e(xbar) is a description of S.

Map Coloring with Logic

Issues

- Need efficient algorithm for computing all solutions in FHL that terminates.
- Would settle for termination on clausal form.
- OSHL[Zhu,Plaisted]
- Model Evolution[Baumgartner]
- Computing the complement via negation worked well for map coloring, but in general it may be expensive when converting to clausal form (CNF to DNF).

Map Coloring with Logic

An Observation about McCarthy

XYZ | next(X,Y) ^ next(Y,Z)

- The valid colorings are entailed

Map Coloring with Logic

Complexity Analysis

- CS157

hue(X) ^ hue(Y) ^ hue(Z) ^ X≠Y ^ Y≠Z

hue(red) red ≠ blue

hue(blue) blue ≠ red

- McCarthy

next(X,Y) ^ next(Y,Z)

next(red,blue)

next(blue,red)

- Compare on trees. C=# of colors, N=# of nodes

Map Coloring with Logic

Asymptotic Complexity Results for Trees

- hue(X) ^ hue(Y) ^ hue(Z) ^ X≠Y ^ Y≠Z
- O(CN)
- hue(X) ^ hue(Y) ^ X≠Y ^ hue(Z) ^ Y≠Z
- O(CN-1)
- next(X,Y) ^ next(Y,Z)
- O(CN-2)
- Same as 3 but indexed on first argument of next.
- O(CN-3)

Map Coloring with Logic

Back to FOL

- If we are using first-order deduction procedures to answer finite Herbrand entailment queries, need a DCA and UNA.
- [Reiter80] showed that for * queries, DCA is unnecessary.

Map Coloring with Logic

Looks can be deceiving

- Suppose there were object constants besides red and blue, e.g. paperclip
- New axioms:

-r1(X) V -r2(X)

-r3(X) V -r2(X)

r1(X) V r2(X) V r3(X) => hue(X)

hue(red)

hue(blue)

-hue(paperclip)

Map Coloring with Logic

Running the algorithm:

- After conversion:

goal(X,Y,Z) <=>

X ≠ paperclip ^

X ≠ Y ^

Y ≠ paperclip ^

Y ≠ Z ^

Z ≠ paperclip

- Linear in the size of -hue(x)
- Not the same as the McCarthy formulation

Map Coloring with Logic

Solving CSPs as DB queries

- Can every CSP be formulated as a DB query?
- Map coloring can be formulated as a DB query.
- When is it beneficial to solve a CSP by translating it into a DB query?
- Under some conditions, it is beneficial for map coloring.
- Can the transformation be automated?

Map Coloring with Logic

Language of Study

- Option 1 (Mathematics): Explore algorithms of the following sort.
- Input: <V, DV, CV>, i.e. a set, a set of functions, and a set of tables.
- Output: <D, (x1,…,xn)>, i.e. a database and a query over that database.
- Option 2 (Logic): Explore algorithms of the following sort.
- Input: Description of a CSP in logic.
- Output: Description of the database and the query in logic.

Map Coloring with Logic

CS157 Orderings

- hue(X) ^ hue(Y) ^ hue(Z) ^ X≠Y ^ Y≠Z
- Walk hue space and check each point.
- X≠Y ^ Y≠Z ^ hue(X) ^ hue(Y) ^ hue(Z)
- Walk ≠ space and check each point.
- hue(X) ^ hue(Y) ^ X≠Y ^ hue(Z) ^ Y≠Z
- Walk hue space, pruning as we go.
- X≠Y ^ hue(X) ^ hue(Y) ^ Y≠Z ^ hue(Z)
- Walk ≠ space, pruning as we go.

Map Coloring with Logic

On the Complexity of Query Evaluation

- Complexity depends heavily on indexing.
- Four options
- Relational: index only on the relation of facts
- Relational + attach : eval ≠ in code
- Arguments: index on reln and each argument position
- Arguments + attach

Map Coloring with Logic

Some Connections

- Some connections to other work
- CWA
- Applicability to Datalog queries

Map Coloring with Logic

Closed World Assumption

- qC2D and CWA are closely related
- In fact, a generalization of qC2D and CWA are nearly equivalent.
- If we could compute CWA efficiently, that might help us compute qC2D efficiently.

Map Coloring with Logic

A Property of qC2D

- Consider adding our solutions to the map coloring axioms.

-r1(X) <= r2(X)

-r2(X) <= r3(X)

r1(r) ^ r2(b) ^ r3(r)

r1(b) ^ r2(r) ^ r3(b)

- These axioms are inconsistent.
- Very similar to the inconsistency that can be produced by Closed World Assumption.

Map Coloring with Logic

Closed World

- Definition (Set-based CWA): Let P={p1,…,pn} be a set of closed sentences.

sCWA[, P] = {-pi | |# pi}

- Example:

= {q(a), q(b)}

P = {q(a), q(c)}

sCWA[, P] = {-q(c)}

- sCWA generalizes the standard definition for CWA

Map Coloring with Logic

sC2D

- For comparison to sCWA, we define sC2D.
- Definition (sC2D): Let P={p1,…,pn} be a set of sentences.

sC2D[, P] = {pi | pi is satisfiable}

- Theorem: Let {p1,…,pn} be a set of ground atoms.

sCWA[, {p1,…,pn}] = sC2D[, {-p1,…,-pn}]

Map Coloring with Logic

An Application

- Suppose we have some set of Datalog sentences and a negative query.

:- not r(X), not s(Y), not p(Z)

- One approach:

goal(X,Y,Z) :- univ(X), univ(Y), univ(Z),

not r(X), not s(Y), not p(Z)

- Cost: |univ|3
- If we had an explicit definition for the negative theory, the cost might be substantially less.

Map Coloring with Logic

An Application (2)

Data:

p(a,b) p(b,a) p(c,a) p(d,a)

p(a,c) p(b,c) p(c,b) p(d,b)

p(a,d) p(b,d) p(c,d) p(d,c)

Query: univ(X), univ(Y), not p(X,Y)

Cost: |univ|2

Query: univ(X), not p(X,X)

Cost: |univ|

Map Coloring with Logic

In Map Coloring

Expression for invalid colors

X=Y V Y=Z

Negation of that expression

X#Y ^ Y#Z

Final reformulation

goal(X,Y,Z) <= X#Y ^ Y#Z

More generally

Expression for entailed negations of query

(xbar)

Negation of that expression

-(xbar)

Final reformulation

goal(xbar) <= -(xbar)

Computing the ComplementMap Coloring with Logic

Questions

- Every relation in a theory with a DCA can be defined entirely in terms of equality.
- Question: When can we use fullfinds to compute that definition? (logically justified)
- Question: When should we use fullfinds to compute that definition? (computability/complexity justified)

Map Coloring with Logic

Consistency

- No coloring of the map is entailed by the axioms.
- But both colorings are consistent with the axioms.
- How do we compute consistency?
- With a DCA, consistency is decidable.
- How do we compute consistency efficiently?
- Sort of sounds like abduction.

Map Coloring with Logic

Abduction

- Definition (Abduction): explains |= in the language L if and only if
- |=
- is consistent
- is in L
- Let be the map coloring axioms
- r1(r) ^ r2(b) ^ r3(r) explains

|= XYZ. r1(X) ^ r2(Y) ^ r3(Z)

Map Coloring with Logic

Attempt #3

- Residue attempts to compute abductive consequences

-r1(X) <= r2(X)

-r2(X) <= r3(X)

goal(X,Y,Z) <= r1(X) ^ r2(Y) ^ r3(Z)

? (fullresidue \'(goal ?x ?y ?z) \'mapc)

Call: (GOAL ?X ?Y ?Z)

| Call: (R1 ?X)

| Fail: (R1 ?X)

Fail: (GOAL ?X ?Y ?Z)

- Equality does not help.

Map Coloring with Logic

An observation

- The constraints only talk about the negative part of the theory.

-r1(X) <= r2(X) or -r1(X) | -r2(X)

-r2(X) <= r3(X) or -r2(X) | -r3(X)

- The query only talks about the positive part of the theory.

XYZ.r1(X) ^ r2(Y) ^ r3(Z)

- If we could define the negative part of the theory and then negate it, we would have an explicit definition for the positive theory, on which we could use entailment tools.

Map Coloring with Logic

Reformulation and Completion

- Temporarily add the goal to the original axioms.

-r1(X) <= r2(X)

-r2(X) <= r3(X)

goal(X,Y,Z) <=> r1(X) ^ r2(Y) ^ r3(Z)

- Capture the negative query in terms of equality.

-goal(X,Y,Z) <= X=Y | Y=Z

- Apply predicate completion

-goal(X,Y,Z) <=> X=Y | Y=Z

goal(X,Y,Z) <=> X#Y ^ Y#Z

- Assuming UNAxioms, ask for goal(X,Y,Z).

Map Coloring with Logic

Step 4

goal(X,Y,Z) <= X#Y ^ Y#Z

r # b

b # r

? (fullfindx \'(?x ?y ?z) \'(goal ?x ?y ?z) \'mapcc)

Call: (GOAL ?X ?Y ?Z)

| Call: (NOT (= ?X ?Y))

| Exit: (NOT (= R B))

| Call: (NOT (= B ?Z))

| Exit: (NOT (= B R))

Exit: (GOAL R B R)

Map Coloring with Logic

Step 2

-r1(X) <= r2(X)

-r2(X) <= r3(X)

goal(X,Y,Z) <= r1(X) ^ r2(Y) ^ r3(Z)

-goal(X,Y,Z) <= -r1(X)

-goal(X,Y,Z) <= -r2(X)

-goal(X,Y,Z) <= -r3(X)

r1(X) <= goal(X,Y,Z)

r2(Y) <= goal(X,Y,Z)

r3(Z) <= goal(X,Y,Z)

? (fullfinds \'(?x ?y ?z) \'(not (goal ?x ?y ?z)) \'mapc)

((?u ?u ?w) (?u ?w ?w))

Map Coloring with Logic

Back to the CSP Version

V: {v1,v2,v3}

DV: {{r, b},{r, b},{r, b}}

CV:

goal(X,Y,Z) <= X#Y ^ Y#Z

r # b

b # r

Map Coloring with Logic

CS157 Formulation (3)

Premises: Models:

X=red | X=blue

-r1(X) <= r2(X)

-r2(X) <= r3(X)

X.r1(X)

X.r2(X)

X.r3(X)

Satisfaction Query: XYZ |r1(X) ^ r2(Y) ^ r3(Z)

- All resolution-style t.p. return something like

In general, adding these sentences

may cause inconsistency.

k1, k2, k3

Map Coloring with Logic

Pereira, Porto, and McCarthy

- Peireira and Porto [1980] treated map coloring in the context of Logic Programming

goal <= next(V1,V2), next(V1,V4), next(V1,V5), next(V2,V3), next(V3,V4), next(V3,V5), next(V3, V7), next(V4,V5), next(V5,V6), next(V6,V7)

next(r,y) next(y,b) next(b,y)

next(r,b) next(y,r) next(b,r)

Map Coloring with Logic

Why does this work?

- Reformulation (Step 2) rewrites the negation of the query in terms of just equality.
- For larger map colorings, there will be O(n2) answers to -goal(X,Y,Z), where n = # of regions.
- At an abstract level, it enumerates all the cases where

-goal(X,Y,Z) is directly entailed

- Everything else is thrown into goal.

Map Coloring with Logic

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