The linear-algebraic lambda-calculus and the algebraic lambda-calculus are untyped lambda-calculi extended with arbitrary linear combinations of terms. The former presents the axioms of linear algebra in the form of a rewrite system, while the latter uses equalities. When given by rewrites, algebraic lambda-calculi are not confluent unless further restrictions are added. We provide a type system for the linear-algebraic lambda-calculus enforcing strong normalisation, which gives back confluence. The type system allows an abstract interpretation in System F.
The confluence of untyped λ-calculus with unconditional rewriting is now well un- derstood. In this paper, we investigate the confluence of λ-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Müller and Dougherty for unconditional rewriting. Two cases are considered, whether β-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing β-reduction to weakly normalizing β-reduction. We also provide examples sh…
We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critical pairs and it also (partially) works even if the system is not terminating modulo E. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into a termina…
In this paper we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorically as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewrite systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure…
On the topic of probabilistic rewriting, there are several works studying both termination and confluence of different systems. While working with a lambda calculus modelling quantum computation, we found a system with probabilistic rewriting rules and strongly normalizing terms. We examine the effect of small modifications in probabilistic rewriting, affine variables, and strategies on the overall confluence in this strongly normalizing probabilistic calculus.
Conditional term rewriting is an intuitive yet complex extension of term rewriting. In order to benefit from the simpler framework of unconditional rewriting, transformations have been defined to eliminate the conditions of conditional term rewrite systems. Recent results provide confluence criteria for conditional term rewrite systems via transformations, yet they are restricted to CTRSs with certain syntactic properties like weak left-linearity. These syntactic properties imply that the transformations are sound for the given CTRS. This paper shows how to use transformations to prove conflue…
We study rewriting systems whose underlying set of terms is equipped with a vector space structure over a given field. We introduce parallel rewriting relations, which are rewriting relations compatible with the vector space structure, as well as rewriting strategies, which consist in choosing one rewriting step for each reducible basis element of the vector space. Using these notions, we introduce the S-confluence property and show that it implies confluence. We deduce a proof of the diamond's lemma, based on strategies. We illustrate our general framework with rewriting systems over rational…
The confluence of untyped lambda-calculus with unconditional rewriting has already been studied in various directions. In this paper, we investigate the confluence of lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Muller and Dougherty for unconditional rewriting. Two cases are considered, whether beta-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing beta-reduction to weakly normalizing be…
The notion of normal forms is ubiquitous in various equivalent transformations. Confluence (CR), one of the central properties of term rewriting systems (TRSs), concerns uniqueness of normal forms. Yet another such property, which is weaker than confluence, is the property of unique normal forms w.r.t. conversion (UNC). Famous examples having UNC but not CR include the TRSs consisting of S,K,I-rules for the combinatory logic supplemented with various pairing rules (de Vrijer, 1999). Recently, automated confluence proof of TRSs has caught attentions leading to investigations of automatable meth…
In this note we give a simple unifying proof of the undecidability of several diagrammatic properties of term rewriting systems that include: local confluence, strong confluence, diamond property, subcommutative property, and the existence of successor. The idea is to code configurations of Turing Machines into terms, and then define a suitable relation on those terms such that the termination of the Turing Machine becomes equivalent to the satisfiability of the diagrammatic property.
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of unicity of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence which is shown to imply the desired uniqueness (even for infinite sequences of reduction) and further properties. We then carry over several criteria from the classical case, such as Newman's lemma, to simplify proving confluence in concrete languages. Using these criteria, we obtain simple proofs of confluence for $λ_1$, an affine probabilistic $λ$-calculus, a…
We study the combination of the following already known ideas for showing confluence of unconditional or conditional term rewriting systems into practically more useful confluence criteria for conditional systems: Our syntactical separation into constructor and non-constructor symbols, Huet's introduction and Toyama's generalization of parallel closedness for non-noetherian unconditional systems, the use of shallow confluence for proving confluence of noetherian and non-noetherian conditional systems, the idea that certain kinds of limited confluence can be assumed for checking the fulfilledne…
We show confluence of a conditional term rewriting system CL-pc${}^1$, which is an extension of Combinatory Logic by Boolean constants. This solves problem 15 from the RTA list of open problems. The proof has been fully formalised in the Coq proof assistant.
On Complexity Bounds and Confluence of Parallel Term Rewriting
LIP (UMR CNRS/ENS Lyon/UCB Lyon1/Inria)（LIP（法国国家科学研究中心/里昂高等师范学院/里昂第一大学/法国国家信息与自动化研究所联合研究单位））；Birkbeck, University of London（伦敦大学伯贝克学院）；University Grenoble Alpes（格勒诺布尔阿尔卑斯大学）
Confluence in abstract parallel category systems is established for net class-rewriting in iterative closed multilevel quotient graph structures with uncountable node arities by multi-dimensional transducer operations in topological metrics defined by alphabetically abstracting net block homomorphism. We obtain minimum prerequisites for the comprehensive connector pairs in a multitude dimensional rewriting closure generating confluence in Participatory algebra for different horizontal and vertical level projections modulo abstraction relations constituting formal semantics for confluence in in…
The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduction or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo. However, they hold if the rewrite system generated by the rewrite rules together with beta-reduction is confluent. But this is too restrictive. To handle the case where non confluence comes from the interference bet…
We study confluence in the setting of higher-order infinitary rewriting, in particular for infinitary Combinatory Reduction Systems (iCRSs). We prove that fully-extended, orthogonal iCRSs are confluent modulo identification of hypercollapsing subterms. As a corollary, we obtain that fully-extended, orthogonal iCRSs have the normal form property and the unique normal form property (with respect to reduction). We also show that, unlike the case in first-order infinitary rewriting, almost non-collapsing iCRSs are not necessarily confluent.
Orthogonality is a discipline of programming that in a syntactic manner guarantees determinism of functional specifications. Essentially, orthogonality avoids, on the one side, the inherent ambiguity of non determinism, prohibiting the existence of different rules that specify the same function and that may apply simultaneously (non-ambiguity), and, on the other side, it eliminates the possibility of occurrence of repetitions of variables in the left-hand side of these rules (left linearity). In the theory of term rewriting systems (TRSs) determinism is captured by the well-known property of c…
In this paper we examine a number of term rewriting system for integer number representations, building further upon the datatype defining systems described in [2]. In particular, we look at automated methods for proving confluence and termination in binary and decimal term rewriting systems for both append and tree constructor functions. We find that some of these term rewriting systems are not strongly terminating, which we resolve with minor changes to these systems. Moreover, most of the term rewriting systems discussed do not exhibit the confluence property, which seems more difficult to …
We show that left-linear generalized semi-monadic TRSs effectively preserving recognizability of finite tree languages (are EPRF-TRSs). We show that reachability, joinability, and local confluence are decidable for EPRF-TRSs.
Inspired by a recent graphical formalism for lambda-calculus based on linear logic technology, we introduce an untyped structural lambda-calculus, called lambda j, which combines actions at a distance with exponential rules decomposing the substitution by means of weakening, contraction and derelicition. First, we prove some fundamental properties of lambda j such as confluence and preservation of beta-strong normalisation. Second, we add a strong bisimulation to lambda j by means of an equational theory which captures in particular Regnier's sigma-equivalence. We then complete this bisimulati…
Encodings of term rewriting systems (TRSs) into graph rewriting systems usually lose global termination, meaning the encodings do not terminate on all graphs. A typical encoding of the terminating TRS rule a(b(x)) -> b(a(x)), for example, may be indefinitely applicable along a cycle of a's and b's. Recently, we introduced PBPO+, a graph rewriting formalism in which rules employ a type graph to specify transformations and control rule applicability. In the present paper, we show that PBPO+ allows for a natural encoding of linear TRS rules that preserves termination globally. This result is a st…
The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. We prove the strong normalization of the reduction relation generated by the beta-rule and the user-defined rules under some general syntactic conditions including confluence. As examples, we show that two important systems satisfy these conditions: a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq, and the Natural Deduction Modulo a large class of equational theories.
Term rewriting is a Turing complete model of computation. When taught to students of computer science, key properties of computation as well as techniques to analyze programs on an abstract level are conveyed. This paper gives a swift introduction to term rewriting and presents several automatic tools to analyze term rewrite systems which were developed by the Computational Logic Group at the University of Innsbruck. These include the termination tool TTT2, the confluence prover CSI, the completion tools mkbTT and KBCV, the complexity tool TcT, the strategy tool AutoStrat, as well as FORT, an …
In the last twenty years, several approaches to higher-order rewriting have been proposed, among which Klop's Combinatory Rewrite Systems (CRSs), Nipkow's Higher-order Rewrite Systems (HRSs) and Jouannaud and Okada's higher-order algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higher-order pattern-matching mechanism, resulting in simply-typed CRSs. Then, we show how the termination criterion de…
The theory of finite and infinitary term rewriting is extensively developed for orthogonal rewrite systems, but to a lesser degree for weakly orthogonal rewrite systems. In this note we present some contributions to the latter case of weak orthogonality, where critial pairs are admitted provided they are trivial. We start with a refinement of the by now classical Compression Lemma, as a tool for establishing infinitary confluence, and hence the property of unique infinitary normal forms, for the case of weakly orthogonal TRSs that do not contain collapsing rewrite rules. That this restriction …
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西米昂·斯托伊洛夫数学研究所）
We develop a general model theoretic semantics to rewriting beyond the usual confluence and termination assumptions. This is based on preordered algebra which is a model theory that extends many sorted algebra. In this framework we characterise rewriting in arbitrary algebras rather than term algebras (called algebraic rewriting) as a persistent adjunction and use this result, on the one hand for proving the soundness and the completeness of an abstract computational model of rewriting that underlies the non-deterministic programming with Maude and CafeOBJ, and on the other hand for developing…
Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms an…
Adding rewriting to a proof assistant based on the Curry-Howard isomorphism, such as Coq, may greatly improve usability of the tool. Unfortunately adding an arbitrary set of rewrite rules may render the underlying formal system undecidable and inconsistent. While ways to ensure termination and confluence, and hence decidability of type-checking, have already been studied to some extent, logical consistency has got little attention so far. In this paper we show that consistency is a consequence of canonicity, which in turn follows from the assumption that all functions defined by rewrite rules …
We tackle the problem of simultaneous transformations of networks represented as graphs. Roughly speaking, one may distinguish two kinds of simultaneous or parallel rewrite relations over complex structures such as graphs: (i) those which transform disjoint subgraphs in parallel and hence can be simulated by successive mere sequential and local transformations and (ii) those which transform overlapping subgraphs simultaneously. In the latter situations, parallel transformations cannot be simulated in general by means of successive local rewrite steps. We investigate this last problem in the fr…
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities betwee…
A key result in the theory of the modal mu-calculus is the disjunctive normal form theorem by Janin & Walukiewicz, stating that every mu-calculus formula is semantically equivalent to a so-called disjunctive formula. These disjunctive formulas have good computational properties and play a pivotal role in the theory of the modal mu-calculus. It is therefore an interesting question what the best normalisation procedure is for rewriting a formula into an equivalent disjunctive formula of minimal size. The best constructions that are known from the literature are automata-theoretic in nature and c…
This thesis is devoted to the study of a calculus that describes the application of conditional rewriting rules and the obtained results at the same level of representation. We introduce the rewriting calculus, also called the rho-calculus, which generalizes the first order term rewriting and lambda-calculus, and makes possible the representation of the non-determinism. In our approach the abstraction operator as well as the application operator are objects of calculus. The result of a reduction in the rewriting calculus is either an empty set representing the application failure, or a singlet…
We examine the relationship between the algebraic lambda-calculus, a fragment of the differential lambda-calculus and the linear-algebraic lambda-calculus, a candidate lambda-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and their set of terms is closed under linear combinations. However, the two languages were built using different approaches: the former is a call-by-name language whereas the latter is call-by-value; the former considers algebraic equalities whereas the latter approaches them through …
We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus with a linear negation of "trivialised" De Morgan duality. Reduction is realised through explicit substitution, based on a symmetric notion of binding of global scope, with rules acting on the entire typing judgement instead of on a specific subterm. Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and the language is shown to be a…
Factorization -- a simple form of standardization -- is concerned with reduction strategies, i.e. how a result is computed. We present a new technique for proving factorization theorems for compound rewriting systems in a modular way, which is inspired by the Hindley-Rosen technique for confluence. Specifically, our technique is well adapted to deal with extensions of the call-by-name and call-by-value lambda-calculi. The technique is first developed abstractly. We isolate a sufficient condition (called linear swap) for lifting factorization from components to the compound system, and which is…
The infinitary lambda calculi pioneered by Kennaway et al. extend the basic lambda calculus by metric completion to infinite terms and reductions. Depending on the chosen metric, the resulting infinitary calculi exhibit different notions of strictness. To obtain infinitary normalisation and infinitary confluence properties for these calculi, Kennaway et al. extend $β$-reduction with infinitely many `$\bot$-rules', which contract meaningless terms directly to $\bot$. Three of the resulting Böhm reduction calculi have unique infinitary normal forms corresponding to Böhm-like trees. In this paper…
This volume contains a final and revised selection of papers presented at the Seventh Workshop on Intersection Types and Related Systems (ITRS 2014), held in Vienna (Austria) on July 18th, affiliated with TLCA 2014, Typed Lambda Calculi and Applications (held jointly with RTA, Rewriting Techniques and Applications) as part of FLoC and the Vienna Summer of Logic (VSL) 2014. Intersection types have been introduced in the late 1970s as a language for describing properties of lambda calculus which were not captured by all previous type systems. They provided the first characterisation of strongly …
