67 research outputs found
Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic
We present a new syntactical proof that first-order Peano Arithmetic with
Skolem axioms is conservative over Peano Arithmetic alone for arithmetical
formulas. This result - which shows that the Excluded Middle principle can be
used to eliminate Skolem functions - has been previously proved by other
techniques, among them the epsilon substitution method and forcing. In our
proof, we employ Interactive Realizability, a computational semantics for Peano
Arithmetic which extends Kreisel's modified realizability to the classical
case.Comment: In Proceedings CL&C 2012, arXiv:1210.289
Wave-Style Token Machines and Quantum Lambda Calculi
Particle-style token machines are a way to interpret proofs and programs,
when the latter are written following the principles of linear logic. In this
paper, we show that token machines also make sense when the programs at hand
are those of a simple quantum lambda-calculus with implicit qubits. This,
however, requires generalising the concept of a token machine to one in which
more than one particle travel around the term at the same time. The presence of
multiple tokens is intimately related to entanglement and allows us to give a
simple operational semantics to the calculus, coherently with the principles of
quantum computation.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441
Probabilistic Operational Semantics for the Lambda Calculus
Probabilistic operational semantics for a nondeterministic extension of pure
lambda calculus is studied. In this semantics, a term evaluates to a (finite or
infinite) distribution of values. Small-step and big-step semantics are both
inductively and coinductively defined. Moreover, small-step and big-step
semantics are shown to produce identical outcomes, both in call-by- value and
in call-by-name. Plotkin's CPS translation is extended to accommodate the
choice operator and shown correct with respect to the operational semantics.
Finally, the expressive power of the obtained system is studied: the calculus
is shown to be sound and complete with respect to computable probability
distributions.Comment: 35 page
QPCF: higher order languages and quantum circuits
qPCF is a paradigmatic quantum programming language that ex- tends PCF with
quantum circuits and a quantum co-processor. Quantum circuits are treated as
classical data that can be duplicated and manipulated in flexible ways by means
of a dependent type system. The co-processor is essentially a standard QRAM
device, albeit we avoid to store permanently quantum states in between two
co-processor's calls. Despite its quantum features, qPCF retains the classic
programming approach of PCF. We introduce qPCF syntax, typing rules, and its
operational semantics. We prove fundamental properties of the system, such as
Preservation and Progress Theorems. Moreover, we provide some higher-order
examples of circuit encoding
Quantum Programming Made Easy
We present IQu, namely a quantum programming language that extends Reynold's
Idealized Algol, the paradigmatic core of Algol-like languages. IQu combines
imperative programming with high-order features, mediated by a simple type
theory. IQu mildly merges its quantum features with the classical programming
style that we can experiment through Idealized Algol, the aim being to ease a
transition towards the quantum programming world. The proposed extension is
done along two main directions. First, IQu makes the access to quantum
co-processors by means of quantum stores. Second, IQu includes some support for
the direct manipulation of quantum circuits, in accordance with recent trends
in the development of quantum programming languages. Finally, we show that IQu
is quite effective in expressing well-known quantum algorithms.Comment: In Proceedings Linearity-TLLA 2018, arXiv:1904.0615
A "Game Semantical" Intuitionistic Realizability Validating Markov\u27s Principle
We propose a very simple modification of Kreisel\u27s modified realizability in order to computationally realize Markov\u27s Principle in the context of Heyting Arithmetic. Intuitively, realizers correspond to arbitrary strategies in Hintikka-Tarski games, while in Kreisel\u27s realizability they can only represent winning strategies. Our definition, however, does not employ directly game semantical concepts and remains in the style of functional interpretations. As term calculus, we employ a purely functional language, which is Goedel\u27s System T enriched with some syntactic sugar
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
On Natural Deduction in Classical First-Order Logic: Curry-Howard Correspondence, Strong Normalization and Herbrand's Theorem
International audienceWe present a new Curry-Howard correspondence for classical first-order natural deduction. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for raising and catching multiple exceptions, and from the viewpoint of logic, the excluded middle over arbitrary prenex formulas. The machinery will allow to extend the idea of learning -- originally developed in Arithmetic -- to pure logic. We prove that our typed calculus is strongly normalizing and show that proof terms for simply existential statements reduce to a list of individual terms forming a Herbrand disjunction. A by-product of our approach is a natural-deduction proof and a computational interpretation of Herbrand's Theorem
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