In 2002 Jurdzinski and Lorys settled a long-standing conjecture that palindromes are not a Church-Rosser language. Their proof required a sophisticated theory about computation graphs of 2-stack automata. We present their proof in terms of 1-tape Turing machines.We also provide an alternative proof of Buntrock and Otto's result that the set of non-square bitstrings, which is context-free, is not Church-Rosser.
In the 1970s, Nivat [13] began the study of languages defined by Thue systems: see also [5,1]. Book [2] continued the study of Church-Rosser Thue systems, and the theory has been much extended since then [3,9].
We follow the definitions of length-reducing Thue systems, etcetera, in [3]. A Thue system S is Church-Rosser if whenever u * ↔ S v, there exists a string w such that u * →w and v * →w. Equivalently, every congruence class contains exactly one irreducible string. The redexes, reducts, and irreducible strings, with respect to S, are denoted Redexes(S), Reducts(S), and Irred(S).
PAL denotes the set of (bitstring) palindromes: those bitstrings which read the same backwards as forwards, namely, PAL = {x ∈ {0, 1} * : x R = x} where x R is the reversal of x.
Church-Rosser languages will be described below. They are a surprisingly powerful generalisation of congruential languages, which are finite unions of congruence classes of a finite Church-Rosser Thue system.
In [1] it is shown that PAL is not a congruential language. This is proved by contradiction. Otherwise, by definition, PAL is a finite union of congruence classes of a Thue system T . 1However, the linguistic congruence ≡ PAL is the identity relation. It is defined by
x ≡ PAL y ⇐⇒ (def.) (∀u, v)(uxv ∈ PAL ⇐⇒ uyv ∈ PAL).
If x and y are different bitstrings, suppose without loss of generality that |x| ≤ |y| and y ends in 1. Then λx0 |y| y R is not a palindrome but λy0 |y| y R is. Thus x ≡ PAL y ⇐⇒ x = y. But * ↔ T would be a refinement of ≡ PAL , so
Given a Church-Rosser Thue system S, we exhibit a 1-tape Turing machine TM implementing reduction modulo S in a systematic way. While Book’s 2-stack machine is more efficient [3,6], the advantage of studying reductions on a 1-tape Turing machine is that blanks are steadily accumulated, allowing us to see where information has been lost.
‘Turing machine’ will mean a deterministic machine with quintuple instructions and 2-way infinite tape, although the worktape used will be only slightly longer than the input string. An instruction (quintuple) has the form current state, current symbol, new symbol, head movement, new state where the head movement is 1 square left or right (the read/write head moves at every step).
Given a language L such that
on input x the machine TM converts the tape contents to t 1 xt 2 , reduces it, and compares the result to t 3 . Let Σ be the smallest alphabet such that Σ * contains Redexes(S), Reducts(S), {t 1 , t 2 , t 3 }, and L. The machine TM executes reductions systematically. If a string z is reducible, then it has a leftmost redex, i.e., it can be written as wut where u is a redex and no proper prefix of wu is reducible.
The set of such strings wu is regular and one can easily describe a DFA D which recognises this set, and has the property that when it accepts wu, one such redex u, and hence a rule u → v, is determined uniquely by its accepting state. Ties are broken arbitrarily.
Let K be the set of states of D.
The worktape alphabet of TM consists of • Σ, a new blank symbol B, and left and right sentinel characters ¢ and $.
• Compound symbols [a, k] where a ∈ Σ ∪ {B} and k ∈ K (the states of D).
The blank symbols are B and {[B, k] : k ∈ K}.
(2.1) Write h for the following homomorphism.
Let k 0 be the initial state of D and δ the transition function for D. We extend δ to K × (Σ ∪ {B}):
Obviously,
(2.2) Definition The string ¢t 1 xt 2 $ (including endmarkers) is called the initial redex on input x.
The machine TM creates the intial redex, then reduces as often as needed.
• Its configurations are represented in the form αqβ where αβ are the tape contents, including ¢ on the left and $ on the right, β = λ (so $ is the rightmost symbol in β), q is the current state, and the machine is scanning the first symbol of β.
• Except for the sentinel characters, all symbols in β are in Σ ∪ {B} and all symbols in α are compound symbols, and α is historical.
• After ¢t 1 and t 2 $ have been added to the input, h(α) is always irreducible and h(αβ) is always a reduct of t 1 xt 2 (except temporarily during REDUCE phases).
• First, TM moves to the right, appending t 2 $ to x. Then it moves to the left, prefixing ¢t 1 to xt 2 $: the tape contents are now the initial redex ¢t 1 xt 2 $, and the current symbol is ¢. It enters a SHIFT phase.
For the rest of this description αqβ denotes the current configuration, and a is the current symbol.
• In a SHIFT phase, if β = $, then TM enters its final phase, described below. Otherwise, k is an accepting state of D, and the string h(α)a ends in a redex u, so there exists a rule u → v associated with k. TM enters a REDUCE phase.
• In a REDUCE phase, h(α)a ends with a redex u, and TM can select a unique rule u → v to be applied. TM moves left, overwriting the rightmost |v| symbols of αa with v, extending leftwards with blank symbols B, until the square holding the leftmost symbol ℓ of u (or rather, a compound symbol [ℓ, k]) is ove
