On Constructor Rewrite Systems and the Lambda Calculus

We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by- value beta-reduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in a term rewrite system can be simulated by a constant number of beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:0904.412

An Invariant Cost Model for the Lambda Calculus

We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the call-by-value lambda-calculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of usual beta-reduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form).Comment: 19 page

Towards A Theory Of Quantum Computability

We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum Turing machine. The class of quantum computable functions is recursively enumerable, thus opening the door to a quantum computability theory which may follow some of the classical developments

Quantum Turing Machines Computations and Measurements

Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not being fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example being the intrinsic infinite nature of any quantum computation. In this paper we propose a definition of QTM, which extends and unifies the notions of Deutsch and Bernstein and Vazirani. In particular, we allow both arbitrary quantum input, and meaningful superpositions of computations, where some of them are "terminated" with an "output", while others are not. For some infinite computations an "output" is obtained as a limit of finite portions of the computation. We propose a natural and robust observation protocol for our QTMs, that does not modify the probability of the possible outcomes of the machines. Finally, we use QTMs to define a class of quantum computable functions---any such function is a mapping from a general quantum state to a probability distribution of natural numbers. We expect that our class of functions, when restricted to classical input-output, will be not different from the set of the recursive functions.Comment: arXiv admin note: substantial text overlap with arXiv:1504.02817 To appear on MDPI Applied Sciences, 202

The Standard Model for Programming Languages: The Birth of a Mathematical Theory of Computation

International audienceDespite the insight of some of the pioneers (Turing, von Neumann, Curry, Böhm), programming the early computers was a matter of fiddling with small architecture-dependent details. Only in the sixties some form of "mathematical program development" will be in the agenda of some of the most influential players of that time. A "Mathematical Theory of Computation" is the name chosen by John McCarthy for his approach, which uses a class of recursively computable functions as an (extensional) model of a class of programs. It is the beginning of that grand endeavour to present programming as a mathematical activity, and reasoning on programs as a form of mathematical logic. An important part of this process is the standard model of programming languages-the informal assumption that the meaning of programs should be understood on an abstract machine with unbounded resources, and with true arithmetic. We present some crucial moments of this story, concluding with the emergence, in the seventies, of the need of more "intensional" semantics, like the sequential algorithms on concrete data structures. The paper is a small step of a larger project-reflecting and tracing the interaction between mathematical logic and programming (languages), identifying some of the driving forces of this process. To Maurizio Gabbrielli, on his 60th birthda

General Ramified Recurrence is Sound for Polynomial Time

Leivant's ramified recurrence is one of the earliest examples of an implicit characterization of the polytime functions as a subalgebra of the primitive recursive functions. Leivant's result, however, is originally stated and proved only for word algebras, i.e. free algebras whose constructors take at most one argument. This paper presents an extension of these results to ramified functions on any free algebras, provided the underlying terms are represented as graphs rather than trees, so that sharing of identical subterms can be exploited

Phase semantics and decidability of elementary affine logic

AbstractLight, elementary and soft linear logics are formal systems derived from Linear Logic, enjoying remarkable normalization properties. In this paper, we prove decidability of Elementary Affine Logic, EAL. The result is obtained by semantical means, first defining a class of phase models for EAL and then proving soundness and (strong) completeness, following Okada's technique. Phase models for Light Affine Logic and Soft Linear Logic are also defined and shown complete

Coherent coupling between localised and propagating phonon polaritons

Following the recent observation of localised phonon polaritons in user-defined silicon carbide nano-resonators, here we demonstrate coherent coupling between those localised modes and propagating phonon polaritons bound to the surface of the nano-resonator's substrate. In order to obtain phase-matching, the nano-resonators have been fabricated to serve the double function of hosting the localised modes, while also acting as grating for the propagating ones. The coherent coupling between long lived, optically accessible localised modes, and low-loss propagative ones, opens the way to the design and realisation of phonon-polariton based quantum circuits

From 2-sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and 2-sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction---only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalisation, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (following Scott) for formulating sound deduction rules

Position-agnostic autonomous navigation in vineyards with Deep Reinforcement Learning

Precision agriculture is rapidly attracting research to efficiently introduce automation and robotics solutions to support agricultural activities. Robotic navigation in vineyards and orchards offers competitive advantages in autonomously monitoring and easily accessing crops for harvesting, spraying and performing time-consuming necessary tasks. Nowadays, autonomous navigation algorithms exploit expensive sensors which also require heavy computational cost for data processing. Nonetheless, vineyard rows represent a challenging outdoor scenario where GPS and Visual Odometry techniques often struggle to provide reliable positioning information. In this work, we combine Edge AI with Deep Reinforcement Learning to propose a cutting-edge lightweight solution to tackle the problem of autonomous vineyard navigation with-out exploiting precise localization data and overcoming task-tailored algorithms with a flexible learning-based approach. We train an end-to-end sensorimotor agent which directly maps noisy depth images and position-agnostic robot state information to velocity commands and guides the robot to the end of a row, continuously adjusting its heading for a collision-free central trajectory. Our extensive experimentation in realistic simulated vineyards demonstrates the effectiveness of our solution and the generalization capabilities of our agent