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Initiality for Typed Syntax and Semantics
"... Abstract. We give an algebraic characterization of the syntax and semantics of a class of simply–typed languages, such as the language PCF: we characterize simply–typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. For this purp ..."
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Abstract. We give an algebraic characterization of the syntax and semantics of a class of simply–typed languages, such as the language PCF: we characterize simply–typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. For this purpose, we employ techniques developed in two previous works: in [2], we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In [1], we characterize untyped syntax with reduction rules as initial object in a category of models. In the present work, we show that those techniques are modular enough to be combined: we thus characterize simply–typed syntax with reduction rules as initial object in a category. The universal property yields an operator which allows to specify translations — that are semantically faithful by construction — between languages over possibly different sets of types. We specify a language by a 2–signature, that is, a signature on two levels: the syntactic level specifies the types and terms of the language, and associates a type to each term. The semantic level specifies, through inequations, reduction rules on the terms of the language. To any given
HigherOrder Processes, Functions, and Sessions: A Monadic Integration
"... Abstract. In prior research we have developed a CurryHoward interpretation of linear sequent calculus as sessiontyped processes. In this paper we uniformly integrate this computational interpretation in a functional language via a linear contextual monad that isolates sessionbased concurrency. Mo ..."
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Abstract. In prior research we have developed a CurryHoward interpretation of linear sequent calculus as sessiontyped processes. In this paper we uniformly integrate this computational interpretation in a functional language via a linear contextual monad that isolates sessionbased concurrency. Monadic values are open process expressions and are first class objects in the language, thus providing a logical foundation for higherorder session typed processes. We illustrate how the combined use of the monad and recursive types allows us to cleanly write a rich variety of concurrent programs, including higherorder programs that communicate processes. We show the standard metatheoretic result of type preservation, as well as a global progress theorem, which to the best of our knowledge, is new in the higherorder session typed setting. 1
Instances of computational effects: an algebraic perspective
"... Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different ins ..."
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Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different instances of a particular computational effect. We use our framework to give a general account of several notions of computation that had previously been analyzed in terms of monads on presheaf categories: the analysis of local store by Plotkin and Power; the analysis of restriction by Pitts; and the analysis of the pi calculus by Stark. I.
Short Contributions
"... CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. Probability, Networks and Algorithms (PN ..."
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CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. Probability, Networks and Algorithms (PNA) Software Engineering (SEN)
Foundations and Applications of HigherDimensional Directed Type Theory
"... Intuitionistic type theory [43] is an expressive formalism that unifies mathematics and computation. A central concept is the propositionsastypes principle, according to which propositions are interpreted as types, and proofs of a proposition are interpreted as programs of the associated type. Mat ..."
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Intuitionistic type theory [43] is an expressive formalism that unifies mathematics and computation. A central concept is the propositionsastypes principle, according to which propositions are interpreted as types, and proofs of a proposition are interpreted as programs of the associated type. Mathematical propositions are thereby to be understood as specifications, or problem descriptions, that are solved by providing a program that meets the specification. Conversely, a program can, by the same token, be understood as a proof of its type viewed as a proposition. Over the last quartercentury type theory has emerged as the central organizing principle of programming language research, through the identification of the informal concept of language features with type structure. Numerous benefits accrue from the identification of proofs and programs in type theory. First, it provides the foundation for integrating types and verification, the two most successful formal methods used to ensure the correctness of software. Second, it provides a language for the mechanization of mathematics in which proof checking is equivalent to type checking, and proof search is equivalent to writing a program to meet a specification.
An algebraic presentation of predicate logic (extended abstract)
"... Abstract. We present an algebraic theory for a fragment of predicate logic. The fragment has disjunction, existential quantification and equality. It is not an algebraic theory in the classical sense, but rather within a new framework that we call ‘parameterized algebraic theories’. We demonstrate t ..."
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Abstract. We present an algebraic theory for a fragment of predicate logic. The fragment has disjunction, existential quantification and equality. It is not an algebraic theory in the classical sense, but rather within a new framework that we call ‘parameterized algebraic theories’. We demonstrate the relevance of this algebraic presentation to computer science by identifying a programming language in which every type carries a model of the algebraic theory. The result is a simple functional logic programming language. We provide a syntaxfree representation theorem which places terms in bijection with sieves, a concept from category theory. We study presentationinvariance for general parameterized algebraic theories by providing a theory of clones. We show that parameterized algebraic theories characterize a class of enriched monads. 1
Universal Properties of Impure Programming Languages
"... We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums ..."
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We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums and functions — can be characterized by universal properties in the setting of ‘premulticategories’, multicategories where the commutativity law may fail. This leads us to new, universal characterizations of two earlier equational theories of impure programming languages: the premonoidal categories of Power and Robinson, and the monadbased models of Moggi. Our analysis thus puts these earlier abstract ideas on a canonical foundation, bringing them to a new, syntactic level. F.3.2 [Semantics of Pro