We propose a notion of morphisms between tree automata based on game semantics. Morphisms are winning strategies on a synchronous restriction of the linear implication between acceptance games. This leads to split indexed categories, with substitution based on a suitable notion of synchronous tree function. By restricting to tree functions issued from maps on alphabets, this gives a fibration of tree automata. We then discuss the (fibrewise) monoidal structure issued from the synchronous product of automata. We also discuss how a variant of the usual projection operation on automata leads to an existential quantification in the fibered sense. Our notion of morphism is correct (it respects language inclusion), and in a weaker sense also complete.
In classical homotopy theory, the homotopy hypothesis asserts that the fundamental varpi-groupoid construction induces an equivalence between topological spaces and weak varpi-groupoids. In the light of Voevodsky's univalent foundations program, which puts forward an interpretation of types as topological spaces, we consider the question of transposing the homotopy hypothesis to type theory. Indeed such a transposition could stand as a new approach to specifying higher inductive types. Since the formalisation of general weak varpi-groupoids in type theory is a difficult task, we only take a first step towards this goal, which consists in exploring a shortcut through strict varpi-categories. The first outcome is a satisfactory type-theoretic notion of strict varpi-category, which has hsets of cells in all dimensions. For this notion, defining the 'fundamental strict varpi-category' of a type seems out of reach. The second outcome is an 'incoherently strict' notion of type-theoretic varpi-category, which admits arbitrary types of cells in all dimensions. These are the 'wild' varpi-categories of the title. They allow the definition of a 'fundamental wild varpi-category' map, which leads to our (partial) homotopy hypothesis for type theory (stating an adjunction, not an equivalence). All of our results have been formalised in the Coq proof assistant. Our formalisation makes systematic use of the machinery of coinductive types.
We study an extension of Plotkin's call-by-value lambda-calculus by means of two commutation rules (sigma-reductions). Recently, it has been proved that this extended calculus provides elegant characterizations of many semantic properties, as for example solvability. We prove a standardization theorem for this calculus by generalizing Takahashi's approach of parallel reductions. The standardization property allows us to prove that our calculus is conservative with respect to the Plotkin's one. In particular, we show that the notion of solvability for this calculus coincides with that for Plotkin's call-by-value lambda-calculus.
We investigate a framework of Krivine realizability with I/O effects, and present a method of associating realizability models to specifications on the I/O behavior of processes, by using adequate interpretations of the central concepts of `pole' and `proof-like term'. This method does in particular allow to associate realizability models to computable functions. Following recent work of Streicher and others we show how these models give rise to triposes and toposes.
Proof equivalence in a logic is the problem of deciding whether two proofs are equivalent modulo a set of permutation of rules that reflects the commutative conversions of its cut-elimination procedure. As such, it is related to the question of proofnets: finding canonical representatives of equivalence classes of proofs that have good computational properties. It can also be seen as the word problem for the notion of free category corresponding to the logic. It has been recently shown that proof equivalence in MLL (the multiplicative with units fragment of linear logic) is Pspace-complete, which rules out any low-complexity notion of proofnet for this particular logic. Since it is another fragment of linear logic for which attempts to define a fully satisfactory low-complexity notion of proofnet have not been successful so far, we study proof equivalence in MALL- (multiplicative-additive without units fragment of linear logic) and discover a situation that is totally different from the MLL case. Indeed, we show that proof equivalence in MALL- corresponds(under AC_0 reductions)to equivalence of binary decision diagrams, a data structure widely used to represent and analyze Boolean functions efficiently. We show these two equivalent problems to be Logspace-complete. If this technically leaves open the possibility for a complete solution to the question of proofnets for MALL-, the established relation with binary decision diagrams actually suggests a negative solution to this problem.
