“Mr Brown” a simple logic puzzle requiring

common sense

David Mott (ETS, IBM UK)Nov 2014

David Mott (ETS, IBM UK)Nov 2014

International Technology AllianceIn Network & Information Sciences

International Technology AllianceIn Network & Information Sciences

The Puzzle

An Englishman (Mr Salmon), a Welshman (Mr Green), and a Scotsman (Mr Brown) met for lunch one day. One man was wearing a salmon tie, another was wearing a green tie and the third was wearing a brown tie. "Isn't it funny," said Mr Brown to the others, "that not one of us is wearing a tie which matches our name?" "That's true," agreed the man wearing the green tie. Can you now say what colour tie each man was wearing?

Martin Gardner

Basic Domain Knowledge

conceptualise conceptualise a ~ person ~ P.a ~ person ~ P.

conceptualise conceptualise a ~ tie ~ T.a ~ tie ~ T.

conceptualise conceptualise the person P ~ is wearing ~ the tie T.the person P ~ is wearing ~ the tie T.

Can write: the person MrBrown is wearing the tie SalmonTie.

Talking about unidentified entities• Need to talk about things that are just described but not identified:

– “the person wearing the green tie”– “the tie that MrGreen is wearing”

• Must use a descriptor, not a name:– the person “GreenTieWearer”– the tie “MrGreensTie”

• Useful to distinguish between the entities that are real and the entities that are descriptive:

– the real person MrGreen– the actor GreenTieWearer

– the real tie GreenTie– the unknown tie MrGreensTie

• But descriptive things are still subtypes of the basic entity– the unknown tie MrGreensTie is a tie.– the actor GreenTieWearer is a person.

Descriptive Formulation

conceptualise conceptualise a ~ real person ~ P that is a person and a ~ real person ~ P that is a person and is a real thing.is a real thing.

conceptualise conceptualise an ~ actor ~ A that is a person and an ~ actor ~ A that is a person and is a descriptive thing.is a descriptive thing.

conceptualiseconceptualise a ~ real tie ~ T that is a tie and a ~ real tie ~ T that is a tie and is a real thing.is a real thing.

conceptualise conceptualise an ~ unknown tie ~ T that is a tie and an ~ unknown tie ~ T that is a tie and is a descriptive thing.is a descriptive thing.

thing

person tie

real thing

descriptive thing

real person

real tie

actor unknown tie

is wearing

Basic Entities

An Englishman (Mr Salmon), a Welshman (Mr Green), and a Scotsman (Mr Brown) met for lunch one day. One man was wearing a salmon tie, another was wearing a green tie and the third was wearing a brown tie.

there is a real person named MrBrown.there is a real person named MrBrown.there is a real person named MrGreen.there is a real person named MrGreen.there is a real person named MrSalmon.there is a real person named MrSalmon.

there is a real tie named BrownTie.there is a real tie named BrownTie.there is a real tie named GreenTie.there is a real tie named GreenTie.there is a real tie named SalmonTie.there is a real tie named SalmonTie.

the real person MrGreen is wearing the unknown tie MrGreensTie.the real person MrGreen is wearing the unknown tie MrGreensTie.the real person MrBrown is wearing the unknown tie MrBrownsTie.the real person MrBrown is wearing the unknown tie MrBrownsTie.the real person MrSalmon is wearing the unknown tie MrSalmonsTie.the real person MrSalmon is wearing the unknown tie MrSalmonsTie.

the actor GreenTieWearer is wearing the real tie GreenTie.the actor GreenTieWearer is wearing the real tie GreenTie.the actor BrownTieWearer is wearing the real tie BrownTie.the actor BrownTieWearer is wearing the real tie BrownTie.the actor SalmonTieWearer is wearing the real tie SalmonTie.the actor SalmonTieWearer is wearing the real tie SalmonTie.

Tie Colour Constraint

The puzzle has the following utterance by Mr Green:

…not one of us is wearing a tie which matches our name…

the unknown tie MrGreensTie cannot be the real tie GreenTie. the unknown tie MrGreensTie cannot be the real tie GreenTie. the unknown tie MrBrownsTie cannot be the real tie BrownTie. the unknown tie MrBrownsTie cannot be the real tie BrownTie. the unknown tie MrSalmonsTie cannot be the real tie SalmonTie. the unknown tie MrSalmonsTie cannot be the real tie SalmonTie.

the actor GreenTieWearer cannot be the real person MrGreen. the actor GreenTieWearer cannot be the real person MrGreen. the actor BrownTieWearer cannot be the real person MrBrown. the actor BrownTieWearer cannot be the real person MrBrown. the actor SalmonTieWearer cannot be the real person MrSalmon.the actor SalmonTieWearer cannot be the real person MrSalmon.

this is a fairly “pre-interpreted” formulation; we could instead model the colours of the ties and the “colour

names” and use a rule to infer the above

Problem Solving Strategy• Model as constraint-based problem:

– unknown entities (MrGreensTie) have a range of possibities (GreenTie, … )– constraints rule out possibilities:

• the unknown tie MrGreensTie cannot be the real tie GreenTie. – constraints may propagate to other constraints via rules– some facts may assign a possibility to an unknown entity:

• the descriptive thing T is the same as the unknown thing T1– properties of T are propagated to T1 and vice versa

– if only one possibility remains then this must be the correct answer

• For more complex problems could include hypothesise-and-test:– for an unknown entity that has no assignment, make an assumption of a

possible assignment– infer consequences, including propagating constraints– if no possibility remains for an unknown entity, then the facts are inconsistent

and the assumptions that led to the inconsisteny must be removed

Constraint Rules • A specific tie can only be worn by one person[ only_one_person_per_tie ]if ( the person P is wearing the tie T ) and ( the person P1 cannot be the person P ) and ( the person P1 is wearing the tie T1) then ( the tie T1 cannot be the tie T ).

• Each tie description is unique (MrGreensTie and MrBrownsTie must be different)

[ unk_tie_diff ]if ( there is an unknown tie named H1 ) and ( there is an unknown tie named H2 ) and ( the thing H1 # the thing H2 )then ( the thing H1 cannot be the thing H2 ).

The full formulation contains more constraint rules, but these are the only ones used in the

solution

Rules to select single possibility

• If a tie cannot be Green or Brown, then it must be Salmon[ lastTie1 ]if ( the unknown tie T cannot be the real tie GreenTie and cannot be the real tie BrownTie )then ( the unknown tie T is the same as the real tie SalmonTie ).

There are 6 such rules in total (3 for tie, 3 for person)

Final Clue

"Isn't it funny," said Mr Brown to the others, "that not one of us is wearing a tie which matches our name?" "That's true," agreed the man wearing the green tie.

“Common sense” tells us that two people engaged in dialog are not the same, at least in normal situations. Thus Mr Brown and the person wearing the green tie are not the same.

the person MrBrown cannot be the actor GreenTieWearer.

Thus to formulate the problem, common sense

reasoning is required.

Again this is preinterpreted; we could have modelled the dialog explicitly and inferred that the two speakers were

not the same.

Iteration to Solution

• Solution was found automatically, but required two formulations:– the first formulation did not show a solution, and

specifically missed the intuitive inference that:• the person MrBrown cannot be the actor

GreenTieWearer meant that MrBrown cannot wear the green tie

– a new rule was formulated (only_one_person_per_tie)only_one_person_per_tie)– the second formulation automatically found the solution

Solutionthe real person P is wearing the real tie T.

the real person MrBrown is wearing the real tie SalmonTie.the real person MrGreen is wearing the real tie BrownTie.the real person MrSalmon is wearing the real tie GreenTie.

the unknown tie MrGreensTie cannot be the tie GreenTie.

the unknown tie MrGreensTie cannot be the real tie GreenTie.

the unknown tie MrGreensTie cannot be the tie BrownTie.

the unknown tie MrGreensTie cannot be the tie SalmonTie.

the unknown tie MrGreensTie cannot be the real tie SalmonTie.

the unknown tie MrBrownsTie cannot be the tie GreenTie.

the unknown tie MrBrownsTie cannot be the real tie GreenTie.

the unknown tie MrBrownsTie cannot be the tie BrownTie.

the unknown tie MrBrownsTie cannot be the real tie BrownTie.

the unknown tie MrBrownsTie cannot be the tie SalmonTie.

the unknown tie MrSalmonsTie cannot be the tie GreenTie.

the unknown tie MrSalmonsTie cannot be the tie BrownTie.

the unknown tie MrSalmonsTie cannot be the real tie BrownTie.

the unknown tie MrSalmonsTie cannot be the tie SalmonTie.

the unknown tie MrSalmonsTie cannot be the real tie SalmonTie.

“Crossoff grid”

Rationale

the real person MrBrown is wearing the real tie SalmonTie.

Rationale (2)

the real person MrGreen is wearing the real tie BrownTie.

Rationale (3)

the real person MrSalmon is wearing the real tie GreenTie.

Summary of Formulation

• Representation of people and what ties they are wearing

• The use of the “descriptive thing" strategy to handle the fact that some things in a relationship are not known, but must be spoken about.

• Rules and facts to represent constraints and to propagate constraints

• A basic principle of dialog, that people do not speak to themselves, or at least within this puzzle.

To formulate the problem, common sense

reasoning is required.

Meta-data to reduce rulewriting

• By specifying semantic relations on the entities and relations, the user may avoid having to write rules

• Need to construct meta-rules to interpret the semantic relations and make the required inferences

Example

[ only_one_person_per_tie ]if ( the person P is wearing the tie T ) and ( the person P1 cannot be the person P ) and ( the person P1 is wearing the tie T1) then ( the tie T1 cannot be the tie T ).

the relation concept 'is wearing' is an inverse functional relation.the relation concept 'is wearing' is an inverse functional relation.Semantic relationship

Meta-rule

+

=

Original domain rule

[ inverse_functional_relation ]if ( the inverse functional relation FREL has the sequence ( the thing S1 , and the thing O1 ) as relation realisation and has the sequence ( the thing S2 , and the thing O2 ) as relation realisation ) and ( the thing S1 cannot be the thing S2 )then ( the thing O1 cannot be the thing O2 ).

This work is ongoing