242 research outputs found
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus calculus, which covers in particular all the models
individually introduced in the literature. We prove that the order theory of an
effective model is never r.e.; from this it follows that its equational theory
cannot be l-beta or l-beta-eta. We then show that no effective model living in
the stable or strongly stable semantics has an r.e. equational theory.
Concerning Scott's semantics, we investigate the class of graph models and
prove that no order theory of a graph model can be r.e., and that there exists
an effective graph model whose equational/order theory is minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34
On approximate and algebraic computability over the real numbers
AbstractWe consider algebraic and approximate computations of (partial) real functions ƒ:Rd ↣ R. Algebraic computability is defined by means of (parameter-free) finite algorithmic procedures. The notion of approximate computability is a straightforward generalization of the Ko-Friedman approach, based on oracle Turing machines, to functions with not necessarily recursively open domains.The main results of the paper give characterizations of approximate computability by means of the passing sets of finite algorithmic procedures, i.e., characterizations from the algebraic point of view. Some consequences and also modifications of the concepts are discussed. Finally, two variants of arithmetical hierarchies over the reals are considered and used to classify and mutually compare the domains, graphs and ranges of algebraically resp. approximately computable real functions
Monoids with tests and the algebra of possibly non-halting programs
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou
Decision Problems in Information Theory
Constraints on entropies are considered to be the laws of information theory. Even though the pursuit of their discovery has been a central theme of research in information theory, the algorithmic aspects of constraints on entropies remain largely unexplored. Here, we initiate an investigation of decision problems about constraints on entropies by placing several different such problems into levels of the arithmetical hierarchy. We establish the following results on checking the validity over all almost-entropic functions: first, validity of a Boolean information constraint arising from a monotone Boolean formula is co-recursively enumerable; second, validity of "tight" conditional information constraints is in ???. Furthermore, under some restrictions, validity of conditional information constraints "with slack" is in ???, and validity of information inequality constraints involving max is Turing equivalent to validity of information inequality constraints (with no max involved). We also prove that the classical implication problem for conditional independence statements is co-recursively enumerable
Finding the limit of incompleteness I
In this paper, we examine the limit of applicability of G\"{o}del's first
incompleteness theorem ( for short). We first define the notion " holds for the theory ". This paper is motivated by the following
question: can we find a theory with a minimal degree of interpretation for
which holds. To approach this question, we first examine the following
question: is there a theory such that Robinson's interprets
but does not interpret (i.e. is weaker than
w.r.t. interpretation) and holds for ? In this paper,
we show that there are many such theories based on Je\v{r}\'{a}bek's work using
some model theory. We prove that for each recursively inseparable pair , we can construct a r.e. theory such that
is weaker than w.r.t. interpretation and
holds for . As a corollary, we answer a
question from Albert Visser. Moreover, we prove that for any Turing degree
, there is a theory with Turing
degree such that holds for and is weaker than
w.r.t. Turing reducibility. As a corollary, based on Shoenfield's
work using some recursion theory, we show that there is no theory with a
minimal degree of Turing reducibility for which holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi
Finite Representability of Semigroups with Demonic Refinement
Composition and demonic refinement of binary relations are
defined by \begin{align*} (x, y)\in (R;S)&\iff \exists z((x, z)\in R\wedge (z,
y)\in S)
R\sqsubseteq S&\iff (dom(S)\subseteq dom(R) \wedge
R\restriction_{dom(S)}\subseteq S)
\end{align*} where and
denotes the restriction of to pairs where
.
Demonic calculus was introduced to model the total correctness of
non-deterministic programs and has been applied to program verification.
We prove that the class of abstract
structures isomorphic to a set of binary relations ordered by demonic
refinement with composition cannot be axiomatised by any finite set of
first-order formulas. We provide a fairly simple, infinite,
recursive axiomatisation that defines . We prove that a
finite representable structure has a representation over a
finite base. This appears to be the first example of a signature for binary
relations with composition where the representation class is non-finitely
axiomatisable, but where the finite representations for finite representable
structures property holds
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