je rey heinz (instructor) jon rawski (ta) · computational phonology - class 2 je rey heinz...

Computational Phonology - Class 2

Jeffrey Heinz (Instructor)Jon Rawski (TA)

LSA Summer InstituteUC Davis

June 27, 2019UC Davis ∣ 2019/06/27 J. Heinz ∣ 1

1 Strings

2 String Representations (Word Models)

3 First Order Logic

4 Defining Constraints

5 Defining Transformations

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Developing Logical Languages*

Ingredients

1 A model signature for the structures of interest

2 A logical type (QF, FO, MSO, QFLFP, . . . )

Instructions

• Combine and stir well!

(*This is what de Lacy (2011) calls a “Constraint Definition Language”)

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What are we modeling?

• Strings?

• Trees?

• Syntactic structures?

• Autosegmental structures?

• Prosodic structures?

• . . .

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What are we modeling?

• Strings?

• Trees?

• Syntactic structures?

• Autosegmental structures?

• Prosodic structures?

• . . .

We begin with strings because they are simple. Once weunderstand something how things work in the simple cases, wecan try to understand how they work in the complex cases.

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Part I

What are strings?

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Strings and Stringsets

Assume a finite set of symbols. Traditionally, Σ denotes thisset. Strings are built inductively with a non-commutativeoperation called concatenation.

1 Base case: λ is a string.

2 Inductive case: If u is a string and σ ∈ Σ then u ⋅ σ is astring.

• The string λ is the identity. So for all strings u:u ⋅ λ = λ ⋅ u = u.

• We refer to all strings of finite length with the notation Σ∗.

A stringset is a (possibly infinite) subset of Σ∗.

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Part II

Models

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Word Models

We use the word ‘word’ synonymously with ‘string.’

• A model of a word is a representation of it.

• A relational model contains two kinds of elements.

1 A domain. This is a finite set of elements.2 Some relations over the domain elements.

• Guiding principles:

1 Every word has some model.2 Different words must have different models.

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The successor model

Let Σ = {a, b, c} and suppose we wish to model strings in Σ∗.The successor model’s signature

W◁ = ⟨D,◁,a,b,c⟩

DW — Finite set of elements (positions)

◁W — A binary relation encoding immediatelinear precedence on D

a,b,c — Unary relations (so subsets of D) encodingpositions at which a,b,c occurs

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Example: W◁

Consider the string abbab.The model of abbab under the signature W◁ (denoted M◁

abbab)looks like this.

M◁abbab = ⟨

{1,2,3,4,5},{(1,2), (2,3), (3,4), (4,5)},{1,4},{2,3,5},∅

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Illustrating the successor model of abbab

1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

abbab = ⟨

{1,2,3,4,5},{(1,2), (2,3), (3,4), (4,5)},{1,4},{2,3,5}

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In class exercise

1 Give models for these strings.

1 abc2 cacaca

2 Suppose we removed the unary relations from the signatureso the it looks like this: W† = ⟨D,◁⟩. Can models withsuch a signature distinguish all strings in Σ∗?

3 Suppose we removed the successor relation from thesignature so it looks like this: W‡ = ⟨D,a,b,c⟩. Can modelswith such a signature distinguish all strings in Σ∗?

4 Phonological theories often uses features asrepresentational elements, not segments. How could youdefine a signature for a model that refers to features?What would the model of can [kæn] look like?

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Talking Truth: Models (⊧)

1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2

2 1 ◁ 3

3 ◁ (1,2)

4 ◁ (1,3)

5 a(1)

6 a(2)

7 b(3)

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3

3 ◁ (1,2)

4 ◁ (1,3)

5 a(1)

6 a(2)

7 b(3)

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3 False

3 ◁ (1,2)

4 ◁ (1,3)

5 a(1)

6 a(2)

7 b(3)

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3 False

3 ◁ (1,2) True

4 ◁ (1,3)

5 a(1)

6 a(2)

7 b(3)

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3 False

3 ◁ (1,2) True

4 ◁ (1,3) False

5 a(1)

6 a(2)

7 b(3)

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3 False

3 ◁ (1,2) True

4 ◁ (1,3) False

5 a(1) True

6 a(2)

7 b(3)

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3 False

3 ◁ (1,2) True

4 ◁ (1,3) False

5 a(1) True

6 a(2) False

7 b(3)

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3 False

3 ◁ (1,2) True

4 ◁ (1,3) False

5 a(1) True

6 a(2) False

7 b(3) True

8 b(4)

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

True or False?

1 1 ◁ 2 True

2 1 ◁ 3 False

3 ◁ (1,2) True

4 ◁ (1,3) False

5 a(1) True

6 a(2) False

7 b(3) True

8 b(4) False

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1 2 3 4 5

a b b a b◁ ◁ ◁ ◁

Whenever R(x⃗) is true in Mw we write

Mw ⊧ R(x⃗)

“The model of w satisfies R(x⃗)” or “Mw models R(x⃗)”

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Part III

Defining Constraints with FO Logic

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Formulas of First Order Logic

We let x, y, z, x1, x2, . . . be variables. They range over elementsof the domain.

• Base Cases

1 (x=y) (equality)2 R(x) (for each unary relation R∈W)3 R(x,y) (for each binary relation R∈W)

(binary relations are often written in infix notation as xRy)

• Inductive Cases. If ϕ,ψ are formulas of FO logic so are:

1 ¬ϕ, (ϕ ∧ ψ), (ϕ ∨ ψ),(ϕ⇒ ψ), (ϕ⇔ ψ) (Boolean connectives)

2 (∃x)[ϕ] (existential quantification)3 (∀x)[ϕ] (universal quantification)

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Defining new predicates

Defining new predicates is like writing little programs or scriptsthat can be used again and again. We write them from thebasic aforementioned pieces.

Examples

x ≠ ydef= ¬(x = y)

first(x)def= ¬(∃y)[y ◁ x]

last(x)def= ¬(∃y)[x◁ y]

CxC(x)def= ∃(y)[y ◁ x ∧ cons(x) ∧ cons(y)]

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Interpreting sentence of FO logic (theextensions!)

• Only sentences where every variable is bound can beinterpreted to define stringsets. Variables that are notbound are called free.

• The details of interpretations are in the handout. Theretends to be a lot of bookkeeping to write out thedefinitions, but it is easy to explain with examples.

• In a nutshell: A word w models a sentence of FO logic ϕ ifthe sentence is true of Mw. We write Mw ⊧ ϕ.

qϕy= {w ∈ Σ∗ ∣Mw ⊧ ϕ}

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Examples

1 ϕ = (∀x)[¬CxC(x)]

Words like paba and ana satisfy ϕ but words like pikka andpint do not.

2 ϕ = (∀x)[first(x)⇒ cons(x)]

Words like paba and tana satisfy ϕ but words like asa andilio do not.

3 ϕ = (∃x)[nasal(x)]

Words like pana and mule satisfy ϕ but words like asa andlazi do not.

4 ϕ = (∃x, y, z)[nasal(x) ∧ nasal(y) ∧ nasal(z) ∧ x ≠ y ∧

x ≠ z ∧ y ≠ z]

Words like panaman and mulenumina satisfy ϕ but wordslike asa and munile do not.

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Examples

1 ϕ = (∀x)[¬CxC(x)]

x ≠ z ∧ y ≠ z]

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Examples

1 ϕ = (∀x)[¬CxC(x)]

x ≠ z ∧ y ≠ z]

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Examples

1 ϕ = (∀x)[¬CxC(x)]

x ≠ z ∧ y ≠ z]

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In class exercise

What generalization (=markedness constraint) is this?

1 (∀x, y)[(x◁ y ∧ nasal(x) ∧ cons(y))⇒ voice(y)]

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Interim Summary

We have defined our first Constraint Definition Language!Ingredients

1 A model signature: M◁

2 A logical type: First Order Logic

This logical language is known as First Order with SuccessorFO(◁).

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Part IV

Analysis

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Outstanding Questions

1 What constraints can we write (and not write) withFO(◁)?

2 This defines constraints as functionsf ∶ Σ∗ → {True,False}? How do we count violations?Assign probabilities?

3 This defines constraints. How do we definetransformations?

4 What happens when we change the signature? What othersignatures are there?

5 What happens when we change the logic? What otherlogics are there?

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What constraints can we write (and notwrite) with FO(◁)?

Theorem

A constraint is FO-definable with successor if and only if thereare two natural numbers k and t such that for any two stringsw and v, if w and v contain the same substrings x of lengthk the same number of times counting only up to t, then eitherboth w and v violate the constraint or neither does.

In other words, FO(◁) cannot distinguish two strings whichhave the same number of substrings x of length k (counting upto some threshold t).

(Thomas 1982, “Classifying regular events in symbolic logic”)

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Some Phonology

Kikongo/ku-kinis-il-a/ becomes [kukinisina] ‘to make dance for’

A Common AnalysisThis alternation is motivated by the following constraint:

• *N..L: Laterals cannot follow nasals at any distance.

(Odden 2004)

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*N..L cannot be expressed with FO(◁)

ProofPick any k and t. Compare w = aknak`ak with v = ak`aknak.

count w = ⋊aknak`ak⋉ Notes

1 ⋊ak−1

1 ainaj (for each 0 ≤ i, j ≤ k − 1, i + j = k − 1)1 ai`aj (for each 0 ≤ i, j ≤ k − 1, i + j = k − 1)1 ak−1⋉

count v = ⋊ak`aknak⋉ Notes

1 ⋊ak−1

1 ainaj (for each 0 ≤ i, j ≤ k − 1, i + j = k − 1)1 ai`aj (for each 0 ≤ i, j ≤ k − 1, i + j = k − 1)1 ak−1⋉

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How can we express the constraint *N..L?

Two options

1 Increase the power of the logic

2 Change the representation

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Part V

The Precedence Model

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The precedence model

Let Σ = {a, b, c} and suppose we wish to model strings in Σ∗.The precedence model’s signature

W◁ = ⟨D,<,a,b,c⟩

DW — Finite set of elements (positions)

<W — A binary relation encoding generallinear precedence on D

a,b,c — Unary relations (so subsets of D) encodingpositions at which a,b,c occurs

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Example: W<

Consider the string abbab.The model of abbab under the signature W< (denoted M<

abbab)looks like this.

M<abbab = ⟨

{1,2,3,4,5},{(1,2), (1,3), (1,4), (1,5), (2,3),(2,4), (2,5), (3,4), (3,5), (4,5)}{1,4},{2,3,5},∅

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Illustrating the precedence model of abbab

1 2 3 4 5

a b b a b

abbab = ⟨

{1,2,3,4,5},{(1,2), (1,3), (1,4), (1,5), (2,3),

(2,4), (2,5), (3,4), (3,5), (4,5)}{1,4},{2,3,5},∅

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The constraint *N..L

• *N..L: Laterals cannot follow nasals at any distance.

∗N..Ldef= ∀x, y[nasal(x) ∧ lateral(y)⇒ ¬(x < y)]

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Revisiting Outstanding Questions

1 What constraints can we write (and not write) with FO(<)?

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Revisiting Outstanding Questions

1 What constraints can we write (and not write) with FO(<)?

4 What happens when we change the signature?

What other signatures are there?

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Part VI

Summary

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Summary

1 We learned the successor model for words.

2 We learned how to express constraints in First Order Logicwith this model.

3 We encountered some limitations in the expressivity of thisclass.

4 There are many paths forward from FO(◁).

5 We briefly encountered the precedence model for words.

UC Davis ∣ 2019/06/27 J. Heinz ∣ 34

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