Download PDF

unique readability of well-formed
formulas∗
CWoo†
2013-03-22 2:44:57
Suppose V is an alphabet. Two words w1 and w2 over V are the same if they
have the same length and same symbol for every position of the word. In other
words, if w1 = a1 · · · an and w2 = b1 · · · bm , where each ai and bi are symbols in
V , then n = m and ai = bi . In other words, every word over V has a unique
representation as product (concatenation) of symbols in V . This is called the
unique readability of words over an alphabet.
Unique readability remains true if V is infinite. Now, suppose V is a set of
propositional variables, and suppose we have two well-formed formulas (wffs)
p := αp1 · · · pn and q := βq1 · · · qm over V , where α, β are logical connectives
from a fixed set F of connectives (for example, F could be the set {¬, ∨, ∧, →
, ↔}), and pi , qj are wffs. Does p = q mean α = β, m = n, and pi = qi ? This is
the notion of unique readability of well-formed formulas. It is slightly different
from the unique readability of words over an alphabet, for pi and qj are no
longer symbols in an alphabet, but words themselves.
Theorem 1. Given any countable set V of propositional variables and a set
F of logical connectives, well-formed formulas over V constructed via F are
uniquely readable
Proof. Every p ∈ V has a representation αp1 · · · pn for some n-ary connective α
and wffs pi . The rest of the proof we show that this representation is unique,
establishing unique readability.
Define a function φ : V ∪ F → Z such that φ(v) = 1 for any v ∈ V , and
φ(f ) = 1 − n for any n-ary connective f ∈ F . Defined inductively on the length
of words over V ∪ F , φ can be extended to an integer-valued function φ∗ on the
set of all words on V ∪ F so that φ∗ (w1 w2 ) = φ∗ (w1 ) + φ∗ (w2 ), for any words
w1 , w2 over V ∪ F . We have
1. φ∗ is constant (whose value is 1) when restricted to V .
∗ hUniqueReadabilityOfWellformedFormulasi created: h2013-03-2i by: hCWooi version:
h41723i Privacy setting: h1i hTheoremi h03B05i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
This can be proved by induction on Vi (for the definition of Vi , see the
parent entry). By definition, this is true for any atoms. Assume this is true
for Vi . Pick any p ∈ Vi+1 . Then p = αp1 · · · pn for some n-ary α ∈ F , and
pj ∈ Vi for all j = 1, . . . , n Then φ∗ (p) = φ∗ (α) + φ∗ (p1 ) + · · · + φ∗ (pn ) =
1 − n + n × 1 = 1. Therefore, φ∗ (p) = 1 for all p ∈ V .
2. for any non-trivial suffix s of a wff p, φ∗ (s) > 0. (A suffix of a word w is
a word s such that w = ts for some word t; s is non-trivial if s is not the
empty word)
This is also proved by induction. If p ∈ V0 , then p itself is its only nontrivial final segment, so the assertion is true. Suppose now this is true
for any proposition in Vi . If p ∈ Vi+1 , then p = αp1 · · · pn , where each
pk ∈ Vi . A non-trivial final segment s of p is either p, a final segment
of pn , or has the form tpj · · · pn , where t is a non-trivial final segment of
pj−1 . In the first case, φ∗ (s) = φ∗ (p) = 1. In the second case, φ∗ (s) > 0
from assumption. In the last case, φ∗ (s) = φ∗ (t) + (n − j + 1) > 0.
Now, back to the main proof. Suppose p = q. If p is an atom, so must q, and we
are done. Otherwise, assume p = αp1 · · · pm = βq1 · · · qn = q. Then α = β since
the expressions are words over V ∪F , and α, β ∈ F . Since the two connectives are
the same, they have the same arity: m = n, and we have αp1 · · · pn = αq1 · · · qn .
If n = 0, then we are done. So assume n > 0. Then p1 · · · pn = q1 · · · qn .
We want to show that pn = qn , and therefore p1 · · · pn−1 = q1 · · · qn−1 , and by
induction pi = qi for all i < n as well, proving the theorem.
First, notice that pn is a non-trivial suffix of q1 · · · qn . So pn is either a suffix
of qn or has the form tqj · · · qn , where j ≤ n, and t is a non-trivial suffix of
qj−1 . In the latter case, 1 = φ∗ (pn ) = φ∗ (tqj · · · qn ) = φ∗ (t) + n − j + 1. Then
φ∗ (t) = j − n ≤ 0, contradicting 2 above. Therefore pn is a suffix of qn . By
symmetry, qn is also a suffix of pn , hence pn = qn .
As a corollary, we see that the well-formed formulas of the classical propositional logic, written in Polish notation, are uniquely readable. The unique
readability of wffs using parentheses and infix notation requires a different proof.
Remark. Unique readability will fail if the pi and qj above are not wffs,
even if V is finite. For example, suppose v ∈ V and # is binary, then #v#vv
can be read in three non-trivial ways: combining v and #vv, combining v# and
vv, or combining v#v and v. Notice that only the first combination do we get
wffs.
2