unique readability of well-formed formulas
Suppose is an alphabet. Two words and over are the same if they have the same length and same symbol for every position of the word. In other words, if and , where each and are symbols in , then and . In other words, every word over has a unique representation as product (concatenation) of symbols in . This is called the unique readability of words over an alphabet.
Unique readability remains true if is infinite. Now, suppose is a set of propositional variables, and suppose we have two well-formed formulas (wffs) and over , where are logical connectives from a fixed set of connectives (for example, could be the set ), and are wffs. Does mean , , and ? 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 and are no longer symbols in an alphabet, but words themselves.
Given any countable set of propositional variables and a set of logical connectives, well-formed formulas over constructed via are uniquely readable
Every has a representation for some -ary connective and wffs . The rest of the proof we show that this representation is unique, establishing unique readability.
Define a function such that for any , and for any -ary connective . Defined inductively on the length of words over , can be extended to an integer-valued function on the set of all words on so that , for any words over . We have
is constant (whose value is ) when restricted to .
for any non-trivial suffix of a wff , . (A suffix of a word is a word such that for some word ; is non-trivial if is not the empty word)
This is also proved by induction. If , then itself is its only non-trivial final segment, so the assertion is true. Suppose now this is true for any proposition in . If , then , where each . A non-trivial final segment of is either , a final segment of , or has the form , where is a non-trivial final segment of . In the first case, . In the second case, from assumption. In the last case, .
Now, back to the main proof. Suppose . If is an atom, so must , and we are done. Otherwise, assume . Then since the expressions are words over , and . Since the two connectives are the same, they have the same arity: , and we have . If , then we are done. So assume . Then . We want to show that , and therefore , and by induction for all as well, proving the theorem.
First, notice that is a non-trivial suffix of . So is either a suffix of or has the form , where , and is a non-trivial suffix of . In the latter case, . Then , contradicting 2 above. Therefore is a suffix of . By symmetry, is also a suffix of , hence . ∎
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 and above are not wffs, even if is finite. For example, suppose and is binary, then can be read in three non-trivial ways: combining and , combining and , or combining and . Notice that only the first combination do we get wffs.
|Title||unique readability of well-formed formulas|
|Date of creation||2013-03-22 18:52:32|
|Last modified on||2013-03-22 18:52:32|
|Last modified by||CWoo (3771)|