Intuitionistic ϵ- and τ-calculi

There are several open problems in the study of the calculi which result from adding either of Hilbert's ?- or τ-operators to the first order intuitionistic predicate calculus. This paper provides answers to several of them. In particular, the first complete and sound semantics for these calculi are presented, in both a “quasi-extensional” version which uses choice functions in a straightforward way to interpret the ?- or τ-terms, and in a form which does not require extensionality assumptions. Unlike the classical case, the addition of either operator to intuitionistic logic is non-conservative. Several interesting consequences of the addition of each operator are proved. Finally, the independence of several other schemes in either calculus are also proved, making use of the semantics supplied earlier in the paper.     

Function Spaces of Posets with Projections

This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.More precisely, given a fixed directed index set (I,), we consider triples (D,,(p _i ) _iI ) with (D,) a poset and (p _i ) _iI a monotone net of projections of D. We call them (I,)-pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,)-pop's. Such homomorphisms respect both order and projections.Any (I,)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,)-pop's turns out to form an (I,)-pop itself. We show that its uniformity is the uniformity of uniform convergence. This enables us to prove that properties such as completeness and compactness transfer to function pop's.Concerning categorical properties of (I,)-pop's, we will see that we are in a lucky situation from a computer scientist's point of view: we obtain Cartesian closed categories. Moreover, by a D -construction we get (I,)-pop's that are isomorphic to their own exponent. This yields new models for the untyped -calculus.     

S-Storage Operators

In 1990, J. L. Krivine introduced the notion of storage operator to simulate, for Church integers, the “call by value” in a context of a “call by name” strategy. In the present paper we define for every λ-term S which realizes the successor function on Church integers the notion of S-storage operator. We prove that every storage operator is an S-storage operator. But the converse is not always true.     

Hybrid Dynamics of Stochastic π-Calculus

In this paper we describe a method to map stochastic π-calculus processes in chemical ground form into hybrid automata. Hybrid automata are tools widely employed to model systems characterized by both discrete and continuous evolution and their use in the context of Systems Biology allows us to address rather fundamental issues. Specifically, the key ingredient we use in this work is the possibility granted by hybrid automata to implement a separation of control and molecular terms in biochemical systems. The computational counterpart of our analysis turns out to be related to the determination of conservation properties of the system.      

The particle in a box problem inq-quantum mechanics

Aq-deformed,q-Hermitian kinetic energy operator is realised and hence aq-Schrödinger equation (q-SE) is obtained. Theq-SE for a particle confined in an infinite potential box is solved and the energy spectrum is found to have an upper bound.     

The basis decision problem in λ-calculus

We show that the problem of deciding if a finite set of closed terms in normal form is a basis is recursively unsolvable. The restricted problem concerning one element sets is still recursively unsolvable. MSC: 03B40, 03D35.     

ON (q,h)-ANALOGUE OF FRACTIONAL CALCULUS

The paper discusses fractional integrals and derivatives appearing in the so-called (q, h)-calculus which is reduced for h = 0 to quantum calculus and for q = h = 1 to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we give comparisons with the known results and discuss possible extensions to more general settings.     

The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

It is proved that the Stokes operator in L^q -space on an infinite cylindrical domain of , , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section is studied in L ^q where and is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on -boundedness of operator families and on square function estimates involving Muckenhoupt weights.     

Hilbert's ϵ-operator in intuitionistic type theories

We investigate Hilbert's ?-calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed ?-terms. We extend the usual topos semantics for type theories to the ?-operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined” ?-term. MSC: 03B15, 03B20, 03G30.