Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems.

One of the best-known methods for discriminating λ-terms with respect to β-convertibility is due to Corrado Böhm. The idea is to compute the infinitary normal form of a λ-term M, the Böhm Tree (BT) of M. If λ-terms M, N have distinct BTs, then M ≠βN, that is, M and N are not β-convertible. But what if their BTs coincide? For example, all fixed point combinators (FPCs) have the same BT, namely λx.x(x(x(. . .))). We introduce a clocked λ-calculus, an extension of the classical λ-calculus with a unary symbol τ used to witness the β-steps needed in the normalization to the BT. This extension is infinitary strongly normalizing, infinitary confluent and the unique infinitary normal forms constitute enriched BTs, which we call clocked BTs. These are suitable for discriminating a rich class of λ-terms having the same BTs, including the well-known sequence of Böhm's FPCs. We further increase the discrimination power in two directions. First, by a refinement of the calculus: the atomic clocked λ-calculus, where we employ symbols τp that also witness the (relative) positions p of the β-steps. Second, by employing a localized version of the (atomic) clocked BTs that has even more discriminating power.

We consider infinite sequences of symbols, also known as streams, and the decidability question for equality of streams defined in a restricted format. (Some formats lead to undecidable equivalence problems.) This restricted format consists of prefixing a symbol at the head of a stream, of the stream function `zip', and recursion variables. Here `zip' interleaves the elements of two streams alternatingly. The celebrated Thue- Morse sequence is obtained by the succinct `zip-specification' M = 0 : X X = 1 : zip(X, Y) Y = 0 : zip(Y, X) The main results are as follows. We establish decidability of equivalence of zip-specifications, by employing bisimilarity of observation graphs based on a suitably chosen cobasis. Furthermore, our analysis, based on term rewriting and coalgebraic techniques, reveals an intimate connection between zip-specifications and automatic sequences. This leads to a new and simple characterization of automatic sequences. The study of zip-specifications is placed in a wider perspective by employing observation graphs in a dynamic logic setting, yielding yet another alternative characterization of automatic sequences. By the first characterization result, zip-specifications can be perceived as a term rewriting syntax for automatic sequences. For streams σ the following are equivalent: (a) σ can be specified using zip; (b) σ is 2-automatic; and (c) σ has a finite observation graph using the cobasis (hd, even, odd). Here even and odd are defined by even(a : s) = a : odd(s), and odd(a : s) = even(s). The generalization to zip-k specifications (with zip-k interleaving k streams) and to k-automaticity is straightforward. As a natural extension of the class of automatic sequences, we also consider `zip-mix' specifications that use zips of different arities in one specification. The corresponding notion of automaton employs a state-dependent input-alphabet, with a number representation (n)A = dm... d0where the base of digit di is determined by the automaton A on input di-1... d0. Finally we show that equivalence is undecidable for a simple extension of the zip-mix format with projections analogous to even and odd.

Fixed point combinators (and their generalization: looping combinators) are classic notions belonging to the heart of λ-calculus and logic. We start with an exploration of the structure of fixed point combinators (fpc's), vastly generalizing the wellknown fact that if Yis an fpc, Y(SI) is again an fpc, generating the Böhm sequence of fpc's. Using the infinitary λ-calculus we devise infinitely many other generation schemes for fpc's. In this way we find schemes and building blocks to construct new fpc's in a modular way. Having created a plethora of new fixed point combinators, the task is to prove that they are indeed new. That is, we have to prove their β-inconvertibility. Known techniques via Böhm Trees do not apply, because all fpc's have the same Böhm Tree (BT). Therefore, we employ 'clocked BT's', with annotations that convey information of the tempo in which the data in the BT are produced. BT's are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for λ-terms. The corresponding equality is strictly intermediate between =βand =BT, the equality in the classical models of λ-calculus. An analogous approach pertains to Lévy-Longo and Berarducci trees. Finally, we increase the discrimination power by a precision of the clock notion that we call 'atomic clock'.

We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property (UNinf) fails by a simple example of a weakly orthogonal TRS with two collapsing rules. By translating this example, we show that UNinf also fails for the infinitary lambda-beta-eta-calculus. As positive results we obtain the following: Infinitary confluence, and hence UNinf, holds for weakly orthogonal TRSs that do not contain collapsing rules. To this end we refine the compression lemma. Furthermore, we consider the triangle and diamond properties for infinitary developments in weakly orthogonal TRSs, by refining an earlier cluster-analysis for the finite case.

Journal Article Editorial Get access F Kamareddine, F Kamareddine Search for other works by this author on: Oxford Academic Google Scholar JW Klop JW Klop Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 10, Issue 3, June 2000, Pages 321–322, https://doi.org/10.1093/logcom/10.3.321 Published: 01 June 2000

The criterion for confluence using decreasing diagrams is a generalization of several well-known confluence criteria in abstract rewriting, such as the strong confluence lemma. We give a new proof of the decreasing diagram theorem based on a geometric study of infinite reduction diagrams, arising from unsuccessful attempts to obtain a confluent diagram by tiling with elementary diagrams.

In this note, we prove that having unique head-normal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CL-terms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras.

We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the μ-rule. Next, orthogonal term graph rewrite systems, also in the presence of copying and hidden nodes, are shown to be confluent.

A term rewriting system is called complete if it is confluent and terminating.We prove that completeness of TRSS is a "modular" property (meaning that it stays preserved under direct sums), provided the constituent TRSS are left-linear.Here, the direct sum RO S3R ~is the union of TRSS R., RI with disjoint signature.The proof hinges crucially upon the (non)deterministic collapsing behavior of terms from the sum TRS.

Studies cyclic /spl lambda/-graphs. The starting point is to treat a /spl lambda/-graph as a system of recursion equations involving /spl lambda/-terms, and to manipulate such systems in an unrestricted manner, using equational logic, just as is possible for first-order term rewriting. Surprisingly, now the confluence property breaks down in an essential way. Confluence can be restored by introducing a restraining mechanism on the 'copying' operation. This leads to a family of /spl lambda/-graph calculi, which are inspired by the family of /spl lambdaspl sigma/-calculi (/spl lambda/-calculi with explicit substitution). However, these concern acyclic expressions only. In this paper we are not concerned with optimality questions for acyclic /spl lambda/-reduction. We also indicate how Wadsworth's (1978) interpreter can be simulated in the /spl lambda/-graph rewrite rules that we propose.>

Several authors have investigatedthe correspondence between graph rewriting and term rewriting.Almost invariably they have considered only acyclic graphs.Yet cyclic graphs naturally arise from certain optimizations in implementing functional languages.They correspond to infinite terms, and their reductions correspond to transfinite term-reduction sequences, which have recently received detailed attention.We formalize the close correspondence between finitary cyclic graph rewriting and a restricted form of infinitary term rewriting, called rational term rewriting.This subsumes the known relation between finitary acyclic graph rewriting and finitary term rewriting.Surprisingly, the correspondence breaks down for general infinitary rewriting.We present an example showing that infinitary term rewriting is strictly more powerful than infinitary graph .

article Free AccessDecidability of bisimulation equivalence for process generating context-free languages Authors: J. C. M. Baeten Univ. of Amsterdam, Amsterdam, The Netherlands Univ. of Amsterdam, Amsterdam, The NetherlandsView Profile , J. A. Bergstra Univ. of Amsterdam, Amsterdam, The Netherlands; and State Univ. of Utrecht, Utrecht, The Netherlands Univ. of Amsterdam, Amsterdam, The Netherlands; and State Univ. of Utrecht, Utrecht, The NetherlandsView Profile , J. W. Klop CWI, Amsterdam, The Netherlands; and Free Univ., Amsterdam, The Netherlands CWI, Amsterdam, The Netherlands; and Free Univ., Amsterdam, The NetherlandsView Profile Authors Info & Claims Journal of the ACMVolume 40Issue 3July 1993 pp 653–682https://doi.org/10.1145/174130.174141Published:01 July 1993Publication History 96citation724DownloadsMetricsTotal Citations96Total Downloads724Last 12 Months22Last 6 weeks7 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

The authors study the paradigm of asynchronous process communication, as contrasted with the synchronous communication mechanism that is present in process algebra frameworks such as CCS, CSP, and ACP. They investigate semantics and axiomatizations with respect to various observability criteria: bisimulation, traces and abstract traces. The aim is to develop a process theory that can be regarded as a kernel for languages based on asynchronous communication, like data flow, concurrent logic languages, and concurrent constraint programming.>

Readiness and failure semantics are studied in the setting of Algebra of Communicating Processes (ACP). A model of process graphs modulo readiness equivalence, respectively, failure equivalence, is constructed, and an equational axiom system is presented which is complete for this graph model. An explicit representation of the graph model is given, the failure model, whose elements are failure sets. Furthermore, a characterisation of failure equivalence is obtained as the maximal congruence which is consistent with trace semantics. By suitably restricting the communication format in ACP, this result is shown to carry over to subsets of Hoare’s Communicating Sequential Processes (CSP) and Milner’s Calculus of Communicating Systems (CCS). Also, the characterisation implies a full abstraction result for the failure model. In the above we restrict ourselves to finite processes without $\tau $-steps. At the end of the paper a comment is made on the situation for infinite processes with $\tau $-steps: notably we obtain that failure semantics is incompatible with Koomen’s fair abstraction rule, a proof principle based on the notion of bisimulation. This is remarkable because a weaker version of Koomen’s fair abstraction rule is consistent with (finite) failure semantics.

We consider a process semantics intermediate between bi-simulation semantics and readiness semantics, called here ready-trace semantics. The advantage of this semantics is that, while retaining the simplicity of readiness semantics, it is still possible to augment this process model with the mechanism of atomic actions with priority (the θ operator). It is shown that in readiness semantics and a fortiori in failure semantics such an extension with θ is impossible. Ready-trace semantics is considered here in the simple setting of concrete process algebra, that is: without abstraction (no silent moves), moreover for finite processes only. For such finite processes without silent moves a complete axiomatisation of ready-trace semantics is given via the method of process graph transformations.

Summary A λ-theory T is a consistent set of equations between λ-terms closed under derivability. The degree of T is the degree of the set of Gödel numbers of its elements. is the λ-theory axiomatized by the set {M = N∣ M, N unsolvable}. A λ-theory is sensible iff T ⊃ ; for a motivation see [6] and [4]. In §1 it is proved that the theory is Σ20-complete. We present Wadsworth's proof that its unique maximal consistent extension * (= Th(D∞)) is Π20-complete. In §2 it is proved that η (= λη-calculus + ) is not closed under the ω-rule (see [1]). In §3 arguments are given to conjecture that is Π11-complete. This is done by representing recursive sets of sequence numbers as λ-terms and by connecting wellfoundedness of trees with provability in ω. In §4 an infinite set of equations independent over η will be constructed. From this it follows that there are 2ℵ0 sensible theories T such that and 2ℵ0 sensible hard models of arbitrarily high degrees. In §5 some nonprovability results needed in §§1 and 2 are established. For this purpose one uses the theory η extended with a reduction relation for which the Church–Rosser theorem holds. The concept of Gross reduction is used in order to show that certain terms have no common reduct.