je rey heinz (instructor) jon rawski (ta) · computational phonology - class 2 je rey heinz...
TRANSCRIPT
Computational Phonology - Class 2
Jeffrey Heinz (Instructor)Jon Rawski (TA)
LSA Summer InstituteUC Davis
June 27, 2019UC Davis ∣ 2019/06/27 J. Heinz ∣ 1
Today
1 Strings
2 String Representations (Word Models)
3 First Order Logic
4 Defining Constraints
5 Defining Transformations
UC Davis ∣ 2019/06/27 J. Heinz ∣ 2
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”)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 3
What are we modeling?
• Strings?
• Trees?
• Syntactic structures?
• Autosegmental structures?
• Prosodic structures?
• . . .
UC Davis ∣ 2019/06/27 J. Heinz ∣ 4
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.
UC Davis ∣ 2019/06/27 J. Heinz ∣ 4
Part I
What are strings?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 5
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∗.
UC Davis ∣ 2019/06/27 J. Heinz ∣ 6
Part II
Models
UC Davis ∣ 2019/06/27 J. Heinz ∣ 7
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.
UC Davis ∣ 2019/06/27 J. Heinz ∣ 8
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
UC Davis ∣ 2019/06/27 J. Heinz ∣ 9
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},∅
⟩
UC Davis ∣ 2019/06/27 J. Heinz ∣ 10
Illustrating the successor model of abbab
1 2 3 4 5
a b b a b◁ ◁ ◁ ◁
M◁
abbab = ⟨
{1,2,3,4,5},{(1,2), (2,3), (3,4), (4,5)},{1,4},{2,3,5}
⟩
UC Davis ∣ 2019/06/27 J. Heinz ∣ 11
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?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 12
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Talking Truth: Models (⊧)
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)”
UC Davis ∣ 2019/06/27 J. Heinz ∣ 13
Part III
Defining Constraints with FO Logic
UC Davis ∣ 2019/06/27 J. Heinz ∣ 14
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 15
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)]
UC Davis ∣ 2019/06/27 J. Heinz ∣ 16
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}
UC Davis ∣ 2019/06/27 J. Heinz ∣ 17
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.
UC Davis ∣ 2019/06/27 J. Heinz ∣ 18
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.
UC Davis ∣ 2019/06/27 J. Heinz ∣ 18
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.
UC Davis ∣ 2019/06/27 J. Heinz ∣ 18
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.
UC Davis ∣ 2019/06/27 J. Heinz ∣ 18
In class exercise
What generalization (=markedness constraint) is this?
1 (∀x, y)[(x◁ y ∧ nasal(x) ∧ cons(y))⇒ voice(y)]
UC Davis ∣ 2019/06/27 J. Heinz ∣ 19
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(◁).
UC Davis ∣ 2019/06/27 J. Heinz ∣ 20
Part IV
Analysis
UC Davis ∣ 2019/06/27 J. Heinz ∣ 21
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?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 22
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?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 22
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?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 22
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?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 22
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?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 22
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”)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 23
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)
UC Davis ∣ 2019/06/27 J. Heinz ∣ 24
*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
3 ak
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
3 ak
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⋉
UC Davis ∣ 2019/06/27 J. Heinz ∣ 25
How can we express the constraint *N..L?
Two options
1 Increase the power of the logic
2 Change the representation
UC Davis ∣ 2019/06/27 J. Heinz ∣ 26
Part V
The Precedence Model
UC Davis ∣ 2019/06/27 J. Heinz ∣ 27
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
UC Davis ∣ 2019/06/27 J. Heinz ∣ 28
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},∅
⟩
UC Davis ∣ 2019/06/27 J. Heinz ∣ 29
Illustrating the precedence model of abbab
1 2 3 4 5
a b b a b
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},∅
⟩
UC Davis ∣ 2019/06/27 J. Heinz ∣ 30
The constraint *N..L
• *N..L: Laterals cannot follow nasals at any distance.
∗N..Ldef= ∀x, y[nasal(x) ∧ lateral(y)⇒ ¬(x < y)]
UC Davis ∣ 2019/06/27 J. Heinz ∣ 31
Revisiting Outstanding Questions
1 What constraints can we write (and not write) with FO(<)?
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?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 32
Revisiting Outstanding Questions
1 What constraints can we write (and not write) with FO(<)?
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 other signatures are there?
5 What happens when we change the logic? What otherlogics are there?
UC Davis ∣ 2019/06/27 J. Heinz ∣ 32
Part VI
Summary
UC Davis ∣ 2019/06/27 J. Heinz ∣ 33
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