Honours Project Proposal
Logically Equivalent Coloured Linear Orders
Ruaan Kellerman
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Elementary equivalence
The syntax of first-order logic has the logical connectives ¬ (negation), ∨ (disjunction),
∧ (conjunction), → (implication) and ↔ (biconditional), along with universal quantification (∀) and existential quantification (∃) over individual variables. Using these
connectives, one can form first-order sentences. For example, in the language of linear
orders, the following first-order sentence asserts that every element has an immediate
predecessor:
∀x∃y (y < x ∧ ∀z (¬ (y < z ∧ z < x))) .
The quantifier rank of a first-order sentence is the largest number of nested quantifiers
in that sentence. The sentence above has quantifier rank 3. Two structures A and B are
called n-equivalent when they satisfy the same first-order sentences of quantifier rank
at most n. A and B are called elementarily equivalent when they are n-equivalent for
every natural number n. Hence two structures are elementarily equivalent when they
cannot be distinguished using first-order sentences. Elementary equivalence is a weaker
type of structural equivalence than isomorphism: two structures can be non-isomorphic
but elementarily equivalent.
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Ehrenfeucht-Fraı̈ssé games
One of the standard methods for proving that two structures are n-equivalent is the
Ehrenfeucht-Fraı̈ssé game. Let A and B be two relational structures. An n-round
Ehrenfeucht-Fraı̈ssé game on A and B is played by two players, Player I and Player II,
as follows.
For each of the n rounds of the game, Player I chooses any element from either the
structure A or the structure B and Player II responds by choosing any element from the
structure not used by Player I during that round. At the end of the game we therefore
have elements a1 , a2 , . . . , an from A and b1 , b2 , . . . , bn from B that were chosen by the
two players during the game. Player II wins the game provided the map ai 7→ bi is a
local isomorphism, otherwise Player I wins the game.
The value of Ehrenfeucht-Fraı̈ssé games is seen from the following theorem.
Theorem: Player II has a winning strategy for the n-round Ehrenfeucht-Fraı̈ssé
game on A and B if and only if A is n-equivalent to B.
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Example: Consider the structures A = (A; <) and B = (B; <), where A = {a, b, c}
with a < b < c and B = {p, q, r, s} with p < q < r < s. The following is a winning
strategy for Player II for the 2-round Ehrenfeucht-Fraı̈ssé game on A and B.
Round 1.
Player I chooses a1 ∈ A (respectively b1 ∈ B). Player II responds with b1 ∈ B
(respectively a1 ∈ A) according to Figure 1.
Choice of
Player I
a
b
c
Choice of
Player I
p
q
r
s
Response of
Player II
p
q
s
Response of
Player II
a
b
b
c
Figure 1: Player II’s round 1 response. This is also Player II’s round 2 response if Player
I chose a, c, p or s in round 1.
Round 2.
Player I chooses a2 ∈ A (respectively b2 ∈ B). Player II responds with b2 ∈ B
(respectively a2 ∈ A) as follows.
Case 1: (Player I chose a, c, p or s in round 1.)
The round 2 response of Player II is tabulated in Figure 1.
Case 2: (Player I chose b or q in round 1.)
The round 2 response of Player II is tabulated in Figure 2.
Choice of
Player I
a
b
c
Choice of
Player I
p
q
r
s
Response of
Player II
p
q
s
Response of
Player II
a
b
c
c
Figure 2: Player II’s round 2 response if Player I chose b or q in round 1.
Case 3: (Player I chose r in round 1.)
The round 2 response of Player II is tabulated in Figure 3.
Now with a1 , a2 , b1 and b2 as chosen above, it follows that:
• a1 = a2 if and only if b1 = b2
• a1 < a2 if and only if b1 < b2
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Choice of
Player I
a
b
c
Choice of
Player I
p
q
r
s
Response of
Player II
p
r
s
Response of
Player II
a
a
b
c
Figure 3: Player II’s round 2 response if Player I chose r in round 1.
• a2 < a1 if and only if b2 < b1
Hence the map ai 7→ bi is a local isomorphism so Player II wins the game. It follows
that the structures A and B are 2-equivalent, even though one has three elements in its
domain while the other has four elements in its domain.
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Coloured linear orders
A coloured linear order is a structure of the form L = (L; <, c1 , . . . , ck ), where (L; <) is
a linear order and where every ci (called a colour ) is a unary predicate on L. While the
logical equivalence of linear orders without colours has been extensively studied, that of
coloured linear orders is fairly new and much work remains to be done. Adding colours
to linear orders substantially increases the complexity of the Ehrenfeucht-Fraı̈ssé games
that are used to analyse their logical equivalence.
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Goals of the project
In this project, the student will first have to familiarise themself with the syntax of firstorder logic, with basic first-order model theory, and with Ehrenfeucht-Fraı̈ssé games.
The student will then be expected to investigate the logical equivalence of specific
coloured linear orders. As the example for uncoloured linear orders above demonstrates,
designing a winning strategy for Player II can be hard work even in simple cases. Due to
the combinatorial nature of Ehrenfeucht-Fraı̈ssé games, a solid working understanding
of basic combinatorics will be useful. The project must be typeset using LATEX. The
project will be of interest to a student who wishes to learn more about mathematical
logic.
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