Conditional rewriting logic as a unified model of concurrency

Within the context of an algebraic theory of processes, an equational specification of process cooperation is provided. Four cases are considered: free merge or interleaving, merging with communication, merging with mutual exclusion of tight regions, and synchronous process cooperation. The rewrite system behind the communication algebra is shown to be confluent and terminating (modulo its permutative reductions). Further, some relationships are shown to hold between the four concepts of merging. © 1984 Academic Press, Inc.

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Process algebra for synchronous communication

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

http://types10.mimuw.edu.pl/content/slides/day1/HenkBarendregt.pdf

Lambda Calculus with Types

Combinatory reduction systems, or CRSs for short, were designed to combine the usual first-order format of term rewriting with the presence of bound variables as in pure λ-calculus and various typed λ-calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and left-linear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the well-known confluence proof for λ-calculus by Tait and Martin-Lof. There is a well-known connection between the parallel reduction featuring in the latter proof and the concept of “developments”, and a classical lemma in the theory of λ-calculus is that of “finite developments”, a strong normalization result. It turns out that the notion of “parallel reduction” used in Aczel's proof gives rise to a generalized form of developments which we call “superdevelopments” and on which we will briefly comment.

Combinatory reduction systems

Handbook of logic in computer science.

In this paper we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known ‘confluence by orthogonality’ results.

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Weak Orthogonality Implies Confluence: The Higher Order Case

The Spectra of Words.- On the Undecidability of Coherent Logic.- Lob's Logic Meets the ?-Calculus.- A Characterisation of Weak Bisimulation Congruence.- Bohm's Theorem, Church's Delta, Numeral Systems, and Ershov Morphisms.- Explaining Constraint Programming.- Sharing in the Weak Lambda-Calculus.- Term Rewriting Meets Aspect-Oriented Programming.- Observing Reductions in Nominal Calculi Via a Graphical Encoding of Processes.- Primitive Rewriting.- Infinitary Rewriting: From Syntax to Semantics.- Reducing Right-Hand Sides for Termination.- Reduction Strategies for Left-Linear Term Rewriting Systems.- Higher-Order Rewriting: Framework, Confluence and Termination.- Timing the Untimed: Terminating Successfully While Being Conservative.- Confluence of Graph Transformation Revisited.- Compositional Reasoning for Probabilistic Finite-State Behaviors.- Finite Equational Bases in Process Algebra: Results and Open Questions.- Skew and ?-Skew Confluence and Abstract Bohm Semantics.- A Mobility Calculus with Local and Dependent Types.- Model Theory for Process Algebra.- Expression Reduction Systems and Extensions: An Overview.- Axiomatic Rewriting Theory I: A Diagrammatic Standardization Theorem.

Processes, Terms and Cycles: Steps on the Road to Infinity: Essays Dedicated to Jan Willem Klop on the Occasion of his 60th Birthday (Lecture Notes in Computer Science)

This paper surveys a part of the theory ofs-reduction in?-calculus which might aptly be calledperpetual reductions. The theory is concerned withperpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from?-terms (when possible), and withperpetual redexes, i.e., redexes whose contraction in?-terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in?-calculus and type theory.

Perpetual Reductions in λ-Calculus

In this paper two formats of higher-order rewriting are compared: Combinatory Reduction Systems introduced by Klop and Higher-order Rewrite Systems defined by Nipkow. Although it always has been obvious that both formats are closely related to each other, up to now the exact relationship between them has not been clear. This was an unsatisfying situation since it meant that proofs for much related frameworks were given twice. We present two translations, one from Combinatory Reduction Systems into Higher-Order Rewrite Systems and one vice versa, based on a detailed comparison of both formats. Since the translations are very ‘neat’ in the sense that the rewrite relation is preserved and (almost) reflected, we can conclude that as far as rewrite theory is concerned, Combinatory Reduction Systems and Higher-Order Rewrite Systems are equivalent, the only difference being that Combinatory Reduction Systems employ a more ‘lazy’ evaluation strategy. Moreover, due to this result it is the case that some syntactic properties derived for the one class also hold for the other.

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Femke van Raamsdonk

Papers

Combinatory reduction systems

Weak Orthogonality Implies Confluence: The Higher Order Case

Processes, Terms and Cycles: Steps on the Road to Infinity: Essays Dedicated to Jan Willem Klop on the Occasion of his 60th Birthday (Lecture Notes in Computer Science)

Perpetual Reductions in λ-Calculus

Comparing combinatory reduction systems and higher-order rewrite systems