Church-Rosser Property Research Articles

An (almost) fuzzy logic of action and preferences, its quasi-model interpretations, and the problem of its decidability

The logical depiction of the concept of action forms a unique method of grasping its semantic content. It finds its reflection in different proposals of epistemic and deontic-logic systems describing actions as the operational behaviour of an agent as a game played over a given game tree. This perspective establishes some unique connection between actions and the agent's preferences. One of the logical-game-theoretic approaches to their modelling has been elaborated in van Benthem's school in the common area of game-theory, epistemic and fixed-point logic. In reference to some ideas of van Benthem's school – the article aims to propose a new (almost) fuzzy logic system AFLAP for actions and preferences describing a game-theoretic behaviour of an agent over a given game tree. Nevertheless, the backward induction strategy is idealised and exchanged for the Church-Rosser property, and a piece of fuzziness is introduced to the system by considering two unique ‘epsilon’ relations, Move and BI, with their approximation sequences. An abstract model in the form of the so-called quasi-model is proposed for such a system. It forms a multi-level (potentially infinite) construction built due to the pattern recently elaborated and enhanced by Wolter-Kurucz's school. This construction is exploited in the article as a convenient basis for proving the decidability of AFLAP by showing its ‘abstract’ finite model property. This task is performed by constructing an additional workable quasi-model of the size dependent on the size of the set of subformulae of a given formula of AFLAP. Finally, some R-programming-oriented applications of the earlier theoretic considerations are presented.

A dependently typed calculus with pattern matching and erasure inference

Some parts of dependently typed programs constitute evidence of their type-correctness and, once checked, are unnecessary for execution. These parts can easily become asymptotically larger than the remaining runtime-useful computation, which can cause normally linear-time programs run in exponential time, or worse. We should not make programs run slower by just describing them more precisely. Current dependently typed systems do not erase such computation satisfactorily. By modelling erasure indirectly through type universes or irrelevance, they impose the limitations of these means to erasure. Some useless computation then cannot be erased and idiomatic programs remain asymptotically sub-optimal. In this paper, we explain why we need erasure, that it is different from other concepts like irrelevance, and propose a dependently typed calculus with pattern matching with erasure annotations to model it. We show that erasure in well-typed programs is sound in that it commutes with reduction. Assuming the Church-Rosser property, erasure furthermore preserves convertibility in general. We also give an erasure inference algorithm for erasure-unannotated or partially annotated programs and prove it sound, complete, and optimal with respect to the typing rules of the calculus. Finally, we show that this erasure method is effective in that it can not only recover the expected asymptotic complexity in compiled programs at run time, but it can also shorten compilation times.

Capturing associations in graphs

This paper proposes a class of graph association rules, denoted by GARs, to specify regularities between entities in graphs. A GAR is a combination of a graph pattern and a dependency; it may take as predicates ML (machine learning) classifiers for link prediction. We show that GARs help us catch incomplete information in schemaless graphs, predict links in social graphs, identify potential customers in digital marketing, and extend graph functional dependencies (GFDs) to capture both missing links and inconsistencies. We formalize association deduction with GARs in terms of the chase, and prove its Church-Rosser property. We show that the satisfiability, implication and association deduction problems for GARs are coNP-complete, NP-complete and NP-complete, respectively, retaining the same complexity bounds as their GFD counterparts, despite the increased expressive power of GARs. The incremental deduction problem is DP-complete for GARs versus coNP-complete for GFDs. In addition, we provide parallel algorithms for association deduction and incremental deduction. Using real-life and synthetic graphs, we experimentally verify the effectiveness, scalability and efficiency of the parallel algorithms.

Unifying logic rules and machine learning for entity enhancing

This paper proposes a notion of entity enhancing, which unifies entity resolution and conflict resolution, to identify tuples that refer to the same real-world entity and at the same time, correct semantic inconsistencies. We propose to unify rule-based and machine learning (ML) methods for entity enhancing, by embedding ML classifiers as predicates in logic rules. We model entity enhancing by extending the chase. We show that the chase warrants correctness justification and the Church-Rosser property. Moreover, we settle fundamental problems associated with entity enhancing, including the enhancing, consistency, satisfiability, and implication problems, ranging from NP-complete and coNP-complete to Π 2 -complete. Taken together, these provide a new theoretical framework for unifying entity resolution and conflict resolution.

Propositional quantifiers in labelled natural deduction for normal modal logic

Abstract This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs.

Dependencies for Graphs

This article proposes a class of dependencies for graphs, referred to as graph entity dependencies (GEDs). A GED is defined as a combination of a graph pattern and an attribute dependency. In a uniform format, GEDs can express graph functional dependencies with constant literals to catch inconsistencies, and keys carrying id literals to identify entities (vertices) in a graph. We revise the chase for GEDs and prove its Church-Rosser property. We characterize GED satisfiability and implication, and establish the complexity of these problems and the validation problem for GEDs, in the presence and absence of constant literals and id literals. We also develop a sound, complete and independent axiom system for finite implication of GEDs. In addition, we extend GEDs with built-in predicates or disjunctions, to strike a balance between the expressive power and complexity. We settle the complexity of the satisfiability, implication, and validation problems for these extensions.

On Church-Rosser Property of Notion of βδ-Reduction for Canonical Notion of δ-Reduction

In this paper the canonical notions of δ-reduction for typed λ-terms are considered. Typed λ-terms use variables of any order and constants of order ≤ 1, where constants of order 1 are strongly computable, monotonic functions with indeterminate values of arguments. The canonical notions of δ-reduction are the notions of δ-reduction that are used in the implementations of functional programming languages. It is shown that for the main canonical notion of δ-reduction the notion of βδ-reduction has the Church-Rosser property. It is also shown that there exists a canonical notion of δ-reduction such that the notion of βδ-reduction does not have Church-Rosser property.

Reduction operators and completion of rewriting systems

We propose a functional description of rewriting systems where reduction rules are represented by linear maps called reduction operators. We show that reduction operators admit a lattice structure. Using this structure we define the notions of confluence and of Church–Rosser property. We show that these notions are equivalent. We give an algebraic formulation of completion and show that such a completion exists using the lattice structure. We interpret the confluence for reduction operators in terms of Gröbner bases. Finally, we introduce generalised reduction operators relative to non totally ordered sets.

Strict coherence of conditional rewriting modulo axioms

Conditional rewriting modulo axioms with rich types makes specifications and declarative programs very expressive and succinct and is used in all well-known rule-based languages. However, the current foundations of rewriting modulo axioms have focused for the most part on the unconditional and untyped case. The main purpose of this work is to generalize the foundations of rewriting modulo axioms to the conditional order-sorted case. A related goal is to simplify such foundations. In particular, even in the unconditional case, the notion of strict coherence proposed here makes rewriting modulo axioms simpler and easier to understand. Properties of strictly coherent conditional theories, like operational equi-termination of the →R/B and →R,B relations and general conditions for the conditional Church–Rosser property modulo B are also studied.

Multiset rewriting over Fibonacci and Tribonacci numbers

We show how techniques from the formal logic, can be applied directly to the problems studied completely independently in the world of combinatorics, the theory of integer partitions. We characterize equinumerous partition ideals in terms of the minimal elements generating the complementary order filters. Here we apply a general rewriting methodology to the case of filters having overlapping minimal elements. In addition to a ‘bijective proof ’ for Zeckendorf-like theorems – that every positive integer is uniquely representable within the Fibonacci, Tribonacci and k-Bonacci numeration systems, we establish ‘bijective proofs’ for a new series of partition identities related to Fibonacci, Tribonacci and k-step Fibonacci numbers. The main result is proved with the help of a multiset rewriting system such that the system itself and the system consisting of its reverse rewriting rules, both have the Church–Rosser property, which provides an explicit bijection between partitions of two different types (represented by the two normal forms).

Least upper bounds on the size of confluence and church-rosser diagrams in term rewriting and λ-calculus

We study confluence and the Church-Rosser property in term rewriting and λ-calculus with explicit bounds on term sizes and reduction lengths. Given a system R , we are interested in the lengths of the reductions in the smallest valleys t → * s ′ * ← t ′ expressed as a function: —for confluence a function vs R ( m , n ) where the valleys are for peaks t ← s → * t ′ with s of size at most m and the reductions of maximum length n , and —for the Church-Rosser property a function cvs R ( m , n ) where the valleys are for conversions t ↔ * t ′ with t and t ′ of size at most m and the conversion of maximum length n . For confluent Term Rewriting Systems (TRSs), we prove that vs R is a total computable function, and for linear such systems that cvs R is a total computable function. Conversely, we show that every total computable function is the lower bound on the functions vs R ( m , n ) and cvs R ( m , n ) for some TRS R : In particular, we show that for every total computable function φ: N → N there is a TRS R with a single term s such that vs R (| s |, n ) ≥ φ( n ) and cvs R ( n , n ) ≥ φ( n ) for all n . For orthogonal TRSs R we prove that there is a constant k such that: (a) vs R ( m , n ) is bounded from above by a function exponential in k and (b) cvs R ( m , n ) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Similarly, for λ-calculus, we show that vs R ( m , n ) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy.

Call-by-Value and Call-by-Name Dual Calculi with Inductive and Coinductive Types

This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it has the duality of inductive and coinductive types, that is, the duality of terms and coterms for inductive and coinductive types, and the duality of their reduction rules. Its strong normalization is also proved, which is shown by translating it into a second-order dual calculus. The strong normalization of the second-order dual calculus is proved by translating it into the second-order symmetric lambda calculus. This paper then introduces a call-by-value system and a call-by-name system of the dual calculus with inductive and coinductive types, and shows the duality of call-by-value and call-by-name, their Church-Rosser properties, and their strong normalization. Their strong normalization is proved by translating them into the non-deterministic dual calculus with inductive and coinductive types.

Some properties of the -calculus

In this paper, we present the -calculus which at the typed level corresponds to the full classical propositional natural deduction system. The Church–Rosser property of this system is proved using the standardisation and the finiteness developments theorem. We also define the leftmost reduction and prove that it is a winning strategy.

On the Church-Rosser and coherence properties of conditional order-sorted rewrite theories

In the effort to bring rewriting-based methods into contact with practical applications both in programing and in formal verification, there is a tension between: (i) expressiveness and generality—so that a wide range of applications can be expressed easily and naturally—, and (ii) support for formal verification, which is harder to get for general and expressive specifications. This paper answers the challenge of successfully negotiating the tension between goals (i) and (ii) for a wide class of Maude specifications, namely: (a) equational order-sorted conditional specifications Σ,E∪A, corresponding to functional programs modulo axioms such as associativity and/or commutativity and/or identity axioms; and (b) order-sorted conditional rewrite theories R=(Σ,E∪A,R,ϕ), corresponding to concurrent programs modulo axioms A. For Maude functional programs the key formal property checked is the Church-Rosser property. For concurrent declarative programs in rewriting logic, the key property checked is the coherence between rules and equations modulo the axioms A. Such properties are essential, both for the executability purposes and as a basis for verifying many other properties, such as, for example, proving inductive theorems of a functional program, or correct model checking of temporal logic properties for a concurrent program. This paper develops the mathematical foundations on which the checking of these properties (or ground versions of them) is based, presents two tools, the Church-Rosser Checker (CRC) and the Coherence Checker (ChC) supporting the verification of these properties, and illustrates with examples a methodology to establish such properties using the proof obligations returned by the tools.

On the Church-Rosser and coherence properties of conditional order-sorted rewrite theories

In the effort to bring rewriting-based methods into contact with practical applications both in programing and in formal verification, there is a tension between: (i) expressiveness and generality—so that a wide range of applications can be expressed easily and naturally— and (ii) support for formal verification, which is harder to get for general and expressive specifications. This paper answers the challenge of successfully negotiating the tension between goals (i) and (ii) for a wide class of Maude specifications, namely: (a) equational order-sorted conditional specifications (Σ,E∪A), corresponding to functional programs modulo axioms such as associativity and/or commutativity and/or identity axioms and (b) order-sorted conditional rewrite theories R=(Σ,E∪A,R,ϕ), corresponding to concurrent programs modulo axioms A. For Maude functional programs the key formal property checked is the Church-Rosser property. For concurrent declarative programs in rewriting logic, the key property checked is the coherence between rules and equations modulo the axioms A. Such properties are essential, both for executability purposes and as a basis for verifying many other properties, such as, for example, proving inductive theorems of a functional program, or correct model checking of temporal logic properties for a concurrent program. This paper develops the mathematical foundations on which the checking of these properties (or ground versions of them) is based, presents two tools, the Church-Rosser Checker (CRC) and the Coherence Checker (ChC) supporting the verification of these properties, and illustrates with examples a methodology to establish such properties using the proof obligations returned by the tools.

Light linear logics with controlled weakening: Expressibility, confluent strong normalization

Starting from Girard’s seminal paper on light linear logic (LLL), a number of works investigated systems derived from linear logic to capture polynomial time computation within the computation-as-cut-elimination paradigm.The original syntax of LLL is too complicated, mainly because one has to deal with sequents which do not just consist of formulas but also of ‘blocks’ of formulas.We circumvent the complications of ‘blocks’ by introducing a new modality ∇ which is exclusively in charge of ‘additive blocks’. One of the most interesting features of this purely multiplicative ∇ is the possibility of the second-order encodings of additive connectives.The resulting system (with the traditional syntax), called Easy-LLL, is still powerful to represent any deterministic polynomial time computations in purely logical terms.Unlike the original LLL, Easy-LLL admits polynomial time strong normalization together with the Church–Rosser property, namely, cut elimination terminates in a unique way in polytime by any choice of cut reduction strategies.

Product of Graphs and Hybrid Logic

Left and right commutativity and the Church-Rosser and reverse Church-Rosser properties are necessary conditions for a graph (frame) to be a (non-trivial) product of two other graphs, but their conjunction is not a sufficient condition. This work presents a fifth property, called H-V intransitivity, that, when added to the four previous properties, results in a necessary and sufficient condition for a finite and connected graph to be a product. Then, we show that although the first four properties can be defined in a modal logic (the reverse Church-Rosser property requires a converse modality), H-V intransitivity is not modally definable. We also show that no necessary and sufficient condition for a graph to be a product can be modally definable. Finally, we present a formula in a hybrid language that defines H-V intransitivity.

The λ-calculus with constructors: Syntax, confluence and separation

AbstractWe present an extension of the λ(η)-calculus with a case construct that propagates through functions like a head linear substitution, and show that this construction permits to recover the expressiveness of ML-style pattern matching. We then prove that this system enjoys the Church–Rosser property using a semi-automatic ‘divide and conquer’ technique by which we determine all the pairs of commuting subsystems of the formalism (considering all the possible combinations of the nine primitive reduction rules). Finally, we prove a separation theorem similar to Böhm's theorem for the whole formalism.

Simplified Reducibility Proofs of Church-Rosser for β- and βη-reduction

The simplest proofs of the Church Rosser Property are usually done by the syntactic method of parallel reduction. On the other hand, reducibility is a semantic method which has been used to prove a number of properties in the λ -calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we concentrate on simplifying a semantic method based on reducibility for proving Church-Rosser for both β - and βη -reduction. Interestingly, this simplification results in a syntactic method (so the semantic aspect disappears) which is nonetheless projectable into a semantic method. Our contributions are as follows: • We give a simplification of a semantic proof of CR for β -reduction which unlike some common proofs in the literature, avoids any type machinery and is solely carried out in a completely untyped setting. • We give a new proof of CR for βη -reduction which is a generalisation of our simple proof for β -reduction. • Our simplification of the semantic proof results into a syntactic proof which is projectable into a semantic method and can hence be used as a bridge between syntactic and semantic methods.

Big-step normalisation

AbstractTraditionally, decidability of conversion for typed λ-calculi is established by showing that small-step reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflects and preserves conversion. We apply our approach to the simply typed λ-calculus with explicit substitutions and βη-equality, a system which is not strongly normalising. We also show how the construction can be extended to system T with the usual β-rules for the recursion combinator. Our approach is practical, since it does verify an actual implementation of normalisation which, unlike normalisation by evaluation, is first order. An important feature of our approach is that we are using logical relations to establish equational soundness (identity of normal forms reflects the equational theory), instead of the usual syntactic reasoning using the Church–Rosser property of a term rewriting system.