Semantics and Logical Form

Reference: Chapter 8 of Allen

Aim:

To describe a language for representing logical forms - that is, intermediate representations on the way to transforming a parse tree into the final meaning representation. Logical forms must be able to encode possible ambiguities of meaning of a particular parse of a sentence.

Keywords: co-agent, compositional semantics, exists, experiencer, failure of substitutivity, FOPC, forall, instrument, logical form, logical operator, modal, MOST1, patient, PLUR, predicate operator, semantics, substitutivity, term, THE, thematic role, theme, victim

Plan:

Definition of compositional semantics

Word senses and ambiguity

Logical form language - terms, predicates, propositions, logical operators, quantifiers, predicate operators, modal operators.
Ambiguity in logical forms
Verbs and states in logical forms - thematics roles
Logical forms for speech acts and for embedded sentences

Semantics

Syntax concerns structure; semantics concerns "meaning".
Semantics is often assumed to be compositional: the meaning of a phrase like three green boxes is constructed from the meanings of the words: three, green, and boxes.
Compositional semantics is violated in idioms like kick the bucket which can mean "die", and he still has all his marbles, meaning he has not become senile (lost his memory).
Similarly, the meaning of a sentence comes from the meanings of its phrases.
Constraints between words or phrases frequently affect meaning:

The pig grunted ...

The pig grunted

at the feminist.

The pig grunted

at the demonstrator.

The pig grunted

at the farmer.

pig = { PIG-ANIMAL, MALE-CHAUVINIST-PIG, POLICEPERSON}

Facets of Meaning

relationships between objects, entities, etc.
referents for phrases like you, him, the red gate - i.e. objects that correspond to the words in the phrases;
common sense inferences from basic facts conveyed
speaker's intent: the door is open could be a bare statement, a complaint, a tacit request to close it, ...

Logical Form

It is possible to consider meaning in the absence of context.
The representation of meaning in the absence of context is termed logical form.
In programming language semantics, meaning is expressed with respect to a model of a computing device. The meaning of a particular PL statement (i.e. command) is expressed by giving the effect of that statement on the model.

Model-based Semantics

Modelling is used in NL semantics, too.
NL semantic models are often related to those developed for predicate logic (or modal, sorted, temporal, epistemic, and higher-order logics).

Word Senses and Ambiguity

Words may have more than one sense.
For example, ball could mean a social event (SOCIAL-BALL), or a spherical toy (TOY-BALL):
Some words, like take, have many different senses (some say 57 senses for take).
If the intended sense is not clear, we have "word sense ambiguity".

Word Senses and Ambiguity 2

At the logical form level, words are quite often ambiguous, as it may take context to decide the intended sense.

Jack ran the race

Is Jack in charge of the race (as in Fred ran the company) or did he participate in it (as in Harry ran from the burning house)?

Word Senses and Ambiguity 3

There are other types of ambiguity:

referential ambiguity: Jon is angry with Jim. He bites him.

structural ambiguity: Happy cats and dogs live on the farm. syntactic structure

Every man loves a woman. semantic structure

there can be mixes: The major exhibits age.

Co-occurrence constraints on meaning: sometimes ambiguity can be resolved by the company the word keeps, or the syntax of the sentence: Jack ran in the park vs Jack ran in the election.

The Basic Logical Form Language

This section defines a formal language of logical forms, resembling FOPC (first order predicate calculus):

terms

constants or expressions that describe objects: FIDO1, JACK1

predicates

constants or expressions that describe relations or properties: BITES1.

Each predicate has an associated number of arguments - BITES1 is binary (unary = 1 argument; ternary = 3 arguments; n-ary = n arguments).

Logical Form Language 2

propositions

a predicate followed by the appropriate number of arguments: (BITES1 FIDO1 JACK1), (DOG1 FIDO1) - Fido is a dog.

More complex propositions can be constructed using logical operators:

(NOT (LOVES1 SUE1 JACK1))
(& (BITES1 FIDO1 JACK1) (DOG1 FIDO1))

Note that and does not always "translate" as logical & - e.g. it may suggest temporal sequence: I went home and had a drink compared to I had a drink and went home.

Logical Form Language Part 3

quantifiers

In FOPC, only forall and exists.
English has vaguer quantifiers, too: most, many, a few, a, the, ...

Variables are introduced here, as in FOPC.

However variables in logical form language persist beyond the "scope" of the quantifier.

A man came in. He went to the table.

The first sentence introduces a new object of type MAN. The He in the second sentence refers to this object.

Quantifiers continued

NL quantifiers are typically restricted in the range of objects that the variable ranges over. In Most dogs bark the variable in the MOST1 quantifier is restricted to DOG1 objects:
(MOST1 d1 : (DOG1 d1) (BARKS1 d1))

the and a give rise to important NL quantifiers - the dog barks has logical form

(THE x : (DOG1 x) (BARKS1 x))

which would be true only if we can determine a unique dog in context, and that dog barks. How we find the unique dog is discussed in the section on Reference.

Logical Form Language Part 4

predicate operator

We also need a way to handle plurals as in the dogs bark.

A new type of thing called a predicate operatoris introduced that takes a predicate as an argument and produces a new predicate.

For plurals, PLUR: if DOG1 is true of any dog, then (PLUR DOG1) is true of any set of dogs with more than one member:

(THE x : ((PLUR DOG1) x) (BARKS1 x)).

Logical Form Language Part 5

modal operator

used for verbs like believe, know, want, for tense, and other purposes. Sue believes Jack is happy becomes

(BELIEVE SUE1 (HAPPY JACK1)).

Modal operators may exhibit failure of substitutivity:

JACK1 may = JOHN22 (i.e. the individual known as Jack may also be called John, e.g. by other people)

However, Sue believes John is happy may not be true, e.g. because Sue may not know that JACK1 = JOHN22.

Logical Form Language Part 6

Tenses: use modals PRES, PAST, FUT:

(PRES (SEES1 JOHN1 FIDO1))
(PAST (SEES1 JOHN1 FIDO1))
(FUT (SEES1 JOHN1 FIDO1))

Again, substitutivity may fail.

JOHN1 may = MINISTER1, and (PAST (OWNS1 JOHN1 FIDO1)) may be true, but (PAST (OWNS1 MINISTER1 FIDO1)) can still be false because John was not minister when he owned Fido.

Converting Logical Forms to a Prolog-type Notation

The logical form language so far presented is based on a Lisp-like notation (following that used in the book by James Allen).
To convert a Lisp list to a Prolog term:
Example:
(OWNS1 o1 (NAME m1 "Mary") (THE p1 PIZZA))
1. take the first item in the list and place it outside (before) the leading parenthesis, and separate the rest of the items with commas.
  
  ⇒ OWNS1(o1, (NAME m1 'Mary'), (THE p1 PIZZA))

Converting Logical Forms to Prolog 2

If there are any lists among the items, recursively convert them to terms, as above.

⇒ OWNS1(o1, NAME(m1, 'Mary'), THE(p1, PIZZA))
Look at the role played by the items in the Lisp-style list.
- If they are constants, change them so that they begin with lower case letters (it may make more sense to down-case the whole word).
- If they are variables, change them so that they begin with upper case letters.
⇒ owns1(O1, name(M1, 'Mary'), the(P1, pizza))

Encoding Ambiguity in the Logical Form

Because ambiguity often cannot be resolved at the logical form level, it must be possible to represent ambiguities in the logical form language.
Sue watched the ball becomes

(THE b1: ({BALL-DANCE BALL-SPHERE} b1) (PAST (WATCH1 SUE1 b1)))

Encoding Ambiguity in the Logical Form 2

Every child loves a pet becomes

(LOVES <EVERY c1 (CHILD1 c1)> <A p1 (PET1 p1)>)
The quantifiers look like terms. This represents the two possibilities:

(EVERY c1 : (CHILD1 c1) (A p1 : (PET1 p1) (LOVES1 c1 p1)))
(A p1 : (PET1 p1) (EVERY c1 : (CHILD1 c1) (LOVES1 c1 p1)))

Abbreviation: <EVERY c1 CHILD1>= (EVERY c1 : (CHILD1 c1))

Encoding Ambiguity in the Logical Form 3

Every child didn't run becomes

(<NOT RUN1> <EVERY c1 CHILD>), encompassing

(NOT (EVERY c1 : (CHILD c1) (RUN1 c1))) and
(EVERY c1 : (CHILD c1) (NOT (RUN1 c1))).

Proper Names and Pronouns

Proper names need to be interpreted in context - John could refer to many different Johns.
This is handled in logical forms via

(NAME <variable> <name>): for example John ran becomes

(<PAST RUN1> (NAME j1 "John")).

Proper Names and Pronouns 2

Similarly for pronouns and related phenomena like here and yesterday, we use

(PRO <variable> <proposition>).

Every man liked him becomes

(<PAST LIKE1> <EVERY m1 MAN1> (PRO m2 (HE1 m2)))
where HE1 is the sense for the words he and him.
As with <EVERY c1 CHILD>, (PRO m2 (HE1 m2)) can be shortened to (PRO m2 HE1) as HE1 is a unary predicate.

Verbs and States in Logical Form

1. John broke it
2. John broke it with the hammer

This seems to say that there are two predicates for broke, one binary and one ternary. Messy.
A suitable solution is to convert the "breaking" into an event and assert suitable properties of that event.

1. (exists e1: (BREAK e1 (NAME j1 "John") (PRO i1 IT1))

2. (exists e1: (& (BREAK e1 (NAME j1 "John") (PRO i1 IT1))

(INSTR e1 <THE h1 HAMMER>)))

Verbs and States in Logical Form 2

Extra modifiers can be added as needed, e.g. for in the hallway
Part of the intuition here is that a breaking has three fundamental roles associated with it - an agent, a theme or victim or object, and an instrument.

Verbs and States in Logical Form 3

It is common to label the objects in the logical form with these roles or cases:

2'. (exists e1: (& (BREAK e1)

(AGENT e1 (NAME j1 "John"))

(THEME e1 (PRO i1 IT1))

(INSTR e1 <THE h1 HAMMER>)))

Verbs and States in Logical Form 4

Because the pattern in 1 and 2 above is common, it is usual to leave out the "(exists e1 :" and the "& (" and write 2', for example, as

2''. (BREAK e1 [AGENT (NAME j1 "John"]

[THEME (PRO i1 IT1)]

[INSTR <THE h1 HAMMER>])

Strictly the BREAK should be <PAST BREAK>.

Verbs and States in Logical Form 5

Similarly, Mary was unhappy could initially be

(<PAST UNHAPPY> (NAME m1 "Mary"))

but how do we handle modifiers like in the meeting?
Answer: Introduce states s and use

(<PAST UNHAPPY> s [EXPERIENCER (NAME m1 "Mary")]

[AT-LOC <THE m2 MEETING>])
Allen tends to switch between various notations as convenient. In particular, the angle brackets < > vs round parentheses ( ) seem to be a bit random.

Thematic Roles

These include AGENT, THEME, and INSTR, already met, and

AT-LOC (sometimes this is specialised by adding ON-LOC, IN-LOC, UNDER-LOC)
FROM-LOC (e.g. from here)
TO-LOC (to the ground)
PATH (along the gorge).
In the domain of ownership, there are

TO-POSS (gave a book to John)
FROM-POSS
AT-POSS(John owns a book).

Thematic Roles 2

In the domain of time, there are

AT-TIME
FROM-TIME
TO-TIME.
In the domain of values, there are

AT-VALUE
FROM-VALUE
TO-VALUE(The temperature reached 43 degrees).

Thematic Roles 3

There are also

EXPERIENCER, for when the subject is not acting (John believed it was raining)
BENEFICIARY (Mary bought a present for her mother)
CO-AGENT (Mary lifted the table with her sister)

Preposition and Thematic Roles

Prepositions cue thematics roles: e.g. with can indicate either INSTRUMENT or CO-AGENT.
When these are mixed, as in ? She ate the ice-cream with a large spoon and her best friend we have a linguistic phenomenon called zeugma
The thematic roles listed above are not the lowest level of meaning: there can, for example, be different sub-types of beneficiary:

[Father is pushing heavy trolley along back of supermarket ...]
Mother: You go and get the ice-cream cones for Daddy.
3-year-old: No! They're for meeeee!

The mother means that "Daddy" will benefit by not having to push the trolley down the isle to the ice-cream cones; the child is thinking about the ultimate benefit of getting to eat the (filled) ice-cream cones.

Speech Acts and Embedded Sentences

Each major sentence type - assertion, yes/no-question, wh-question, command - corresponds to a surface speech act.
We extend the logical form language by wrapping the surface speech act operator around the rest of the logical form: e.g. for the man ate a peach:

(ASSERT (<PAST EAT> e1 [AGENT <THE m1 MAN1>]

[THEME <A p1 PEACH 1>]))

Speech Acts and Embedded Sentences 2

Similarly for Did the man eat a peach?

(Y/N-QUERY (<PAST EAT> e1 [AGENT <THE m1 MAN1>]

[THEME <A p1 PEACH 1>]))
and Eat the peach:

(COMMAND (EAT e1 [THEME <THE p1 PEACH1>]))

Wh-Questions and Beyond

Wh-questions are more complicated, and require a new generalized quantifier WH, with variants for

who <WH p1 PERSON>

what <WH o1 ANYTHING>

which dog <WH d1 DOG1>

and extra quantifiers

HOW-MANY
HOW-MUCH

Wh-Questions and Beyond 2

E.g. What did the man eat?

(WH-QUERY (<PAST EAT> e1 [AGENT <THE m1 MAN1>]

[THEME <WH w1 PHYSOBJ>]))
Finally for embedded sentences like relative clauses:

e.g. The man who ate a peach left.

(ASSERT

(<PAST LEAVE> l1

[AGENT <THE m1 (& (MAN1 m1)

(<PAST EAT> e2

[AGENT m1]

[THEME <A p1 PEACH1>]))]))

Summary: Semantics and Logical Form

We have described a logical form language that includes terms, predicates, propositions, logical operators, quantifiers (including special NL quantifiers such as THE), and shown how this language can be used to represent ambiguous sentences.

CRICOS Provider Code No. 00098G
Copyright (C) Bill Wilson, 2006, except where another source is acknowledged.

referential ambiguity:	Jon is angry with Jim. He bites him.
structural ambiguity:	Happy cats and dogs live on the farm.	syntactic structure
structural ambiguity:	Every man loves a woman.	semantic structure
there can be mixes:	The major exhibits age.

1.	(exists e1:	(BREAK e1 (NAME j1 "John") (PRO i1 IT1))
2.	(exists e1:	(& (BREAK e1 (NAME j1 "John") (PRO i1 IT1))
		(INSTR e1 <THE h1 HAMMER>)))

2'.	(exists e1:	(& (BREAK e1)
		(AGENT e1 (NAME j1 "John"))
		(THEME e1 (PRO i1 IT1))
		(INSTR e1 <THE h1 HAMMER>)))

2''.	(BREAK e1	[AGENT (NAME j1 "John"]
		[THEME (PRO i1 IT1)]
		[INSTR <THE h1 HAMMER>])

(<PAST UNHAPPY> s	[EXPERIENCER (NAME m1 "Mary")]
	[AT-LOC <THE m2 MEETING>])

(ASSERT (<PAST EAT> e1	[AGENT <THE m1 MAN1>]
	[THEME <A p1 PEACH 1>]))

(Y/N-QUERY (<PAST EAT> e1	[AGENT <THE m1 MAN1>]
	[THEME <A p1 PEACH 1>]))

who	<WH p1 PERSON>
what	<WH o1 ANYTHING>
which dog	<WH d1 DOG1>

(WH-QUERY (<PAST EAT> e1	[AGENT <THE m1 MAN1>]
	[THEME <WH w1 PHYSOBJ>]))

(ASSERT
	(<PAST LEAVE> l1
	[AGENT <THE m1 (&	(MAN1 m1)
		(<PAST EAT> e2
		[AGENT m1]
		[THEME <A p1 PEACH1>]))]))

The pig grunted
	at the feminist.

The pig grunted
	at the demonstrator.

The pig grunted
	at the farmer.