Frédéric Blanqui
The expressiveness of dependent type theory can be extended by identifying types modulo some additional computation rules. But, for preserving the decidability of type-checking or the logical consistency of the system, one must make sure that those user-defined rewriting rules preserve typing. In this paper, we give a new method to check that property using Knuth-Bendix completion.
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FGL is a successor to GL, a proof procedure for ACL2 that allows complicated finitary conjectures to be translated into efficient Boolean function representations and proved using SAT solvers. A primary focus of FGL is to allow greater programmability using rewrite rules. While the FGL rewriter is modeled on ACL2's rewriter, we have added several features in order to make rewrite rules more powerful. A particular focus is to make it more convenient for rewrite rules to use information from the syntactic domain, allowing them to replace built-in primitives and meta rules in many cases. Since it…
Subsumptions of Algebraic Rewrite Rules
本文定义代数重写规则间的包含关系为规则态射，使规则与子规则同时应用等价于单独应用包含规则，并通过全局相干变换在DPO等重写环境中实现。
From Rewrite Rules to Axioms in the $λ$$Π$-Calculus Modulo Theory
本文研究$λ$$Π$-演算模理论，提出通过引入异质相等和transport替换转换规则的方法，将重写规则替换为等式公理，同时保持表达能力，证明带重写规则的理论是带公理理论的保守扩展。
The static dependency pair method is a method for proving the termination of higher-order rewrite systems a la Nipkow. It combines the dependency pair method introduced for first-order rewrite systems with the notion of strong computability introduced for typed lambda-calculi. Argument filterings and usable rules are two important methods of the dependency pair framework used by current state-of-the-art first-order automated termination provers. In this paper, we extend the class of higher-order systems on which the static dependency pair method can be applied. Then, we extend argument filteri…
Natural Transformations as Rewrite Rules and Monad Composition
本文提出用字符串重写方法验证单子复合等范畴论构造，通过重写系统的合流性刻画单子，并给出伴随的重写范畴表征。
Every database system contains a query optimizer that performs query rewrites. Unfortunately, developing query optimizers remains a highly challenging task. Part of the challenges comes from the intricacies and rich features of query languages, which makes reasoning about rewrite rules difficult. In this paper, we propose a machine-checkable denotational semantics for SQL, the de facto language for relational database, for rigorously validating rewrite rules. Unlike previously proposed semantics that are either non-mechanized or only cover a small amount of SQL language features, our semantics…
ELAN is a powerful language and environment for specifying and prototyping deduction systems in a language based on rewrite rules controlled by strategies. Timed automata is a class of continuous real-time models of reactive systems for which efficient model-checking algorithms have been devised. In this paper, we show that these algorithms can very easily be prototyped in the ELAN system. This paper argues through this example that rewriting based systems relying on rules and strategies are a good framework to prototype, study and test rather efficiently symbolic model-checking algorithms, i.…
Axioms vs. rewrite rules: from completeness to cut elimination
本文研究自动定理证明中证明搜索与重写结合时的完备性问题，指出其关键在于为演绎与计算结合的扩展证明概念建立切割消去，从而揭示自动定理证明与证明论之间的新联系。
Efficiently Synthesizing Lowest Cost Rewrite Rules for Instruction Selection
针对指令选择重写规则综合现有方法受限单指令ISA且效率低的问题，提出两种优化SMT综合算法，分别生成唯一规则和最低代价规则，在多ISA上实现最高768倍、4004倍的速度提升。
String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric representations of string diagrams, amenable to automated reasoning about diagrammatic theories via graph rewrite systems. In this extended abstract, we show how the power of such rewrite systems can be greatly extended by introducing pattern graphs, which provide a means of expressing infinite families of rewrite rules where certain marked subgraphs, called !-boxes ("…
RP-Rewriter (Retain-Property) is a verified clause processor that can use some of the existing ACL2 rewrite rules to prove conjectures through term rewriting. Optimized for conjectures that can expand into large terms, the rewriter tries to mimic some of the ACL2 rewriting heuristics but also adds some extra features. It can attach side-conditions to terms that help the rewriter retain properties about them and prevent possibly some very expensive backchaining. The rewriter supports user-defined complex meta rules that can return a special structure to prevent redundant rewriting. Additionally…
Rewriting Induction for Existentially Quantified Equations in Logically Constrained Rewriting (Full Version)
本文提出在逻辑约束重写中扩展重写归纳方法，以处理存在量词方程，通过引入存在量词到方程和规则中，实现对约束方程的归纳证明。
For the lambda-calculus with surjective pairing and terminal type, Curien and Di Cosmo were inspired by Knuth-Bendix completion, and introduced a confluent rewriting system that (1) extends the naive rewriting system, and (2) is stable under contexts. The rewriting system has (i) a rule that rewrites term of a terminal type rewrites to a term constant *, unless the term is not *, (ii) rewrite rules for the extensionality of function types and product types, and rewrite rules mediating the rewrite rules (i) and (ii). Curien and Di Cosmo supposed that because of (iii), any reducibility method ca…
We introduce an extension of hedge automata called bidimensional context-free hedge automata. The class of unranked ordered tree languages they recognize is shown to be preserved by rewrite closure with inverse-monadic rules. We also extend the parameterized rewriting rules used for modeling the W3C XQuery Update Facility in previous works, by the possibility to insert a new parent node above a given node. We show that the rewrite closure of hedge automata languages with these extended rewriting systems are context-free hedge languages.
We introduce a new class of graph transformation systems in which rewrite rules can be guarded by universally quantified conditions on the neighbourhood of nodes. These conditions are defined via special graph patterns which may be transformed by the rule as well. For the new class for graph rewrite rules, we provide a symbolic procedure working on minimal representations of upward closed sets of configurations. We prove correctness and effectiveness of the procedure by a categorical presentation of rewrite rules as well as the involved order, and using results for well-structured transition s…
In Martin-Löf's Intensional Type Theory, identity type is a heavily used and studied concept. The reason for that is the fact that it's responsible for the recently discovered connection between Type Theory and Homotopy Theory. The main problem with identity types, as originally formulated, is that they are complex to understand and use. Using that fact as motivation, a much simpler formulation for the identity type was proposed by Queiroz & Gabbay (1994) and further developed by de Queiroz & de Oliveira (2013). In this formulation, an element of an identity type is seen as a sequence of rewri…
A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods
本文利用同调方法为项重写系统等价公理集的数量提供更优下界，并实现程序在64个完整TRS上验证。
We consider problems of access control for update of XML documents. In the context of XML programming, types can be viewed as hedge automata, and static type checking amounts to verify that a program always converts valid source documents into also valid output documents. Given a set of update operations we are particularly interested by checking safety properties such as preservation of document types along any sequence of updates. We are also interested by the related policy consistency problem, that is detecting whether a sequence of authorized operations can simulate a forbidden one. We re…
A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits
该研究提出了首套适用于任意n的多三量子比特Clifford电路完整重写规则，通过推广Selinger范式、构建重写系统并简化规则，实现奇素数维度量子电路片段的完备性结果，清晰呈现相关幺正群结构。
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 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…
We investigate the new, Turing-complete class of layered systems, whose lefthand sides of rules can only be overlapped at a multiset of disjoint or equal positions. Layered systems define a natural notion of rank for terms: the maximal number of non-overlapping redexes along a path from the root to a leaf. Overlappings are allowed in finite or infinite trees. Rules may be non-terminating, non-left-linear, or non-right-linear. Using a novel unification technique, cyclic unification, and the so-alled subrewriting relation, we show that rank non-increasing layered systems are confluent provided t…
This paper is an extended abstract of an analysis of term rewriting where the terms in the rewrite rules as well as the term to be rewritten are compressed by a singleton tree grammar (STG). This form of compression is more general than node sharing or representing terms as dags since also partial trees (contexts) can be shared in the compression. In the first part efficient but complex algorithms for detecting applicability of a rewrite rule under STG-compression are constructed and analyzed. The second part applies these results to term rewriting sequences. The main result for submatching is…
Reaction Systems (RSs) are a successful computational framework inspired by biological systems. A RS pairs a set of entities with a set of reactions over them. Entities can be used to enable or inhibit each reaction, and are produced by reactions. Entities can also be provided by an external context. RS semantics is defined in terms of an (unlabelled) rewrite system: given the current set of entities, a rewrite step consists of the application of all and only the enabled reactions. In this paper we define, for the first time, a labelled transition system for RSs in the structural operational s…
A version of the situation calculus in which situations are represented as first-order terms is presented. Fluents can be computed from the term structure, and actions on the situations correspond to rewrite rules on the terms. Actions that only depend on or influence a subset of the fluents can be described as rewrite rules that operate on subterms of the terms in some cases. If actions are bidirectional then efficient completion methods can be used to solve planning problems. This representation for situations and actions is most similar to the fluent calculus of Thielscher \cite{Thielscher9…
Arithmetic as a theory modulo
本文在演绎模框架下，仅用重写规则提出构造性算术，为算术形式化提供新方法。
Graph rewrite formalisms are a powerful approach to modeling complex molecular systems. They capture the intrinsic concurrency of molecular interactions, thereby enabling a formal notion of mechanism (a partially ordered set of events) that explains how a system achieves a particular outcome given a set of rewrite rules. It is then useful to verify whether the mechanisms that emerge from a given model comply with empirical observations about their mutual interference. In this work, our objective is to determine whether a specific event in the mechanism for achieving X prevents or promotes the …
We address the problem of cyclic termgraph rewriting. We propose a new framework where rewrite rules are tuples of the form $(L,R,τ,σ)$ such that $L$ and $R$ are termgraphs representing the left-hand and the right-hand sides of the rule, $τ$ is a mapping from the nodes of $L$ to those of $R$ and $σ$ is a partial function from nodes of $R$ to nodes of $L$. $τ$ describes how incident edges of the nodes in $L$ are connected in $R$. $τ$ is not required to be a graph morphism as in classical algebraic approaches of graph transformation. The role of $σ$ is to indicate the parts of $L$ to be cloned (…
The treatment of equality as a type in type theory gives rise to an interesting type-theoretic structure known as `identity type'. The idea is that, given terms $a,b$ of a type $A$, one may form the type $Id_{A}(a,b)$, whose elements are proofs that $a$ and $b$ are equal elements of type $A$. A term of this type, $p : Id_{A}(a,b)$, makes up for the grounds (or proof) that establishes that $a$ is indeed equal to $b$. Based on that, a proof of equality can be seen as a sequence of substitutions and rewrites, also known as a `computational path'. One interesting fact is that it is possible to rew…
Cut elimination for Zermelo set theory
本文通过将策梅洛集合论重构为指向图理论并将公理转化为重写规则，在演绎模中实现了其直觉主义片段的正规化，并证明了所得理论是原集合论的保守扩张。
Algorithmic Unverifiability of Safety for Fixed and Recursively Self-Improving Systems
本文证明了AGI安全的核心障碍不是对齐状态的不可能性，而是其结构性不可验证性，通过两个不可判定性定理和Trakhtenbrot墙揭示了工程防御的局限性，并推导出完备性-可靠性-可处理性三难困境。
The treatment of equality as a type in type theory gives rise to an interesting type-theoretic structure known as `identity type'. The idea is that, given terms $a,b$ of a type $A$, one may form the type $Id_{A}(a,b)$, whose elements are proofs that $a$ and $b$ are equal elements of type $A$. A term of this type, $p : Id_{A}(a,b)$, makes up for the grounds (or proof) that establishes that $a$ is indeed equal to $b$. Based on that, a proof of equality can be seen as a sequence of substitutions and rewrites, also known as a `computational path'. One interesting fact is that it is possible to rew…
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 …
Safety, Relative Tightness and the Probabilistic Frame Rule
本文提出了一种概率分离逻辑的语义表述，通过引入安全性概念，建立了相对紧致性属性，从而保证了框架规则的正确性。
This paper deals with model transformation based on attributed graph rewriting. Our contribution investigates a single pushout approach for applying the rewrite rules. The computation of graph attributes is obtained through the use of typed lambda-calculus with inductive types. In this paper we present solutions to cope with single pushout construction for the graph structure and the computations functions. As this rewrite system uses inductive types, the expressiveness of attribute computations is facilitated and appears more efficient than the one based on Sigma-algebras. Some examples showi…
We present two methods for proving confluence of left-linear term rewrite systems. One is hot-decreasingness, combining the parallel/development closedness theorems with rule labelling based on a terminating subsystem. The other is critical-pair-closing system, allowing to boil down the confluence problem to confluence of a special subsystem whose duplicating rules are relatively terminating.
Rewriting and Completeness of Sum-Over-Paths in Dyadic Fragments of Quantum Computing
本文为量子计算中的路径求和形式化方法提出一组新的重写规则，证明其对 Toffoli-Hadamard 片段完备且终止但不合流，并将其推广到用于量子傅里叶变换的二进片段，同时支持任意项的求和与拼接。
Towards Neuro-symbolic Causal Rule Synthesis, Verification, and Evaluation Grounded in Legal and Safety Principles
本文提出一种神经符号因果框架，结合一阶逻辑抽象树、结构因果模型和深度强化学习，通过Meta层缓解目标误指定问题，实现可扩展的规则维护。
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