Church-Rosser Property Research Articles

Remarks on the Church-Rosser Property

AbstractA reduction algebra is defined as a set with a collection of partial unary functions (called reduction operators). Motivated by the lambda calculus, the Church-Rosser property is defined for a reduction algebra and a characterization is given for those reduction algebras satisfying CRP and having a measure respecting the reductions. The characterization is used to give (with 20/20 hindsight) a more direct proof of the strong normalization theorem for the impredicative second order intuitionistic propositional calculus.

A Rewriting System for Categorical Combinators with Multiple Arguments

Categorical combinators have been derived from the study of categorical semantics of lambda calculus, and it has been found that they may be used in implementation of functional languages. In this paper categorical combinators are extended so that functions with multiple arguments can be directly handled, thus making them more suitable for practical computation. A rewriting system named ${\operatorname{CCLM}}\beta $ is formulated for these combinators. This system naturally corresponds to the type-free $\lambda \beta $-calculus. The relationship between these two systems is established, and as a result of this, the Church–Rosser property of ${\operatorname{CCLM}}\beta $ is proved. A similar relationship is also established between the original ${\operatorname{CCL}}\beta $ by Curien and the type-free $\lambda \beta $-calculus with product. Finally the embedding theorem of ${\operatorname{CCLM}}\beta $ into ${\operatorname{CCL}}\beta $ is shown.

Jean-Yves Girard. Proof theory and logical complexity. Volume I. Studies in proof theory, no. 1. Bibliopolis, Naples 1987, also distributed by Humanities Press, Atlantic Highlands, N.J., 503 pp.

Introduction: Elementary Proof Theory. The Fall of Hilbert's Program. Hilbert's Program. Recursive Functions. The First Incompleteness Theorem. The Second Incompleteness Theorem. Exercises. Annex: Intuitionism. Part I: Sigma 0 1 Proof Theory. The Calculus of Sequents. Definitions. Completeness of the Sequent Calculus. The Cut-Elimination Theorem. The Subformula Property. Intuitionistic Sequent Calculus. Herbrand's Theorem. Generalization. Annex: Natural Deduction. The Church-Rosser Property. Strong Normalization. The Semantics of Sequent Calculus. Completeness of the Cut-Free Rules. Three-Valued Models. Three-Valued Logic. Annex: Takeuti's Conjecture. Limitations of Takeuti's Conjecture. Three-Valued Equivalence. Cut-Free Analysis. Three-Valued Semantics and Generalized Logics. Applications of the ``Hauptsatz''. The Interpolation Lemma. The Reflection Schema of PA. Elementary Consistency Proofs. 1-Consistency. Annex: The Hauptsatz in a Concrete Case. Normalization in HA. Normalization for NL 2 J. Part II: Pi 1 1 Proof Theory. Pi 1 1 Formulas and Well-Foundedness. The Projective Hierarchy. Well-Founded Trees. Well-Orders. Equivalents of (Sigma 0 1 -CA * ). Recursive Well-Orders. Hyperarithmetical Sets. Annex: Kleene's 0. Hierarchies Indexed by 0. Paths Through 0. The Classification Problem. The omega-Rule. omega-Logic. The Cut-Elimination Theorem. Bounds for Cut-Elimination. Equivalents for (Sigma 0 1 -CA * ). Annex: The Calculus Lomega 1 omega. Cut-Elimination in Lomega 1 omega. The Ordinal epsilon o and Arithmetic. Ordinal Analysis of PA. Extensions to other Systems. Ordinals and Theories. Annex: Godel's System T. Functional Interpretation. Spector's Interpretation. No Conterexample Interpretation. An Application. Bibliography. Analytical Index.

A typed calculus based on a fragment of linear logic

Linear Logic, we concisely write LL, has been introduced recently by Jean Yves Girard in Theoretical Computer Science 50 (1987). Born from the semantics of second order lambda calculus, LL is more expressive than traditional logic (both classical and intuitionistic). Due to the absence of structural rules and to a particular treatment of negation, which is denoted by ⊥, proofs in LL do not have a “directional character”. The constructive meaning of a proof of A→ B is a function mapping all proofs of A into proofs of B; in LL A⊸ B has a similar meaning, but B ⊥⊸ A ⊥ represents the same formula and has the same proofs: so one of such proofs can map a proof of A into one of B as well as a proof of B ⊥ into one of A ⊥. In this paper we are interested in the multiplicative second order subsystem LL ∗ of linear logic; we introduce a calculus (called v-calculus) whose terms are canonical representations of proofs. The aim of the calculus is to give a better comprehension of the computational aspects of the process of cut-elimination. We prove that the v-calculus obeys strong normalization and the Church-Rosser properties.

Only prime superpositions need be considered in the Knuth-Bendix completion procedure

The Knuth and Bendix test for local confluence of a term rewriting system involves generating superpositions of the left-hand sides, and for each superposition deriving a critical pair of terms and checking whether these terms reduce to the same term. We prove that certain superpositions, which are called composite because they can be split into other superpositions, do not have to be subjected to the critical-pair-joinability test; it suffices to consider only prime superpositions. As a corollary, this result settles a conjecture of Lankford that unblocked superpositions can be omitted. To prove the result, we introduce new concepts and proof techniques which appear useful for other proofs relating to the Church-Rosser property. This test has been implemented in the completion procedures for ordinary term rewriting systems as well as term rewriting systems with associative-commutative operators. Performance of the completion procedures with this test and without the test are compared on a number of examples in the Rewrite Rule Laboratory (RRL) being developed at General Electric Research and Development Center.

Church-Rosser Thue systems and formal languages

Since about 1971, much research has been done on Thue systems that have properties that ensure viable and efficient computation. The strongest of these is the Church-Rosser property, which states that two equivalent strings can each be brought to a unique canonical form by a sequence of length-reducing rules. In this paper three ways in which formal languages can be defined by Thue systems with this property are studied, and some general results about the three families of languages so determined are studied.

Bisimulations and abstraction homomorphisms

We show that the notion of bisimulation equivalence for a class of labelled transition systems (the class of nondeterministic processes) may be restated as one of “reducibility to a same system” via a simple reduction relation. This relation is proved to enjoy some desirable properties, notably the Church-Rosser property. We also show that, when restricted to finite nondeterministic processes, the relation yields unique minimal forms for processes and can be characterised algebraically by a set of reduction rules.

Thue systems as rewriting systems

This paper is a survey of recent results on Thue systems, where the systems are viewed as rewriting systems on strings over a finite alphabet. The emphasis is on Thue systems with the Church-Rosser property, where the notion of reduction is based on length-decreasing rewriting rules. The main effort is expended in outlining the properties of such systems, paying close attention to the issue of the possible decidability of properties and to the issue of the computational complexity of decidable properties. Since Thue systems may also be considered as presentations of monoids, the language of monoids (and groups) is used to describe some of these properties.

The Church-Rosser property for ground term-rewriting systems is decidable

A term-rewriting system is said to be a ground system if no variable occurs in any of its rewrite rules. This paper shows that the Church-Rosser Property (i.e., confluence) problem for ground term-rewriting systems is decidable. This is an affirmative solution to this problem posed by Huet and Oppen (1980).

Church-Rosser property and decidability of monadic theories of unary algebras

Church-Rosser property and decidability of monadic theories of unary algebras

String grammars with disconnecting or a basic root of the difficulty in graph grammar parsing

The complexity of languages generated by so-called context-free string grammars with disconnecting is investigated. The result is then applied to a number of graph grammar models with finite Church Rosser property. In particular, it is shown that these graph grammars can generate NP-complete languages.

On the Church-Rosser property for the direct sum of term rewriting systems

The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that if two term rewriting systems both have the Chruch-Rosser property, then the direct sum of these systems also has this property.

Completion of a Set of Rules Modulo a Set of Equations

Abstract Church-Rosser properties are first presented, depending on an arbitrary relation R, an equivalence relation E and a reduction relation $R^E $ used to compute normal forms of R modulo E. Terminating rewriting systems operating on equational congruence classes of terms of a free algebra are then considered. In this framework, the Church–Rosser property is proved decidable for a very general reduction relation which may take into account the left-linearity of rules for efficiency reasons, under the only assumption of existence of a complete and finite unification algorithm for the underlying equational theory, whose congruence classes are assumed to be finite. This extends previous results by Lankford and Ballantyne, Peterson and Stickel, Huet, Jouannaud. A general completion procedure for mixed sets of rules and equations is then presented that generalizes and improves Peterson and Stickel’s one. In addition to computing a Church–Rosser set of rules when it terminates, it yields a semi-decision pro...

Neighborhood-uniform NLC grammars

This paper investigates the basic properties of the class of neighborhood-uniform nodel label controlled (NUNLC) graph grammars. The class of NUNLC grammars is distinguished by requiring a very natural restriction on the connection relations of NLC grammars. The restriction implies “Church-Rosser property” of derivations in an NUNLC grammar, which makes the class of NUNLC grammars “technically easier” to investigate. A number of combinatorial properties of the languages generated by the class of NUNLC grammars are proved. Also, it is demonstrated that a number of basic properties are decidable for the class of NUNLC grammars—many of them are undecidable in the whole class of NLC grammars.

Church-Rosser theorem for typed functional systems

Introduction. This paper contains a new proof of the Church-Rosser theorem for the typed λ-calculus, which also applies to systems with infinitely long terms.The ordinary proof of the Church-Rosser theorem for the general untyped calculus goes as follows (see [1]). If is the binary reduction relation between the terms we define the one-step reduction 1 in such a way that the following lemma is valid.Lemma. For all terms a and b we have: ab if and only if there is a sequence a = a0, …, an = b, n ≥ 0, such that aiiai + 1for 0 ≤ i < n.We then prove the Church-Rosser property for the relation 1 by induction on the length of the reductions. And by combining this result with the above lemma we obtain the Church-Rosser theorem for the relation .Unfortunately when we come to infinite terms the above lemma is not valid anymore. The difficulty is that, assuming the hypothesis for the infinitely many premises of the infinite rule, there may not exist an upper bound for the lengths n of the sequences ai = a0, …, an = bi (i < α); cf. the infinite rule (iv) in §6.A completely new idea in the case of the typed λ-calculus would be to exploit the type structure in the way Tait did in order to prove the normalization theorem. In this we succeed by defining a suitable predicate, the monovaluedness predicate, defined over the type structure and having some nice properties. The key notion permitting to define this predicate is the notion of I-form term (see below). This Tait-type proof has a merit, namely that it can be extended immediately to the case of infinite terms.

Cancellation rules and extended word problems

Generalizations of the extended word problems of Dolev and Yao (1983) are investigated in the context of Thue systems with the Church-Rosser property.

A graph grammar approach to geographical databases

This paper treats the modeling of an important class of databases, i.e. geographical databases, with emphasis on both structural (data definition) and behavioral (data manipulation) aspects. Geometric objects such as polygons, line segments, and points may have different relations among each other (such as order, adjacency, connectivity) and can be represented in a uniform spatial data structure (structure graph). The dynamic behavior is defined by a finite set of consistency-preserving state transitions (productions) where coincidence problems as well as topological properties have to be solved. Moreover, the graph grammar approach can be used to study the synchronization of several concurrent productions (Church-Rosser properties) and offers a framework for implementing a geographical database.

The Church-Rosser property and special Thue systems

In an earlier paper we gave an O(| T| 3) algorithm for testing the Church-Rosser property of Thue systems, where | T| is the total size of the Thue system. Here we improve that bound to O( κ| T|), where κ is the number of rules in T, in the case when the Thue system is special, i.e., when all its rule are of the form (ξ, λ) where λ is the empty string. Also obtained are several results on special Thue systems which may be of independent interest.

An extension of Klop's counterexample to the Church-Rosser property to λ-calculus with other ordered pair combinators

This note provides a short elementary proof of the fact that the reduction sequence discovered by Klop (1979) provides a counterexample to the Church-Rosser property for λ-calculus or combinatory logic with ordered pair combinators. It also extends the counterexample to systems with other forms of ordered pair combinators.

An O(| T| 3) algorithm for testing the Church-Rosser property of thue systems

We give an O(| T| 3) algorithm for testing the Church-Rosser property of Thue systems using the linear string-matching algorithm by Knuth, Morris and Pratt (1977). This improves the earlier bound of O(| T| 6) given by Book and O'Dunlaing (1981). The proposed algorithm uses a reduction algorithm for finding a normal form of a string which is based on building a trie for matching a finite set of patterns, as proposed by Aho and Corasick (1975).