Combining a monad and a comonad

We give a systematic treatment of distributivity for a monad and a comonad as arises in giving category theoretic accounts of operational and denotational semantics, and in giving an intensional denotational semantics. We do this axiomatically, in terms of a monad and a comonad in a 2-category, giving accounts of the Eilenberg–Moore and Kleisli constructions. We analyse the eight possible relationships, deducing that two pairs are isomorphic, but that the other pairs are all distinct. We develop those 2-categorical definitions necessary to support this analysis.     

Coalgebraic semantics for timed processes

We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate.     

Call-by-push-value: Decomposing call-by-value and call-by-name

We present the call-by-push-value (CBPV) calculus, which decomposes the typed call-by-value (CBV) and typed call-by-name (CBN) paradigms into fine-grain primitives. On the operational side, we give big-step semantics and a stack machine for CBPV, which leads to a straightforward push/pop reading of CBPV programs. On the denotational side, we model CBPV using cpos and, more generally, using algebras for a strong monad. For storage, we present an O’Hearn-style “behaviour semantics’’ that does not use a monad. We present the translations from CBN and CBV to CBPV. All these translations straightforwardly preserve denotational semantics. We also study their operational properties: simulation and full abstraction. We give an equational theory for CBPV, and show it equivalent to a categorical semantics using monads and algebras. We use this theory to formally compare CBPV to Filinski’s variant of the monadic metalanguage, as well as to Marz’s language SFPL, both of which have essentially the same type structure as CBPV. We also discuss less formally the differences between the CBPV and monadic frameworks.     

Domain semantics of possibility computations

The purpose of this paper is to introduce a kind of computational model, called possibility computation, to deal with non-determinism. Both its denotational and logical semantics in the framework of domain theory are established and their equivalence is verified. In this approach, the denotational semantics will be set up by assigning to programs possibility state transformers, i.e., Scott-continuous functions from the domain of input states to the possibility powerdomain of the codomain of final states. This possibility powerdomain of a dcpo will consist of all possibility valuations, ordered pointwise, of this dcpo. The logical semantics will be given by fuzzy predicate transformers introduced by the first author and Jung following Dijkstra’s predicate transformers. Two semantics equivalence will be verified in terms of the integration of fuzzy predicates with respect to possibility valuations. The categorical monad of the possibility powerdomain will be investigated.     

Operational and goal-independent denotational semantics for Prolog with cut

In this paper we propose an operational and a denotational semantics for Prolog. We deal with the control rules of Prolog and the cut operator. Our denotational semantics provides a goal-independent semantics. This means that the behaviour of a goal in a program is defined as the evaluation of the goal in the denotation (semantics) of the program. We show how our denotational semantics can be specialised into a computed answer semantics and into a call pattern semantics. Our work provides a basis for a precise abstract interpretation of Prolog programs.     

CPO Models for Compact GSOS Languages

In this paper, we present a general way of giving denotational semantics to a class of languages equipped with an operational semantics that fits the GSOS format of Bloom, Istrail, and Meyer. The canonical model used for this purpose will be Abramsky's domain of synchronization trees, and the denotational semantics automatically generated by our methods will be guaranteed to be fully abstract with respect to the finitely observable part of the bisimulation preorder. In the process of establishing the full abstraction result, we also obtain several general results on the bisimulation preorder (including a complete axiomatization for it), and give a novel operational interpretation of GSOS languages.     

Semantic theories of programs with nested interrupts

In the design of dependable software for embedded and real-time operating systems, time analysis is a crucial but extremely difficult issue, the challenge of which is exacerbated due to the randomness and nondeterminism of interrupt handling behaviors. Thus research into a theory that integrates interrupt behaviors and time analysis seems to be important and challenging. In this paper, we present a programming language to describe programs with interrupts that is comprised of two essential parts: main program and interrupt handling programs. We also explore a timed operational semantics and a denotational semantics to specify the meanings of our language. Furthermore, a strategy of deriving denotational semantics from the timed operational semantics is provided to demonstrate the soundness of our operational semantics by showing the consistency between the derived denotational semantics and the original denotational semantics.     

Probabilistic Ianov's schemes

We present probabilistic Ianov's schemes, studying their semantics and proving the equivalence between operational and denotational ones. We also study the equivalence of schemes relative to them; as usual all these equivalence problems are decidable, and we prove it giving the appropriate decision algorithms.     

On Modeling Coordination via Asynchronous Communication and Enhanced Matching

The paper proposes a theoretical study of a coordination language embodying Linda's asynchronous communication primitive with a refined matching mechanism based on pairs composed of attribute names associated with their values. Computations in this language are described by means of an operational semantics, reporting the whole traces of executions. The non-compositionality of this intuitive operational semantics motivates the design of a compositional and fully abstract denotational semantics, which is then exploited for studying program equivalence in this setting.     

A Provably Correct Translation of the λ-Calculus into a Mathematical Model of C++

We introduce a translation of the simply typed λ-calculus into C++, and give a mathematical proof of the correctness of this translation. For this purpose we develop a suitable fragment of C++ together with a denotational semantics. We introduce a formal translation of the λ-calculus into this fragment, and show that this translation is correct with respect to the denotational semantics. We show as well a completeness result, namely that by translating λ-terms we obtain essentially all C++ terms in this fragment. We introduce a mathematical model for the evaluation of programs of this fragment, and show that the evaluation computes the correct result with respect to this semantics.     

From algebraic semantics to denotational semantics for Verilog

This paper considers how the algebraic semantics for Verilog relates with its denotational semantics. Our approach is to derive the denotational semantics from the algebraic semantics. We first present the algebraic laws for Verilog. Every program can be expressed as a guarded choice that can model the execution of a program. In order to investigate the parallel expansion laws, a sequence is introduced, indicating which instantaneous action is due to which exact parallel component. A head normal form is defined for each program by using a locality sequence. We provide a strategy for deriving the denotational semantics based on head normal form. Using this strategy, the denotational semantics for every program can be calculated. Program equivalence can also be explored by using the derived denotational semantics. A short version of this paper appeared in Proc. ICECCS 2006: 11th IEEE International Conference on Engineering of Complex Computer Systems [48]. This work is partially supported by the National Basic Research Program of China (No. 2005CB321904), the National High Technology Research and Development Program of China (No. 2007AA010302) and the National Natural Science Foundation of China (No. 90718004). Jonathan Bowen is a visiting professor at King’s College London and an emeritus professor at London South Bank University.     

Semantics of plan revision in intelligent agents

In this paper, we give an operational and denotational semantics for a meta-language of the 3APL agent programming language. With this meta-language, various 3APL interpreters can be programmed. We prove equivalence of the operational and denotational semantics. Furthermore, we give an operational semantics for object-level 3APL. Using this semantics, we relate the 3APL meta-language to object-level 3APL by providing a specific interpreter, the semantics of which will prove to be equivalent to object-level 3APL.     

Two-dimensional linear algebra

We introduce two-dimensional linear algebra, by which we do not mean two-dimensional vector spaces but rather the systematic replacement in linear algebra of sets by categories. This entails the study of categories that are simultaneously categories of algebras for a monad and categories of coalgebras for comonad on a category such as SymMon_s, the category of small symmetric monoidal categories. We outline relevant notions such as that of pseudo-closed 2-category, symmetric monoidal Lawvere theory, and commutativity of a symmetric monoidal Lawvere theory, and we explain the role of coalgebra, explaining its precedence over algebra in this setting. We outline salient results and perspectives given by the dual approach of algebra and coalgebra, extending to two dimensions the study of linear algebra.     

The semantics of the combination of atomized statements and parallel choice

In this paper we define a uniform language that is an extension of the language underlying the process algebraPA. One of the main extensions of this language overPA is given by so-called atomizing brackets. If we place these brackets around a statement then we treat this statement as an atomic action. Put differently, these brackets remove all interleaving points. We present a transition system for the language and derive its operational semantics. We show that there are several options for defining a transition system such that the resulting operational semantics is a conservative extension of the semantics forPA. We define a semantic domain and a denotational model for the language. Next we define a closure operator on the semantic domain and show how to use this closure operator to derive a fully abstract denotational semantics. Then the algebraic theory of the language is considered. We define a collection of axioms and a term rewrite system based on these axioms. Using this term rewrite system we are able to identify normal forms for the language. It is shown that these axioms capture the denotational equality. It follows that if two terms are provably equal then they have the same operational semantics. Finally, we show how to extend the axiomatization in order to axiomatize its operational equivalence.     

Non-Standard Semantics for Program Slicing

In this paper we generalize the notion of compositional semantics to cope with transfinite reductions of a transition system. Standard denotational and predicate transformer semantics, even though compositional, provide inadequate models for some known program manipulation techniques. We are interested in the systematic design of extended compositional semantics, observing possible transfinite computations, i.e. computations that may occur after a given number of infinite loops. This generalization is necessary to deal with program manipulation techniques modifying the termination status of programs, such as program slicing. We include the transfinite generalization of semantics in the hierarchy developed in 1997 by P. Cousot, where semantics at different levels of abstraction are related with each other by abstract interpretation. We prove that a specular hierarchy of non-standard semantics modeling transfinite computations of programs can be specifiedin such a way that the standard hierarchy can be derived by abstract interpretation. We prove that non-standard transfinite denotational and predicate transformer semantics can be both systematically derived as solutions of simple abstract domain equations involving the basic operation of reduced power of abstract domains. This allows us to prove the optimality of these semantics, i.e. they are the most abstract semantics in the hierarchy which are compositional and observe respectively the terminating and initial states of transfinite computations, providing an adequate mathematical model for program manipulation.     

A non-standard semantics for program slicing and dependence analysis

We introduce a new non-strict semantics for a simple while language. We demonstrate that this semantics allows us to give a denotational definition of variable dependence and neededness, which is consistent with program slicing. Unlike other semantics used in variable dependence, our semantics is substitutive. We prove that our semantics is preserved by traditional slicing algorithms.     

Tabular表达式的指称语义研究

Tabular 表达式是一种采用表格化结构组织函数或关系的形式化描述工具,在需求工程领域中具有广泛的应用,为Tabular 表达式建立形式的语义模型是非常必要的.针对Tabular 表达式通用模型,给出了Tabular 表达式的形式文法及指称语义.通过定义形式文法中各语法单元的语义指派方程,描述了Tabular 表达式的指称语义,分别对传统类型Tabular 表达式和新类型Tabular 表达式中一些典型表类型的指称语义进行了描述,并与其他几种Tabular 表达式的语义描述方法进行了比较.分析结果表明：该语义描述方法不仅准确描述了Tabular 表达式的语义,而且不再受Tabular 表达式模型和Tabular 表达式类型的限制,打破了现有方法的局限性,是一种非常有效的方法.     

Relating confluence,innermost-confluence and outermost-confluence properties of term rewriting systems

Innermost-confluence is important in giving call-by-value and denotational semantics and outermost-confluence is important in giving call-by-need and lazy semantics of programs. In this paper, we give a few sets of sufficient conditions under which the properties of confluence, innermost-confluence and outermost-confluence coincide. Confluence and innermost-confluence coincide for weakly innermost normalizing overlay systems and confluence and outermost-confluence coincide for outermost normalizing left-linear overlay systems. In general, every weakly innermost (outermost) normalizingconfluent system isinnermost (outermost) confluent but the converse is not true.     

Denotational and operational semantics for prolog

The semantics of PROLOG programs is usually given in terms of the model theory of first-order logic. However, this does not adequately characterizethe computational behavior of PROLOG programs. PROLOG implementations typically use a sequential evaluation strategy based on the textual order of clauses and literals in a program, as well as nonlogical features like cut. In this work we develop a denotational semantics that captures thecomputational behavior of PROLOG. We present a semantics for “cut-free” PROLOG, which is then extended to PROLOG with cut. For each case we develop a congruence proof that relates the semantics to a standard operational interpreter. As an application of our denotational semantics, we show the correctness of some standard “folk” theorems regarding transformations on PROLOG programs.