457 research outputs found
Abstract machines for dialogue games
The notion of abstract Boehm tree has arisen as an operationally-oriented distillation of works on game semantics, and has been investigated in two papers. This paper revisits the notion, providing more syntactic support and more examples (like call-by-value evaluation) illustrating the generality of the underlying computing device. Precise correspondences between various formulations of the evaluation mechanism of abstract Boehm trees are established
An Embedding of the BSS Model of Computation in Light Affine Lambda-Calculus
This paper brings together two lines of research: implicit characterization
of complexity classes by Linear Logic (LL) on the one hand, and computation
over an arbitrary ring in the Blum-Shub-Smale (BSS) model on the other. Given a
fixed ring structure K we define an extension of Terui's light affine
lambda-calculus typed in LAL (Light Affine Logic) with a basic type for K. We
show that this calculus captures the polynomial time function class FP(K):
every typed term can be evaluated in polynomial time and conversely every
polynomial time BSS machine over K can be simulated in this calculus.Comment: 11 pages. A preliminary version appeared as Research Report IAC CNR
Roma, N.57 (11/2004), november 200
On Classical PCF, Linear Logic and the MIX Rule
We study a classical version of PCF from a semantical point of view. We define a general notion of model based on categorical models of Linear Logic, in the spirit of earlier work by Girard, Regnier and Laurent. We give a concrete example based on the relational model of Linear Logic, that we present as a non-idempotents intersection type system, and we prove an Adequacy Theorem using ideas introduced by Krivine. Following Danos and Krivine, we also consider an extension of this language with a MIX construction introducing a form of must non-determinism; in this language, a program of type integer can have more than one value (or no value at all, raising an error). We propose a refinement of the relational model of classical PCF in which programs of type integer are single valued; this model rejects the MIX syntactical constructs (and the MIX rule of Linear Logic)
Lecture notes on the lambda calculus
This is a set of lecture notes that developed out of courses on the lambda
calculus that I taught at the University of Ottawa in 2001 and at Dalhousie
University in 2007 and 2013. Topics covered in these notes include the untyped
lambda calculus, the Church-Rosser theorem, combinatory algebras, the
simply-typed lambda calculus, the Curry-Howard isomorphism, weak and strong
normalization, polymorphism, type inference, denotational semantics, complete
partial orders, and the language PCF.Comment: 120 pages. Added in v2: section on polymorphis
Thin Games with Symmetry and Concurrent Hyland-Ong Games
We build a cartesian closed category, called Cho, based on event structures.
It allows an interpretation of higher-order stateful concurrent programs that
is refined and precise: on the one hand it is conservative with respect to
standard Hyland-Ong games when interpreting purely functional programs as
innocent strategies, while on the other hand it is much more expressive. The
interpretation of programs constructs compositionally a representation of their
execution that exhibits causal dependencies and remembers the points of
non-deterministic branching.The construction is in two stages. First, we build
a compact closed category Tcg. It is a variant of Rideau and Winskel's category
CG, with the difference that games and strategies in Tcg are equipped with
symmetry to express that certain events are essentially the same. This is
analogous to the underlying category of AJM games enriching simple games with
an equivalence relations on plays. Building on this category, we construct the
cartesian closed category Cho as having as objects the standard arenas of
Hyland-Ong games, with strategies, represented by certain events structures,
playing on games with symmetry obtained as expanded forms of these arenas.To
illustrate and give an operational light on these constructions, we interpret
(a close variant of) Idealized Parallel Algol in Cho
Game semantics for first-order logic
We refine HO/N game semantics with an additional notion of pointer
(mu-pointers) and extend it to first-order classical logic with completeness
results. We use a Church style extension of Parigot's lambda-mu-calculus to
represent proofs of first-order classical logic. We present some relations with
Krivine's classical realizability and applications to type isomorphisms
Bar recursion is not computable via iteration
We show that the bar recursion operators of Spector and Kohlenbach,
considered as third-order functionals acting on total arguments, are not
computable in Goedel's System T plus minimization, which we show to be
equivalent to a programming language with a higher-order iteration construct.
The main result is formulated so as to imply the non-definability of bar
recursion in T + min within a variety of partial and total models, for instance
the Kleene-Kreisel continuous functionals. The paper thus supplies proofs of
some results stated in the book by Longley and Normann.
The proof of the main theorem makes serious use of the theory of nested
sequential procedures (also known as PCF Boehm trees), and proceeds by showing
that bar recursion cannot be represented by any sequential procedure within
which the tree of nested function applications is well-founded.Comment: 43 pages, 5 figure
A Semantic analysis of control
This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment.
Abramsky has suggested that an intensional hierarchy of computational features such as state, and their fully abstract models, can be captured as violations of the constraints on strategies in the basic functional model. Non-local control flow is shown to fit into this framework as the violation of strong and weak `bracketing' conditions, related to linear behaviour.
The language muPCF (Parigot's mu_lambda with constants and recursion) is adopted as a simple basis for higher-type, sequential computation with access to the flow of control. A simple operational semantics for both call-by-name and call-by-value evaluation is described. It is shown that dropping the bracketing condition on games models of PCF yields fully abstract models of muPCF.
The games models of muPCF are instances of a general construction based on a continuations monad on Fam(C), where C is a rational cartesian closed category with infinite products. Computational adequacy, definability and full abstraction can then be captured by simple axioms on C.
The fully abstract and universal models of muPCF are shown to have an effective presentation in the category of Berry-Curien sequential algorithms. There is further analysis of observational equivalence, in the form of a context lemma, and a characterization of the unique functor from the (initial) games model, which is an isomorphism on its (fully abstract) quotient. This establishes decidability of observational equivalence for finitary muPCF, contrasting with the undecidability of the analogous relation in pure PCF
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