By Andrej Bauer, Matija Pretnar (auth.), Reiko Heckel, Stefan Milius (eds.)

This ebook constitutes the refereed lawsuits of the fifth foreign convention on Algebra and Coalgebra in desktop technology, CALCO 2013, held in Warsaw, Poland, in September 2013. The 18 complete papers offered including four invited talks have been conscientiously reviewed and chosen from 33 submissions. The papers hide subject matters within the fields of summary types and logics, really expert versions and calculi, algebraic and coalgebraic semantics, process specification and verification, in addition to corecursion in programming languages, and algebra and coalgebra in quantum computing. The e-book additionally comprises 6 papers from the CALCO instruments Workshop, co-located with CALCO 2013 and devoted to instruments in response to algebraic and/or coalgebraic principles.

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Theory and Applications of Category Theory 18(22), 665–732 (2007) 18. : An intuitionistic theory of types (1972), published in Twenty-Five Years of Constructive Type Theory 19. : Intuitionistic type theory. Bibliopolis Naples (1984) 20. : The view from the left. Journal of Functional Programming 14(1), 69–111 (2004) 21. : A ﬁnite axiomatisation of inductive-inductive deﬁnitions. , Seisenberger, M. ) Logic, Construction, Computation, Ontos Mathematical Logic, vol. 3, pp. 259–287. Ontos Verlag (2012) 22.

This observation is crucial for the interpretation of IR codes as functors. Indeed, given a type D, which we think as the discrete (possibly large) set of its terms, we interpret IR codes as functors Fam D → Fam D. Theorem 4 (IR functors). Let D : type. Every code γ : IR(D) induces a functor γ : Fam D → Fam D Positive Inductive-Recursive Deﬁnitions 23 Proof. We deﬁne γ : Fam D → Fam D by induction on the structure of the code. We ﬁrst give the action on objects: ι c (X, P ) = (1, λ . c) σA f (X, P ) = f a (X, P ) a :A F (P ◦ g) (X, P ) δA F (X, P ) = g :A→X We now give the action on morphisms.

E. an element u : U , and a function f : T u → U representing the A-indexed family of sets B. The decoding function T maps elements of U according to the description above: the code for natural numbers decodes to the set of natural numbers N while an element (u, f ) of the right summand decodes to the Σ-type it denotes. g. Martin-L¨ of’s computability predicates [18] or Aczel’s Frege structures [3]. Lately the use of inductive-recursive deﬁnitions to encode invariants in ordinary data structures has also been considered [11].