Related papers: Introduction to clarithmetic II
Logics with team semantics provide alternative means for logical characterization of complexity classes. Both dependence and independence logic are known to capture non-deterministic polynomial time, and the frontiers of tractability in…
In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel…
We introduce a set of eight universal Rules of Inference by which computer programs with known properties (axioms) are transformed into new programs with known properties (theorems). Axioms are presented to formalize a segment of Number…
A $\lambda$-calculus is introduced in which all programs can be evaluated in probabilistic polynomial time and in which there is sufficient structure to represent sequential cryptographic constructions and adversaries for them, even when…
The paper is a contribution both to the theoretical foundations and to the actual construction of efficient automatizable proof procedures for non-classical logics. We focus here on the case of finite-valued logics, and exhibit: (i) a…
This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered…
The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these…
It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the…
This paper presents a plausible reasoning system to illustrate some broad issues in knowledge representation: dualities between different reasoning forms, the difficulty of unifying complementary reasoning styles, and the approximate nature…
We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…
We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation…
We provide the first (non-labelled) sequent calculi for bimodal provability logics with "usual" provability predicates. In particular, we introduce calculi for the logics CS, CSM and ER. Additionally, we present non-wellfounded versions of…
There is a cognitive limit in Human Mind. This cognitive limit has played a decisive role in almost all fields including computer sciences. The cognitive limit replicated in computer sciences is responsible for inherent Computational…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…
The satisfactory development of Quaternionic Analysis has indicated new solutions for physical and mathematical problems. It is worth mentioning the fact that quaternions possess four dimensions, and in this way they may be considered as…
Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambda-calculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms…
Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with…