A Higher-Order Calculus for Graph Transformation

This paper presents a formalism for defining higher-order systems based on the notion of graph transformation (by rewriting or interaction). The syntax is inspired by the Combinatory Reduction Systems of Klop. The rewrite rules can be used to define first-order systems, such as graph or term-graph rewriting systems, Lafont's interaction nets, the interaction systems of Asperti and Laneve, the non-deterministic nets of Alexiev, or a process calculus. They can also be used to specify higher-order systems such as hierarchical graphs and proof nets of Linear Logic, or to specify the operational semantics of graph-based languages.     

直觉模糊神经网络的函数逼近能力

运用直觉模糊集理论,建立了自适应神经-直觉模糊推理系统（ANIFIS）的控制模型,并证明了该模型具有全局逼近性质.首先将Zadeh模糊推理神经网络变为直觉模糊推理网络,建立一个多输入单输出的T-S型ANIFIS模型;然后设计了系统变量的属性函数和推理规则,确定了各层的输入输出计算关系,以及系统输出结果的合成计算表达式;最后通过证明所建模型的输出结果计算式满足Stone-Weirstrass定理的3个假设条件,完成了该模型的全局逼近性证明.     

Focusing and polarization in linear,intuitionistic, and classical logics

A focused proof system provides a normal form to cut-free proofs in which the application of invertible and non-invertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cut-free proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.     

A New Correctness Criterion for Cyclic Proof Nets

We define proof nets for cyclic multiplicative linear logic as edge bi-coloured graphs. Our characterization is purely graph theoretical and works without further complication for proof nets with cuts, which are usually harder to handle in the non-commutative case. This also provides a new characterization of the proof nets for the Lambek calculus (with the empty sequence) which simply are a restriction on the formulas to be considered (which are asked to be intuitionistic).     

Abstract Data Type Specification in the Affirm System

This paper describes the data type definition facilities of the AFFIRM system for program specification and verification. Following an overview of the system, we review the rewrite rule concepts that form the theoretical basis for its data type facilities. The main emphasis is on methods of ensuring convergence (finite and unique termination) of sets of rewrite rules and on the relation of this property to the equational and inductive proof theories of data types.     

On goal-directed provability in classical logic

One of the key features of logic programming is the notion of goal-directed provability. In intuitionistic logic, the notion of uniform proof has been used as a proof-theoretic characterization of this property. Whilst the connections between intuitionistic logic and computation are well known, there is no reason per se why a similar notion cannot be given in classical logic. In this paper we show that there are two notions of goal-directed proof in classical logic, both of which are suitably weaker than that for intuitionistic logic. We show the completeness of this class of proofs for certain fragments, which thus form logic programming languages. As there are more possible variations on the notion of goal-directed provability in classical logic, there is a greater diversity of classical logic programming languages than intuitionistic ones. In particular, we show how logic programs may contain disjunctions in this setting. This provides a proof-theoretic basis for disjunctive logic programs, as well as characterising the “disjunctive” nature of answer substitutions for such programs in terms of the provability properties of the classical connectives Λ and Λ.     

Rewriting with Equivalence Relations in ACL2

Traditionally, a conditional rewrite rule directs replacement of one term by another term that is provably equal to it, perhaps under some hypotheses. This paper generalizes the notion of rewrite rule to permit the connecting relation to be merely an equivalence relation. We then extend the algorithm for applying rewrite rules. Applications of these generalized rewrite rules are only admissible in certain equivalential contexts, so the algorithm tracks which equivalence relations are to be preserved and admissible generalized rewrite rules are selected according to this context. We introduce the notions of congruence rule and refinement rule. We also introduce the idea of generated equivalences, corresponding to a new equivalence relation generated by a set of pre-existing ones. Generated equivalences are used to give the rewriter broad access to admissible generalized rewrite rules. We discuss the implementation of these notions in the ACL2 theorem prover. However, the discussion does not assume familiarity with ACL2, and these ideas can be applied to other reasoning systems as well.     

Reduction rules for time Petri nets

 The goal of net reduction is to increase the effectiveness of Petri-net-based real-time program analysis. Petri-net-based analysis, like all reachability-based methods, suffers from the state explosion problem. Petri net reduction is one key method for combating this problem. In this paper, we extend several rules for the reduction of ordinary Petri nets to work with time Petri nets. We introduce a notion of equivalence among time Petri nets, and prove that our reduction rules yield equivalent nets. This notion of equivalence guarantees that crucial timing and concurrency properties are preserved. Received September 12, 1994/July 4, 1995     

Reduction rules for time Petri nets

The goal of net reduction is to increase the effectiveness of Petri-netbased real-time program analysis. Petri-net-based analysis, like all reachabilitybased methods, suffers from the state explosion problem. Petri net reduction is one key method for combating this problem. In this paper, we extend several rules for the reduction of ordinary Petri nets to work with time Petri nets. We introduce a notion of equivalence among time Petri nets, and prove that our reduction rules yield equivalent nets. This notion of equivalence guarantees that crucial timing and concurrency properties are preserved.     

Knuth－Bendix过程的发散现象的研究

Ｋｎｕｔｈ－Ｂｅｎｄｉｘ完备过程不终止的起因研究得很少．本文研究引起不终止的重写规则的结构性质，提出了相容交叉规则对的概念，推广了文献［６］的结论，并提出了为构造系统检验该过程是否不终止的方法．     

On Symbolic Verification of Weakly Extended PAD

We consider the verification problem of a class of infinite-state systems called wPAD. These systems can be used to model programs with (possibly recursive) procedure calls and dynamic creation of parallel processes. They correspond to PAD models extended with an acyclic finite-state control unit, where PAD models can be seen as combinations of prefix rewrite systems (pushdown systems) with context-free multiset rewrite systems (synchronization-free Petri nets). Recently, we have presented symbolic reachability techniques for the class of PAD based on the use of a class of unranked tree automata. In this paper, we generalize our previous work to the class wPAD which is strictly larger than PAD. This generalization brings a positive answer to an open question on decidability of the model checking problem for wPAD against EF logic. Moreover, we show how symbolic reachability analysis of wPAD can be used in (under) approximate analysis of Synchronized PAD, a (Turing) powerful model for multithreaded programs (with unrestricted synchronization between parallel processes). This leads to a pragmatic approach for detecting the presence of erroneous behaviors in these models based on the bounded reachability paradigm where the notion of bound considered here is the number of synchronization actions.     

Correctness of Classical Compiler Optimizations using CTL

In this paper, global compiler optimizations are captured by conditional rewrite rules of the form ( ), where and are program instructions and φ is a condition expressed in a variant of CTL, a formalism well suited to describe properties involving the control flow of a given program. The goal: to formally prove that if the condition φ is satisfied, then the rewrite rule can be applied to the program without changing the semantics of the program. Once a rewrite rule has been proven correct, it can be directly and automatically utilized in an optimizing compiler.The framework is based on joint work with David Lacey, Neil Jones and Eric Van Wyk [6]. The present paper presents a slightly simplified version of the framework, with emphasis on the CTL variants relation to CTL, along with a correctness proof of a transformation eliminating recomputations of available expressions.     

A Toolkit for Generating and Displaying Proof Scores in the OTS/CafeOBJ Method

The OTS/CafeOBJ method can be used to model, specify and verify distributed systems. Specifications are written in equations, which are regarded as rewrite rules and used to verify specifications. The usefulness of the method is demonstrated by applying the method to nontrivial problems such as electronic commerce protocols and railroad signaling systems. In this paper we describe a toolkit called Buffet, which assists verification in the method. Given predicates used to split cases and lemmas, Buffet automatically generates proofs (called proof scores) and checks the proof scores using the CafeOBJ system. Buffet also has facilities to display proof scores generated and verification results on a web browser.     

A graph-theoretic characterization theorem for multiplicative fragment of non-commutative linear logic

It is well known that every proof net of a non-commutative version of MLL (multiplicative fragment of commutative linear logic) can be drawn as a plane Danos–Regnier graph (drawing) satisfying the switching condition of Danos–Regnier [3]. In this paper, we study the reverse direction; we introduce a system MNCLL which is logically equivalent to the multiplicative fragment of cyclic linear logic introduced by Yetter [9], and show that any plane Danos–Regnier graph drawing with one terminal edge satisfying the switching condition represents a unique non-commutative proof net (i.e., a proof net of MNCLL). In the course of proving this, we also give the characterization of the non-commutative proof nets by means of the notion of strong planarity, as well as the notion of a certain long-trip condition, called the stack-condition, of a Danos–Regnier graph, the latter of which is related to Abrusci's balanced long-trip condition [2].     

A Framework for Proof Systems

Linear logic can be used as a meta-logic to specify a range of object-level proof systems. In particular, we show that by providing different polarizations within a focused proof system for linear logic, one can account for natural deduction (normal and non-normal), sequent proofs (with and without cut), and tableaux proofs. Armed with just a few, simple variations to the linear logic encodings, more proof systems can be accommodated, including proof system using generalized elimination and generalized introduction rules. In general, most of these proof systems are developed for both classical and intuitionistic logics. By using simple results about linear logic, we can also give simple and modular proofs of the soundness and relative completeness of all the proof systems we consider.     

Relative Termination via Dependency Pairs

A term rewrite system is terminating when no infinite reduction sequences are possible. Relative termination generalizes termination by permitting infinite reductions as long as some distinguished rules are not applied infinitely many times. Relative termination is thus a fundamental notion that has been used in a number of different contexts, like analyzing the confluence of rewrite systems or the termination of narrowing. In this work, we introduce a novel technique to prove relative termination by reducing it to dependency pair problems. To the best of our knowledge, this is the first significant contribution to Problem #106 of the RTA List of Open Problems. We first present a general approach that is then instantiated to provide a concrete technique for proving relative termination. The practical significance of our method is illustrated by means of an experimental evaluation.     

Proving termination of (conditional) rewrite systems

In this paper an important problem in the domain of term rewriting, the termination of (conditional) rewrite systems, is dealt with. We show that in many applications, well-founded orderings on terms which only make use of syntactic information of a rewrite systemR, do not suffice for proving termination ofR. Indeed sometimes semantic information is needed to orient a rewrite rule. Therefore we integrate a semantic interpretation of rewrite systems and terms into a well-founded ordering on terms: the notion ofsemantic ordering is the first main contribution of this paper. The use and usefulness of the semantic ordering in proving termination is illustrated by means of some realistic examples.Furthermore the concept of semantic information induces a novel approach for proving termination inconditional rewrite systems. The idea is to employ not only semantic information contained in the terms that are to be compared, but also extra (semantic) information contained in the premiss of the conditional equation in which the terms appear. This leads to our second contribution in the termination problem area: the notion ofcontextual ordering andcontextual semantic ordering. Thecontextual approach allows to prove termination of conditional rewrite systems where all classical partial orderings would fail.     

On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroups

The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.     

Linear logic by levels and bounded time complexity

We give a new characterization of elementary and deterministic polynomial time computation in linear logic through the proofs-as-programs correspondence. Girard’s seminal results, concerning elementary and light linear logic, achieve this characterization by enforcing a stratification principle on proofs, using the notion of depth in proof nets. Here, we propose a more general form of stratification, based on inducing levels in proof nets by means of indices, which allows us to extend Girard’s systems while keeping the same complexity properties. In particular, it turns out that Girard’s systems can be recovered by forcing depth and level to coincide. A consequence of the higher flexibility of levels with respect to depth is the absence of boxes for handling the paragraph modality. We use this fact to propose a variant of our polytime system in which the paragraph modality is only allowed on atoms, and which may thus serve as a basis for developing lambda-calculus type assignment systems with more efficient typing algorithms than existing ones.