A graphical approach to relational reasoning

Relational reasoning is concerned with relations over an unspecified domain of discourse. Two limitations to which it is customarily subject are: only dyadic relations are taken into account; all formulas are equations, having the same expressive power as first-order sentences in three variables. The relational formalism inherits from the Peirce-Schröder tradition, through contributions of Tarski and many others.Algebraic manipulation of relational expressions (equations in particular) is much less natural than developing inferences in first-order logic; it may in fact appear to be overly machine-oriented for direct hand-based exploitation.The situation radically changes when one resorts to a convenient representation of relations based on labeled graphs. The paper provides details of this representation, which abstracts w.r.t. inessential features of expressions.Formal techniques illustrating three uses of the graph representation of relations are discussed: one technique deals with translating first-order specifications into the calculus of relations; another one, with inferring equalities within this calculus with the aid of convenient diagram-rewriting rules; a third one with checking, in the specialized framework of set theory, the definability of particular set operations. Examples of use of these techniques are produced; moreover, a promising approach to mechanization of graphical relational reasoning is outlined.     

Temporal relational data model

This paper incorporates a temporal dimension to nested relations. It combines research in temporal databases and nested relations for managing the temporal data in nontraditional database applications. A temporal data value is represented as a temporal atom; a temporal atom consists of two parts: a temporal set and a value. The temporal atom asserts that the value is valid over the time duration represented by its temporal set. The data model allows relations with arbitrary levels of nesting and can represent the histories of objects and their relationships. Temporal relational algebra and calculus languages are formulated and their equivalence is proved. Temporal relational algebra includes operations to manipulate temporal data and to restructure nested temporal relations. Additionally, we define operations to generate a power set of a relation, a set membership test, and a set inclusion test, which are all derived from the other operations of temporal relational algebra. To obtain a concise representation of temporal data (temporal reduction), collapsed versions of the set-theoretic operations are defined. Procedures to express collapsed operations by the regular operations of temporal relational algebra are included. The paper also develops procedures to completely flatten a nested temporal relation into an equivalent 1 NF relation and back to its original form, thus providing a basis for the semantics of the collapsed operations by the traditional operations on 1 NF relations     

Language-Theoretic and Finite Relation Models for the (Full) Lambek Calculus

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus \({\mathbf {FL}}_\perp \). We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts lattice of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents (henceforth: \(\mathbf L \) and \(\mathbf L1 \)), thus solving a problem left open by Pentus (Ann Pure Appl Log 75:179–213, 1995).     

A sound and complete theory of graph transformations for service programming with sessions and pipelines

Graph transformation techniques, the Double-Pushout (DPO) approach in particular, have been successfully applied in the modeling of concurrent systems. In this area, a research thread has addressed the definition of concurrent semantics for process calculi. In this paper, we propose a theory of graph transformations for service programming with sophisticated features such as sessions and pipelines. Through graph representation of CaSPiS, a recently proposed process calculus, we show how graph transformations can cope with advanced features of service-oriented computing, such as several logical notions of scoping together with the interplay between linking and containment. We first exploit a graph algebra and set up a graph model that supports graph transformations in the DPO approach. Then, we show how to represent CaSPiS processes as hierarchical graphs in the graph model and their behaviors as graph transformation rules. Finally, we provide the soundness and completeness results of these rules with respect to the reduction semantics of CaSPiS.     

Modelling Calculi with Name Mobility using Graphs with Equivalences

In the theory of graph rewriting, the use of coalescing rules, i.e., of rules which besides deleting and generating graph items, can coalesce some parts of the graph, turns out to be quite useful for modelling purposes, but, at the same time, problematic for the development of a satisfactory partial order concurrent semantics for rewrites. Rewriting over graphs with equivalences, i.e., (typed hyper)-graphs equipped with an equivalence over nodes provides a technically convenient replacement of graph rewriting with coalescing rules, for which a truly concurrent semantics can be easily defined. The expressivity of such a formalism is tested in a setting where coalescing rules typically play a basic role: the encoding of calculi with name passing as graph rewriting systems. Specifically, we show how the (monadic fragment) of the solo calculus, one of the dialect of those calculi whose distinctive feature is name fusion, can be encoded as a rewriting system over graph with equivalences.     

A pure labeled transition semantics for the applied pi calculus

The applied pi calculus proposed by Abadi and Fournet is successful in the analysis of security protocols. Its semantics mainly depends on several structural rules. Structural rules are convenient for specification, but inefficient for implementation. In this paper, we establish a new semantics for applied pi calculus based upon pure labeled transition system and propose a new formulation of labeled bisimulation. We prove that the new labeled bisimularity coincides with observational equivalence. A zero-knowledge protocol is given as an example to illustrate the effectiveness of this new semantics.     

Efficient relational calculation for software analysis

Calculating with graphs and relations has many applications in the analysis of software systems, for example, the detection of design patterns or patterns of problematic design and the computation of design metrics. These applications require an expressive query language, in particular, for the detection of graph patterns, and an efficient evaluation of the queries even for large graphs. In this paper, we introduce RML, a simple language for querying and manipulating relations based on predicate calculus, and CrocoPat, an interpreter for RML programs. RML is general because it enables the manipulation not only of graphs (i.e., binary relations), but of relations of arbitrary arity. CrocoPat executes RML programs efficiently because it internally represents relations as binary decision diagrams, a data structure that is well-known as a compact representation of large relations in computer-aided verification. We evaluate RML by giving example programs for several software analyses and CrocoPat by comparing its performance with calculators for binary relations, a Prolog system, and a relational database management system.     

On the semantics of parsing actions

Parsers, whether constructed by hand or automatically via a parser generator tool, typically need to compute some useful semantic information in addition to the purely syntactic analysis of their input. Semantic actions may be added to parsing code by hand, or the parser generator may have its own syntax for annotating grammar rules with semantic actions. In this paper, we take a functional programming view of such actions. We use concepts from the semantics of mostly functional programming languages and adapt them to give meaning to the actions of the parser. Specifically, the semantics is inspired by the categorical semantics of lambda calculi and the use of premonoidal categories for the semantics of effects in programming languages. This framework is then applied to our leading example, the transformation of grammars to eliminate left recursion. The syntactic transformation of left-recursion elimination leads to a corresponding semantic transformation of the actions for the grammar. We prove the semantic transformation correct and relate it to continuation passing style, a widely studied transformation in lambda calculi and functional programming. As an idealization of the input language of parser generators, we define a call-by-value calculus with first-order functions and a type-and-effect system where the effects are given by sequences of grammar symbols. The account of left-recursion elimination is then extended to this calculus.     

Semantics preserving SPARQL-to-SQL translation

Most existing RDF stores, which serve as metadata repositories on the Semantic Web, use an RDBMS as a backend to manage RDF data. This motivates us to study the problem of translating SPARQL queries into equivalent SQL queries, which further can be optimized and evaluated by the relational query engine and their results can be returned as SPARQL query solutions. The main contributions of our research are: (i) We formalize a relational algebra based semantics of SPARQL, which bridges the gap between SPARQL and SQL query languages, and prove that our semantics is equivalent to the mapping-based semantics of SPARQL; (ii) Based on this semantics, we propose the first provably semantics preserving SPARQL-to-SQL translation for SPARQL triple patterns, basic graph patterns, optional graph patterns, alternative graph patterns, and value constraints; (iii) Our translation algorithm is generic and can be directly applied to existing RDBMS-based RDF stores; and (iv) We outline a number of simplifications for the SPARQL-to-SQL translation to generate simpler and more efficient SQL queries and extend our defined semantics and translation to support the bag semantics of a SPARQL query solution. The experimental study showed that our proposed generic translation can serve as a good alternative to existing schema dependent translations in terms of efficient query evaluation and/or ensured query result correctness.     

Causal dynamic inference

We suggest a general logical framework for causal dynamic reasoning. As a first step, we introduce a uniform structural formalism and assign it two kinds of semantics, abstract dynamic models and relational models. The corresponding completeness results are proved. As a second step, we extend the structural formalism to a two-sorted state-transition calculus, and prove its completeness with respect to the associated relational semantics.     

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.     

Tuple calculus: Formal definition and conversion from first-order calculus

In classical database theory, relational calculus has long been used in expressing query formulae and integrity constraints. In fact, relational calculus formulae are much easier to deal with than first-order formulae when evaluating queries and validating database updates in the database environment. In deductive databases, however, first-order calculus is preferred because it is convenient when proof procedures are involved. Since both situations should coexist in advanced information systems, it is very desirable to devise a conversion procedure between relational calculus and first-order calculus. In this paper, interpretation of first-order formulae in the database environment is discussed first, then tuple calculus, an extension of relational calculus, is presented. This extension enables us to describe query formulae and general rules necessary in advanced information systems, in particular, dealing with complex objects. Finally, a conversion algorithm from first-order formulae into tuple calculus formulae is presented. Several application issues are also included.     

Datalog extension for nested relations

The nested relational model allows relations that are not in first normal form. This paper gives an extension of Datalog rules for nested relations. In our approach, nested Datalog is a natural extension of Datalog introduced for the relational data model. A nested Datalog program has a hierarchical structure of rules and subprograms to manipulate relation values of nested relations. We introduce a new category of predicate symbols, the variable predicate symbols to refer to tuples of subrelations. The notion of soundness, safety and consistency is defined to avoid undesirable nested Datalog programs. The evaluation of nested Datalog is given in terms of the nested relational algebra. Finally, we relate the expressive power of nonrecursive nested Datalog to the power of nested relational algebra and safe nested tuple relational calculus.     

次协调数据库中的域关系演算

次协调数据库的数据模型是用来处理数据库中两类不确定信息,即不完全信息和不一致信息(矛盾信息)。关系演算语言是表达关系数据模型中的数据操作的一种方式。域关系演算是以域为变量进行的关系演算。文中提出了一种4值的域关系演算来查询次协调数据库,它的语法与普通关系上的2值域关系演算相似,但是这种新的4值语义能够有效的查询不完全信息和不一致信息。这为次协调数据库中的类SQL语言及实现提供了理论依据,进而为次协调数据库的应用打下坚实基础。     

Classical Logic with Partial Functions

We introduce a semantics for classical logic with partial functions, in which ill-typed formulas are guaranteed to have no truth value, so that they cannot be used in any form of reasoning. The semantics makes it possible to mix reasoning about types and preconditions with reasoning about other properties. This makes it possible to deal with partial functions with preconditions of unlimited complexity. We show that, in spite of its increased complexity, the semantics is still a natural generalization of first-order logic with simple types. If one does not use the increased expressivity, the type system is not stronger than classical logic with simple types. We will define two sequent calculi for our semantics, and prove that they are sound and complete. The first calculus follows the semantics closely, and hence its completeness proof is fairly straightforward. The second calculus is further away from the semantics, but more suitable for practical use because it has better proof theoretic properties. Its completeness can be shown by proving that proofs from the first calculus can be translated.     

Integration of object-oriented programming languages and database systems in KOPERNIK

KOPERNIK is an object-oriented database system, that allows uniform specification of database requests and application programs. The user interface is based on Smalltalk, and the object-oriented data model is represented in terms of classes and messages. Techniques are discussed for implementing such a model on top of an underlying relational database system. Those parts of application programs that cannot be translated into a relational language are handled by a Smalltalk processor. The semantics of the database requests is defined in terms of a meta-model and meta-messages, using an object-oriented approach. Hence we derive rules for translation of database requests into SQL queries over a binary relational view, introduced as an intermediate level between the underlying database and our conceptual view.     

Domain-specific discrete event modelling and simulation using graph transformation

Graph transformation is being increasingly used to express the semantics of domain-specific visual languages since its graphical nature makes rules intuitive. However, many application domains require an explicit handling of time to accurately represent the behaviour of a real system and to obtain useful simulation metrics to measure throughputs, utilization times and average delays. Inspired by the vast knowledge and experience accumulated by the discrete event simulation community, we propose a novel way of adding explicit time to graph transformation rules. In particular, we take the event scheduling discrete simulation world view and provide rules with the ability to schedule the occurrence of other rules in the future. Hence, our work combines standard, efficient techniques for discrete event simulation (based on the handling of a future event set) and the intuitive, visual nature of graph transformation. Moreover, we show how our formalism can be used to give semantics to other timed approaches and provide an implementation on top of the rewriting logic system Maude.     

The equational theory of Kleene lattices

Languages and families of binary relations are standard interpretations of Kleene algebras. It is known that the equational theories of these interpretations coincide and that the free Kleene algebra is representable both as a relation and as a language algebra. We investigate the identities valid in these interpretations when we expand the signature of Kleene algebras with the meet operation. In both cases, meet is interpreted as intersection. We prove that in this case, there are more identities valid in language algebras than in relation algebras (exactly three more in some sense), and representability of the free algebra holds for the relational interpretation but fails for the language interpretation. However, if we exclude the identity constant from the algebras when we add meet, then the equational theories of the relational and language interpretations remain the same, and the free algebra is representable as a language algebra, too. The moral is that only the identity constant behaves differently in the language and the relational interpretations, and only meet makes this visible.     

A Polymorphic Environment Calculus and its Type-Inference Algorithm

The polymorphic environment calculus is a polymorphic lambda calculus which enables us to treat environments as first-class citizens. In the calculus, environments are formalized as explicit substitutions, and the substitutions are included in the set of terms of the calculus. First, we introduce an untyped environment calculus, and we present a semantics of the calculus as a translation into the lambda calculus. Second, we propose a polymorphic type system for the environment calculus based on Damas-Milner's ML-polymorphic type system. In ML, polymorphism is allowed only in let-expressions; in the polymorphic environment calculus, polymorphism is provided with environment compositions. We prove a subject-reduction theorem for the type system. Third, a type-inference algorithm is given to the polymorphic environment calculus, and we establish its soundness, termination, and principal-typing theorem.     

Specifying and computing preferred plans

In this paper, we address the problem of specifying and computing preferred plans using rich, qualitative, user preferences. We propose a logical language for specifying preferences over the evolution of states and actions associated with a plan. We provide a semantics for our first-order preference language in the situation calculus, and prove that progression of our preference formulae preserves this semantics. This leads to the development of PPlan, a bounded best-first search planner that computes preferred plans. Our preference language is amenable to integration with many existing planners, and beyond planning, can be used to support a diversity of dynamical reasoning tasks that employ preferences.