A taste of a taste of rewrite systems

Rewriting is the process of repeatedly replacing terms in a formula with other terms. For example, the rule \(\neg\neg x \rightarrow x\) can be applied to the formula \(\neg \neg \neg \neg a\) to get \(a\). This seems important but I needed a bit more background to understand it. These are my notes on some of the concepts in the survey paper A Taste of Rewrite Systems.

Example

Rewriting is quite general, so let’s start with the Grecian Urn example.

An urn contains 150 black and 75 white beans. You remove two beans. If they are the same colour, add a black bean to the urn. If they are different colours, put the white one back.

If you repeat the process of removing two and putting one back, what is the colour of the last bean? Or does it depend?

The system is expressed as the following four rules:

\[black\ black \rightarrow black \\\\ white\ white \rightarrow black \\\\ white\ black \rightarrow white \\\\ black\ white \rightarrow white\]

A possible sequence of (a smaller urn of) beans might therefore be:

\[black\ white\ black\ (black\ white) \\\\ black\ white\ (black\ white) \\\\ black\ (white\ white) \\\\ (black\ black) \\\\ black\]

where the last two beans on each line, in brackets, are substituted according to the rules.

The study of rewriting tries to answer questions like this: For which types of systems are the answer predetermined, and which systems terminates? Actually doing the rewriting is just computation, and can in fact be used to interpret other programming languages.

Formulae and terms

A well-formed formula is a sequence of symbols in a formal language, for example \(black\ white\ black\) in the urn example, or \(\neg \neg \neg \neg a\) in propositional logic.

A formula is composed of terms. A term is either a variable, a constant, or a term built from the function symbol and other terms, for example \(f(g(a,b),c)\), where \(a\), \(b\), and \(c\) are constants, \(g\) is a binary (takes two terms) function symbol, and \(f\) also. Usually I would think of \(f\) or \(g\) as taking two arguments, but it seems the word “argument” is carefully avoided in the wikipedia article, so maybe they only become arguments when assigned semantics. Prolog talks about terms as function arguments though.

Termination

A system is terminating if there are no sequences of derivations that are infinite. For example, if you had to put back two white beans if you took out two black beans and vice versa, then it is possible to use those rules forever.

The original urn example, on the other hand, is terminating since the number of beans decreases every round.

A termination function can be used to characterise terminating systems. The paper lists seven ways in which a function can be a termination function. I repeat them here each with an example.

  • A function that returns the outermost function symbol of a term, with symbols ordered by some precedence. For example for \(f(g(a,b),c)\), return \(f\), where \(g\succ f\succ c\succ b\succ a\) could be a precedence (although the ordering only has to be partial).
  • A function that extracts an immediate subterm at a position that can depend on the function symbol. For example, a function that always extracts the second subterm for \(f\). For \(f(g(a,b),c)\) it would extract \(c\).
  • A function that extracts an immediate subterm of some rank, where the path rank is defined recursively below. For example, if we extract the largest subterm, then for \(f(g(a,b),c)\) the options are \(g(a,b)\) or \(c\), and the largest one would depend on the path rank.
  • A homomorphism from terms to some well-founded set of values. A homomorphism is a map (basically a function) between two sets so that operations (stuff like addition and multiplication but the algebraic abstractions) are preserved. For example, in our case, a homomorphism from terms to the natural numbers would be the size (number of symbols), or depth (how nested the terms are). A well-founded set is the normal set in ZFC, a set that does not contain infinitely nested sets. For example \(\{\{a\}, \{b\}\}\) is well founded, but \(\{\{\dots, a\}, \{b\}\}\) is not if the dots represent an infinitely deep recursion. The natural numbers are well-founded.
  • A monotonic homomorphism with the strict subterm property, from terms to some well-founded set. We already know what a homomorphism (structure preserving map) is and what a well-founded set is (e.g. natural numbers). A homomorphism is monotonic with respect to the ordering if the value it assigns to a term \(f(\dots s \dots)\) is greater than the value it assigns to \(f(\dots t \dots)\) if the value of \(s\) is greater than \(t\). For example, size (number of symbols) has this property: \(f(g(a,b),c)\) has 4 symbols, and \(f(g(h(a,b)),c)\) has 5, so the latter is larger than the former. But the first subterm of each \(g(a,b)\) has 3 and \(g(h(a,b))\) has 4, so whenever these subterms are larger, the terms are larger also. This property is strict if the value is strictly larger (\(\succ\) and not just \(\succeq\)).
  • A strictly monotonic homomorphism with the strict subterm property. As above, but the value assigned to the outer term is strictly larger when the inner terms are larger. The previous function allowed outer terms to be equal or larger if the inner terms were larger.
  • A constant function.

These conditions define a whole range of possible functions that can be used as termination functions.

Now we use these functions to define a path ordering: Let \(\tau_0 \dots \tau_k\) each be a (possibly different) termination function. Let the first \(i-2\) be strict monotonic homomorphisms specifically, and the \(i-1\)th function either strict or not strict. The path ordering defined by the precedence relation \(s\succeq t\), where \(s=f(s_1,\dots,s_m)\) and \(t=g(t_1,\dots,t_m)\) are terms and their subterms, if either of the following conditions hold:

  1. \(s_i \succeq t\) for some \(s_i\), or
  2. \(s \succ t_1,\dots,t_n\) and \(\langle \tau_1(s),\dots,\tau_k(s) \rangle\) is lexicographically greater or equal to \(\langle \tau_1(t),\dots,\tau_k(t) \rangle\), in other words if the first elements are greater, or if equal (not greater and not smaller), the second elements and so forth.

Dershowitz remarks that this definition combines syntactic (the first three types of termination functions) and semantic considerations (the others).

Let me try to rewrite this definition in my own words: We want to be able to say which terms comes before which other terms, given a bunch of termination functions that we choose. The induced path order defines a precedence by comparing any two terms \(s\) and \(t\). A term succeeds another term in two cases. Firstly, if at least one of its subterms succeeds the function symbol of the other term. Secondly, if its function symbol succeeds all the subterms of the other term and the first few termination functions of the term succeeds that of the other term.

For example, let us compare \(s=g(a,b)\) and \(t=g(a,c)\) using two termination functions, \(\tau_1\) is the number of symbols homomorphism, and \(\tau_2\) a function that returns the second subterm of \(g\).

We now recursively evaluate the conditions:

  1. Does a subterm of \(s\) succeed \(t\)?
    • Does \(a\) succeed \(g(a,c)\)?
      1. No subterms.
      2. Does \(a\) succeed \(a\) and \(c\)? No, \(a\nsucc a\).
    • Does \(b\) succeed \(g(a,c)\)?
      1. No subterms.
      2. Does \(b\) succeed \(a\) and \(c\)? No, \(b\nsucc c\).
  2. Does \(s\) succeed \(a\) and \(c\) and is \(\langle \tau_1(s), tau_2(s) \rangle \succeq \langle \tau_1(t), tau_2(t) \rangle\)?
    • Does \(g(a,b)\) succeed \(a\)?
      1. Does a subterm of \(g(a,b)\) succeed \(a\)? Yes – \(b\succ a\).
    • Does \(g(a,b)\) succeed \(c\)?
      1. Does a subterm of \(g(a,b)\) succeed \(a\)? No – neither \(a\) nor \(b\) succeeds \(c\).
    • We do not even have to check the termination functions in this case as none of the conditions are met.

We now have an ordering on terms. With that, the following theorem can be stated: A rewrite system terminates if \(l \sigma \succ r \sigma\), in the path ordering \(\succ\), for all rules \(l \rightarrow r\), and also \(\tau(l \sigma) = \tau(r \sigma)\) for each of the nonmonotonic homomorphisms among its termination functions.

In other words, first we do any substitution on each rule by substituting the variables in the left-hand side of the rule with something concrete, and then do the same substitution on the right-hand side. If the right-hand side now precedes (is smaller than) the right hand side, it means that each time the rule is applied, the value of the resulting term will be smaller than previously and so it must eventually get to the minimum value and terminate. The second condition states that the value assigned to both sides of the rule must neither be smaller or larger after the substitution for some of the termination functions. Remember that the monotonic homomorphisms are the first \(\tau_1,\dots,\tau_{i-1}\) functions, so the nonmonotonic homomorphisms must be some of the last substitution functions used in the lexicographical comparison. It is not clear to me at the moment why this condition is necessary, but I assume it will become clear when delving into the proof of the theorem, which I will not go into now.

So for example, for the rule \(g(x,c) \rightarrow g(x,a)\), where \(x\) is a variable, and the termination functions as in the example above, it doesn’t matter what we set \(x\) to, the value of the right-hand side will decrease since the second argument to \(g\) decreases from \(c\) to \(a\).

A system can therefore be proven to terminate if we can find termination functions so that applying any rule decreases the value of the result in the path ordering.

Conclusion

The paper further discusses Confluence, Completion, Validity, Satisfiability, Associativity and Commutativity, Conditional Rewriting, Programming, and Theorem Proving. I can post about this another day.

Written on January 6, 2021