12,006 research outputs found
Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional programming are
tricky to handle because of their cyclicity. This paper presents an
investigation of categorical, algebraic, and computational foundations of
cyclic datatypes. Our framework of cyclic datatypes is based on second-order
algebraic theories of Fiore et al., which give a uniform setting for syntax,
types, and computation rules for describing and reasoning about cyclic
datatypes. We extract the "fold" computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby, the rules
are correct by construction. We prove strong normalisation using the General
Schema criterion for second-order computation rules. Rather than the fixed
point law, we particularly choose Bekic law for computation, which is a key to
obtaining strong normalisation. We also prove the property of "Church-Rosser
modulo bisimulation" for the computation rules. Combining these results, we
have a remarkable decidability result of the equational theory of cyclic data
and fold.Comment: 38 page
Globular: an online proof assistant for higher-dimensional rewriting
This article introduces Globular, an online proof assistant for the
formalization and verification of proofs in higher-dimensional category theory.
The tool produces graphical visualizations of higher-dimensional proofs,
assists in their construction with a point-and- click interface, and performs
type checking to prevent incorrect rewrites. Hosted on the web, it has a low
barrier to use, and allows hyperlinking of formalized proofs directly from
research papers. It allows the formalization of proofs from logic, topology and
algebra which are not formalizable by other methods, and we give several
examples
Deriving Bisimulation Congruences: A 2-Categorical Approach
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-pushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisimulation on the LTS derived via GRPOs is a congruence, provided that sufficiently many GRPOs exist. The theory is applied to a simple subset of CCS and the resulting LTS is compared to one derived using a procedure proposed by Sewell
Deriving Bisimulation Congruences using 2-Categories
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-pushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisimulation on the LTS derived via GRPOs is a congruence, provided that sufficiently many GRPOs exist. The theory is applied to a simple subset of CCS and the resulting LTS is compared to one derived using a procedure proposed by Sewell
Initial Algebra Semantics for Cyclic Sharing Tree Structures
Terms are a concise representation of tree structures. Since they can be
naturally defined by an inductive type, they offer data structures in
functional programming and mechanised reasoning with useful principles such as
structural induction and structural recursion. However, for graphs or
"tree-like" structures - trees involving cycles and sharing - it remains
unclear what kind of inductive structures exists and how we can faithfully
assign a term representation of them. In this paper we propose a simple term
syntax for cyclic sharing structures that admits structural induction and
recursion principles. We show that the obtained syntax is directly usable in
the functional language Haskell and the proof assistant Agda, as well as
ordinary data structures such as lists and trees. To achieve this goal, we use
a categorical approach to initial algebra semantics in a presheaf category.
That approach follows the line of Fiore, Plotkin and Turi's models of abstract
syntax with variable binding
Coherence for Skew-Monoidal Categories
I motivate a variation (due to K. Szlach\'{a}nyi) of monoidal categories
called skew-monoidal categories where the unital and associativity laws are not
required to be isomorphisms, only natural transformations. Coherence has to be
formulated differently than in the well-known monoidal case. In my (to my
knowledge new) version, it becomes a statement of uniqueness of normalizing
rewrites. I present a proof of this coherence theorem and also formalize it
fully in the dependently typed programming language Agda.Comment: In Proceedings MSFP 2014, arXiv:1406.153
Nested Term Graphs (Work In Progress)
We report on work in progress on 'nested term graphs' for formalizing
higher-order terms (e.g. finite or infinite lambda-terms), including those
expressing recursion (e.g. terms in the lambda-calculus with letrec). The idea
is to represent the nested scope structure of a higher-order term by a nested
structure of term graphs.
Based on a signature that is partitioned into atomic and nested function
symbols, we define nested term graphs both in a functional representation, as
tree-like recursive graph specifications that associate nested symbols with
usual term graphs, and in a structural representation, as enriched term graph
structures. These definitions induce corresponding notions of bisimulation
between nested term graphs. Our main result states that nested term graphs can
be implemented faithfully by first-order term graphs.
keywords: higher-order term graphs, context-free grammars, cyclic
lambda-terms, higher-order rewrite systemsComment: In Proceedings TERMGRAPH 2014, arXiv:1505.0681
Termination orders for 3-dimensional rewriting
This paper studies 3-polygraphs as a framework for rewriting on
two-dimensional words. A translation of term rewriting systems into
3-polygraphs with explicit resource management is given, and the respective
computational properties of each system are studied. Finally, a convergent
3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In
order to prove these results, it is explained how to craft a class of
termination orders for 3-polygraphs.Comment: 30 pages, 35 figure
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