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

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)

Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings (including this one) [15, 26, 41, 66]; three workshop proceedings [33, 47, 77]; ve special journal issues [5, 88, 24, 40, 67]; more than ten surveys [2, 7, 27, 28, 44, 56, 57, 76, 82, 81]; one edited collection of papers [1]; four monographs [3, 12, 55, 65]; and seven books (four of them still in progress) [8, 9, 35, 54, 60, 75, 84]. To encourage and stimulate continued progress in this area, we have collected (with the help of colleagues) a number of problems that appear to us to be of interest and regarding which we do not know the answer. Questions on rewriting and other equational paradigms have been included; many have not aged su ciently to be accorded the appellation \open problem". We have limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementations. Previous lists of questions in this area include one distributed by Leo Marcus and one of us (Dershowitz) at the Sixth International Conference on Automated Deduction (New York, 1982), the questions posed in a set of lecture notes on \Term Rewriting Systems" by one of us (Klop) for a seminar on reduction machines (Ustica, 1985), another list by one of us (Jouannaud) in the Bulletin of the European Association for Theoretical Computer Science (Number 31, 1987), and electronic postings to the distribution list (rewriting@crin.crin.fr) maintained by Pierre Lescanne. We use primarily terminology and notation of [27].

/pdf/open-problems-in-rewriting-40owgff2o6.pdf

Open Problems in Rewriting

/pdf/descendants-and-origins-in-term-rewriting-36vpnpf0kj.pdf

Descendants and origins in term rewriting

We consider the equational theory λπ of λ-calculus extended with constants π, π0, π1 and axioms for surjective pairing: π0(πXY) = X, π1(πXY) = Y, π(π0X)(π1X) = X. Two reduction systems yielding the equality of λπ are introduced; the first is not confluent and, for the second, confluence is an open problem. It is shown, however, that in both systems each term possessing a normal form has a unique normal form. Some additional properties and problems in the syntactical analysis of λπ and the corresponding reduction systems are discussed.

/pdf/unique-normal-forms-for-lambda-calculus-with-surjective-mwefeldcxp.pdf

Unique normal forms for Lambda calculus with surjective pairing

Consideration is given to the equational theory lambda pi of lambda calculus extended with constants pi , pi /sub 0/, pi /sub 1/ and axioms for subjective pairing: pi /sub 0/( pi XY)=X, pi /sub 1/( pi XY)=Y, pi ( pi /sub 0/X)( pi /sub 1/X)=X. The reduction system that one obtains by reading the equations are reductions (from left to right) is not Church-Rosser. Despite this failure, the author obtains a syntactic consistency proof of lambda pi and shows that it is a conservative extension of the pure lambda calculus. >

R. C. de Vrijer

Papers

Unique normal forms for Lambda calculus with surjective pairing

Extending the lambda calculus with surjective pairing is conservative