By Thomas Ehrhard, Jean-Yves Girard, Paul Ruet, Philip Scott

Linear good judgment is a department of evidence thought which supplies subtle instruments for the research of the computational facets of proofs. those instruments comprise a duality-based express semantics, an intrinsic graphical illustration of proofs, the advent of well-behaved non-commutative logical connectives, and the suggestions of polarity and focalisation. those a number of features are illustrated the following via introductory tutorials in addition to extra specialized contributions, with a specific emphasis on functions to machine technology: denotational semantics, lambda-calculus, good judgment programming and concurrency thought. the amount is rounded-off by way of invited contributions on new issues rooted in contemporary advancements of linear good judgment. The booklet derives from a summer season tuition that used to be the climax of the ecu education and Mobility of Researchers venture 'Linear common sense in desktop Science'. it's a great creation to a few of the main energetic learn subject matters within the region.

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**Example text**

Here 0 is the initial object, ® is disjoint union, which acts as both product and coproduct in this category, and ®9 is the symmetrized tensor, and is expressed as a certain quotient (coequalizer). For example, X ®s X is the coequalizer of the identity and the symmetry map. In general, the n'th symmetric group acts on the n-fold tensor of X with itself, and the symmetrized tensor is the coequalizer of all of these maps. It was an observation of Barr that this formula works frequently, but certainly not always.

It turns out that this involves categorical versions of Parigot's Aµ-calculus and ideas arising from the notion of continuations in programming language theory. The appropriate categorical framework, called control categories, was developed by P. Selinger in [611. 23 A monoidal (or tensored) category (C, I, ®, a, P, r) is a category C, with functor ® : C x C -+ C, unit object I E ob(C), and A specified isos: aABC : (A (9 B) ®C - A ®(B (9 C), 2A : I ®A rA : A ® I - - A satisfying the following: PI = rI : I ® I - I , as well as: A®(I®C) (A®I)®C 1®2CI A®C IrA®1 = A®C A(B(CD)) - (AB)(CD) - ((AB)C)D 1®a A((BC)D) a®11 a (A(BC))D where we omit ®'s in the second diagram for typographical reasons.

Conversely, C(L), the ccc syntactically generated by a lambda theory L, is essentially the closed term model, viewed as a ccc ( for details, see [51]). We remark that for this to go through, we require that our languages (in this case typed A-calculi) need not be freely generated (in the same sense that deductive systems can be generalized). Moreover, F(g), the free ccc generated by graph 9, is equivalent to C(L(q)), where L(9) is the typed lambda calculus generated by the graph 4 ( analogous to L(C)).