Related papers: Randomising Realisability
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal)…
Randomized mechanisms, which map a set of bids to a probability distribution over outcomes rather than a single outcome, are an important but ill-understood area of computational mechanism design. We investigate the role of randomized…
Reconstructing a hypothetical recurrence equation from the first terms of an infinite sequence is a classical and well-known technique in experimental mathematics. We propose a variation of this technique which can succeed with fewer input…
In this article, we study a notion of the extraction rate of Turing functionals that translate between notions of randomness with respect to different underlying probability measures. We analyze several classes of extraction procedures: a…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. In this…
We develop a second-order extension of intuitionistic modal logic, allowing quantification over propositions, both syntactically and semantically. A key feature of second-order logic is its capacity to define positive connectives from the…
We introduce an extension of first-order logic that comes equipped with additional predicates for reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre- and postconditions in the style of Hoare…
Abstraction and realization are bilateral processes that are key in deriving intelligence and creativity. In many domains, the two processes are approached through rules: high-level principles that reveal invariances within similar yet…
It is conventional wisdom in machine learning and data mining that logical models such as rule sets are more interpretable than other models, and that among such rule-based models, simpler models are more interpretable than more complex…
Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and…
Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed-point logic. Inclusion logic is not axiomatizable in full, but its first-order consequences can be axiomatized. In…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…
Many tasks in statistical and causal inference can be construed as problems of \emph{entailment} in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for \emph{causal}…
Let L be some extension of classical propositional logic. The non-iterated probabilistic logic over L, is the logic PL that is defined by adding non-nested probabilistic operators in the language of L. For example in PL we can express a…
We prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On…
We introduce two notions of effective reducibility for set-theoretical statements, based on computability with Ordinal Turing Machines (OTMs), one of which resembles Turing reducibility while the other is modelled after Weihrauch…
We introduce a version of probabilistic Kleene algebra with angelic nondeterminism and a corresponding class of automata. Our approach implements semantics via distributions over multisets in order to overcome theoretical barriers arising…
We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…