engineering

Noncommutative logic

Noncommutative logic is a engineering topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Noncommutative logic rather than just read about it. In short: Noncommutative logic is an extension of linear logic that combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic orders that may be viewed as a species of structure), and the correctness criterion for its proof nets is given in terms of partial permutations.

Key takeaways

Noncommutative logic belongs to engineering; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Noncommutative logic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Noncommutative logic from memory before moving on to harder problems.

Reference excerpt

Noncommutative logic is an extension of linear logic that combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic orders that may be viewed as a species of structure), and the correctness criterion for its proof nets is given in terms of partial permutations. It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras.

Noncommutativity in logic By extension, the term noncommutative logic is also used by a number of authors to refer to a family of substructural logics in which the exchange rule is inadmissible. The remainder of this article is devoted to a presentation of this acceptance of the term. The oldest noncommutative logic is the Lambek calculus, which gave rise to the class of logics known as categorial grammars. Since the publication of Jean-Yves Girard's linear logic there have been several new noncommutative logics proposed, namely the cyclic linear logic of David Yetter, the pomset logic of Christian Retoré, and the noncommutative logics BV and NEL. Noncommutative logic is sometimes called ordered logic, since it is possible with most proposed noncommutative logics to impose a total or partial order on the formulas in sequents. However this is not fully general since some noncommutative logics do not support such an order, such as Yetter's cyclic linear logic. Although most noncommutative logics do not allow weakening or contraction together with noncommutativity, this restriction is not necessary.

The Lambek calculus

Joachim Lambek proposed the first non-commutative logic in his 1958 paper Mathematics of Sentence Structure to model the combinatory possibilities of the syntax of natural languages. In his subsequent 1961 paper On the calculus of syntactic types, he extended the analysis to cover non-associativity as well. His calculus has since become one of the fundamental formalisms of computational linguistics.

Cyclic linear logic David N. Yetter proposed a weaker structural rule in place of the exchange rule of linear logic, yielding cyclic linear logic. Sequents of cyclic linear logic form a cycle, and so are invariant under rotation, where multipremise rules glue their cycles together at the formulas described in the rules. The calculus supports three structural modalities, a self-dual modality allowing exchange, but still linear, and the usual exponentials (? and !) of linear logic, allowing nonlinear structural rules to be used together with exchange.

Pomset logic Pomset logic was proposed by Christian Retoré in a semantic formalism with two dual sequential operators existing together with the usual tensor product and par operators of linear logic, the first logic proposed to have both commutative and noncommutative operators. A sequent calculus for the logic was given, but it lacked a cut-elimination theorem; instead the sense of the calculus was established through a denotational semantics.

BV and NEL Alessio Guglielmi proposed a variation of Retoré's calculus, BV, in which the two noncommutative operations are collapsed onto a single, self-dual, operator, and proposed a novel proof calculus, the calculus of structures to accommodate the calculus. The principal novelty of the calculus of structures was its pervasive use of deep inference, which it was argued is necessary for calculi combining commutative and noncommutative operators; this explanation concurs with the difficulty of designing sequent systems for pomset logic that have cut-elimination. Lutz Straßburger devised a related system, NEL, also in the calculus of structures in which linear logic with the mix rule appears as a subsystem.

See also Ordered type system, a substructural type system Quantum logic

References

External links Non-commutative logic I: the multiplicative fragment by V. Michele Abrusci and Paul Ruet, Annals of Pure and Applied Logic 101(1), 2000. Logical aspects of computational linguistics by Patrick Blackburn, Marc Dymetman, Alain Lecomte, Aarne Ranta, Christian Retoré and Eric Villemonte de la Clergerie. Papers on Commutative/Non-commutative Linear Logic in the calculus of structures: a research homepage from which the papers proposing BV and NEL are available.

Worked examples

Example 1 — a first encounter with Noncommutative logic

Start with the simplest possible case. Write down what Noncommutative logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Noncommutative logic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Noncommutative logic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Noncommutative logic

In research: Noncommutative logic appears in engineering research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Noncommutative logic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Noncommutative logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Substructural logic, so understanding it makes those chapters shorter.
In everyday life: Look for Noncommutative logic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Noncommutative logic in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Noncommutative logic means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Noncommutative logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Noncommutative logic in simple terms?

Why does Noncommutative logic matter?

Because it connects several engineering ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Noncommutative logic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Noncommutative logic.