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Register a new userLean and Full Congruence Formats for Recursion
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Added by
Rob van Glabbeek
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Authors
Rob van Glabbeek
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In this paper I distinguish two (pre)congruence requirements for semantic equivalences and preorders on processes given as closed terms in a system description language with a recursion construct. A lean congruence preserves equivalence when replacing closed subexpressions of a process by equivalent alternatives. A full congruence moreover allows replacement within a recursive specification of subexpressions that may contain recursion variables bound outside of these subexpressions. I establish that bisimilarity is a lean (pre)congruence for recursion for all languages with a structural operational semantics in the ntyft/ntyxt format. Additionally, it is a full congruence for the tyft/tyxt format.
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Probabilistic transition system specifications (PTSSs) in the $nt mu ftheta / ntmu xtheta$ format provide structural operational semantics for Segala-type systems that exhibit both probabilistic and nondeterministic behavior and guarantee that bisimilarity is a congruence for all operator defined in such format. Starting from the $nt mu ftheta / ntmu xtheta$ format, we obtain restricted formats that guarantee that three coarser bisimulation equivalences are congruences. We focus on (i) Segalas variant of bisimulation that considers combined transitions, which we call here convex bisimulation; (ii) the bisimulation equivalence resulting from considering Park & Milners bisimulation on the usual stripped probabilistic transition system (translated into a labelled transition system), which we call here probability obliterated bisimulation; and (iii) a probability abstracted bisimulation, which, like bisimulation, preserves the structure of the distributions but instead, it ignores the probability values. In addition, we compare these bisimulation equivalences and provide a logic characterization for each of them.
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We present the first session typing system guaranteeing request-response liveness properties for possibly non-terminating communicating processes. The types augment the branch and select types of the standard binary session types with a set of required responses, indicating that whenever a particular label is selected, a set of other labels, its responses, must eventually also be selected. We prove that these extended types are strictly more expressive than standard session types. We provide a type system for a process calculus similar to a subset of collaborative BPMN processes with internal (data-based) and external (event-based) branching, message passing, bounded and unbounded looping. We prove that this type system is sound, i.e., it guarantees request-response liveness for dead-lock free processes. We exemplify the use of the calculus and type system on a concrete example of an infinite state system.
The libraries of proof assistants like Isabelle, Coq, HOL are notoriously difficult to interpret by external tools: de facto, only the prover itself can parse and process them adequately. In the case of Isabelle, an export of the library into a FAIR (Findable, Accessible, Interoperable, and Reusable) knowledge exchange format was already envisioned by the authors in 1999 but had previously proved too difficult. After substantial improvements of the Isabelle Prover IDE (PIDE) and the OMDoc/Mmt format since then, we are now able to deliver such an export. Concretely we present an integration of PIDE and MMT that allows exporting all Isabelle libraries in OMDoc format. Our export covers the full Isabelle distribution and the Archive of Formal Proofs (AFP) -- more than 12 thousand theories and locales resulting in over 65GB of OMDoc/XML. Such a systematic export of Isabelle content to a well-defined interchange format like OMDoc enables many applications such as dependency management, independent proof checking, or library search.
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Lof type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.
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