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Questions tagged [rewriting-systems]

For questions related to (term) rewriting systems (which are reduction systems in which rewrite rules apply to terms).

Term rewriting systems are reduction systems in which rewrite rules apply to terms. Terms are built up from variables and constants using function symbols (or operations). Rules of term rewriting systems have the form $x\rightarrow y$, where both $x$ and $y$ are terms, $x$ is not a variable, and every variable from $y$ occurs in $x$ as well.

Term rewriting systems can be employed to compute arithmetic operations on natural numbers. To this end, each such number has to be encoded as a term. The simplest encoding is the one used in the Peano axioms, based on the constant $0$ (zero) and the successor function $S$.

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In rewiring systems do definitions creates new rewrite laws or an alias? And is this a meaningful question?

Lambda calculus is often introduced as a rewriting or substitution system. Where $\beta$ reduction is described as replacing bound variables with the value that variable is bound to. For example $(\lambda x. x) y \rightarrow y$ is described as…
Q the Platypus
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Bergman's Diamond Lemma: do these rules lead to a normal form?

Last week I was recommended Bergman's Diamond Lemma in these comments. I read through the paper, and was working on an exercise in it on page 193: Examine for termination each of the following singleton reduction-systems on $\mathbf{k}\langle x,…
Dani Hobbes
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Uniqueness of the result of rewritting an algebraic expression using distributivity rule

Let $expr$ be an algebraic expression involving natural numbers, addition operator and multiplication operator, e.g., $$(1+2)\cdot(3+4 \cdot 5)+6.$$ By iteratively applying the distributivity of multiplication over addition to $expr$, that is,…
abebebebahabe
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Consequences of difference between "strong" and weak Church-Rosser property

An Abstract rewriting system is a set A, whose elements are usually called objects, together with a binary relation on A, traditionally denoted by $\rightarrow$. An object $x \in A$ is called reducible if there exist some other $y \in A$ and…
Vitaly Olegovitch
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Missing parentheses in $s(k (s I I))(s(\lambda y. s(k y))(\lambda y. s I I)$ leads to interesting error in an nLab page. Need a double check.

I think I found an error in the nLab page on partial combinatory algebra in the Example combinators section: Finally, consider the classical construction of the fixed-point combinator, $Y = \lambda y. (\lambda x. y(x x))(\lambda x. y(x x))$. We…
joseville
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What do "soundness" and "completeness" mean?

Soundness and completeness seem to occur in multiple scenarions: In mathematical logic they are used to describe the relationship between syntax and semantics of logic systems. In relational databases, The Armstrong's axioms are sound in…
Tim
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Equivalence of optimization problems - uniqueness of solution

Let $S(x)=c(x)+Q(x)K := \{c(x) + Q(x)z \mid z \in K \}$ where $K$ is a nonempty closed and convex subset and $c,Q$ continuous. Moreover, let $\phi(x,y) = \langle F(x), y-x \rangle + \frac{\alpha}{2} ||x-y||^2 $ with $F$ continuous. Consider the…
sunflower234
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Mitchell Foundations for PL 2.3.4 (observational equivalence)

Background. The language is PCF, with observable types $\text{bool}$ and $\text{nat}$. $\text{eval}$ is the partial function on PCF terms such that $\text{eval}(M) = N$ iff $N$ is the unique normal form of $M$. observational equivalence: for all…
emesupap
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Associahedron, but with swaps

The associahedron has edges of the form $a(bc)\rightarrow (ab)c.$ But I also want to include the possibility of swapping adjacent entries by doing operations like $a(bc) \rightarrow a(cb).$ I was wondering if there was some type of combinatorial…
Richard Southwell
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Fewest applications of associativity

By repeatedly applying the basic associativity law $(x+y)+z = x+(y+z)$, one can get from any one expression with binary addition to any other with the variables in the same order. Specifically, given two expressions only involving binary addition…
TomKern
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Non-identifiability of a periodic function

This paper (Page 657, Section 2.3) says: It is well known that for a given function there might be more than one representation. For example, a purely harmonic function can also be represented as a function having time varying amplitude and time…
Shanks
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Does $T(\Sigma,V)$, the set of terms for signature $\Sigma$ and a set $V$ of variables, belong to syntax or semantics?

Universal algebra has syntax and semantics parts. A signature $\Sigma$ belongs to syntax. Does $T(\Sigma,V)$, the set of terms for signature $\Sigma$ and a set $V$ of variables, belong to syntax or semantics? In Baader's Term Rewriting Systems…
Tim
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What are the relations and differences between formal systems, rewriting systems, formal grammars and automata?

I learned from Herre & Schroeder-Heister's "Formal Languages and Systems" that A formal system is based on a formal language $L$, endowing it with a consequence operation $C: 2^L\to 2^L$. https://en.wikipedia.org/wiki/Rewriting says In…
Tim
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a locally confluent and terminatting rewrite system is complete

I want to prove that every locally confluent rewrite system is confluent. Since I know very little about rewrite systems and logic, I tried looking at it as a digraph with no external infinite paths and with the locally confluent property but I…
allizdog
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What is this constant 'C' in integration? Why is it different when I integrate using different techniques?

Method 1: Method 2: In these two images, you will see that I have integrated $\sin^3 x$ using different techniques. As you can see I get different answers. I asked my teacher why this is and he said it is because the constants '$C$' are different…
Brilyn Bangura
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