Dynamic Complexity (as introduced by Patnaik and Immerman [13]) tries to express how hard it is to update the solution to a problem when the input is changed slightly. It considers the changes required to some stored data structure (possibly a massive database) as small quantities of data (or a tuple) are inserted or deleted from the database (or a structure over some vocabulary). The main diff...

We offer a view on how probability is related to logic. Specifically, we argue against the widely held belief that standard classical logics have no direct way of modelling the certainty of assumptions in theories and no direct way of stating the certainty of theorems proved from these (uncertain) assumptions. The argument rests on the observation that probability densities, being functions, ca...

This paper analyses the arguments of a group of philosophers – Davidson [7, 6], Kim [9], Bennett [1] and Parsons [15] – who have worked on the semantics of actions. They argue that inference patterns involving adverbs require a semantics, formalised in first-order logic, in which actions are first-class individuals. In particular, equalities between actions must be meaningful. (We should note, ...

Tennenbaum’s Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to modeltheoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standar...

A study is carried out of the elementary theory of quotients of symmetric groups in a similar spirit to [10]. Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(μ) on an infinite cardinal μ are all of the form Sκ(μ) = the subgroup consisting of elements whose support has cardinality < κ, for some κ ≤ μ. A many-sorted structure Mκλμ is defined wh...

A Liouville function is an analytic function H : C → C with a Taylor series ∑ ∞ n=1 xn/an such the an’s form a “very fast growing” sequence of integers. In this paper we exhibit the complete first-order theory of the complex field expanded with H .

Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four or more is shown to be intractable. Complexity is tied-in with the expressivity of a relation al...

scan is an algorithm for reducing existential second-order logic formulae to equivalent simpler formulae, often first-order logic formulae. It is provably impossible for such a reduction to first-order logic to be successful for every second-order logic formula which has an equivalent first-order formula. In this paper we show that scan successfully computes the first-order equivalents of all S...

The behavioural semantics of specifications with higher-order logical formulae as axioms is analyzed. A characterization of behavioural abstraction via behavioural satisfaction of formulae in which the equality symbol is interpreted as indistinguishability, which is due to Reichel and was recently generalized to the case of first-order logic by Bidoit et al, is further generalized to this case....