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### Abstract

A *residuated transition system* (RTS) is
a transition system that is equipped with a certain partial binary
operation, called *residuation*, on transitions.
Using the residuation operation, one can express nuances, such as a
distinction between nondeterministic and concurrent choice, as well as
partial commutativity relationships between transitions, which are not
captured by ordinary transition systems. A version of residuated
transition systems was introduced in previous work by the author, in
which they were called â€śconcurrent transition systemsâ€ť in view of the
original motivation for their definition from the study of
concurrency. In the first part of the present article, we give a
formal development that generalizes and subsumes the original
presentation. We give an axiomatic definition of residuated transition
systems that assumes only a single partial binary operation as given
structure. From the axioms, we derive notions of â€śarrowâ€ś (transition),
â€śsourceâ€ť, â€śtargetâ€ť, â€śidentityâ€ť, as well as â€ścompositionâ€ť and â€śjoinâ€ť of
transitions; thereby recovering structure that in the previous work
was assumed as given. We formalize and generalize the result, that
residuation extends from transitions to transition paths, and we
systematically develop the properties of this extension. A significant
generalization made in the present work is the identification of a
general notion of congruence on RTSâ€™s, along with an associated
quotient construction.

In the second part of this article, we use the RTS framework to formalize several results in the theory of reduction in Churchâ€™s Î»-calculus. Using a de Bruijn index-based syntax in which terms represent parallel reduction steps, we define residuation on terms and show that it satisfies the axioms for an RTS. An application of the results on paths from the first part of the article allows us to prove the classical Church-Rosser Theorem with little additional effort. We then use residuation to define the notion of â€śdevelopmentâ€ť and we prove the Finite Developments Theorem, that every development is finite, formalizing and adapting to de Bruijn indices a proof by de Vrijer. We also use residuation to define the notion of a â€śstandard reduction pathâ€ť, and we prove the Standardization Theorem: that every reduction path is congruent to a standard one. As a corollary of the Standardization Theorem, we obtain the Leftmost Reduction Theorem: that leftmost reduction is a normalizing strategy.

### Topics

- Computer science/Automata and formal languages
- Computer science/Concurrency
- Computer science/Programming languages/Lambda calculi