Probabilities of first-order sentences on sparse random relational structures: An application to definability on random CNF formulas

We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary s and a first-order theory T for s composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite s-structures that satisfy T and show that first-order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in s is linear. It is also shown that these limit probabilities are well behaved with respect to several parameters that represent the density of tuples in each relation R in the vocabulary s¿. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random k-CNF formula on n variables, where each possible clause occurs with probability ~c/nk-1¿, independently any first-order property of k-CNF formulas that implies unsatisfiability does almost surely not hold as n tends to infinity.Peer ReviewedPostprint (published version

The Toll Walk Transit Function of a Graph: Axiomatic Characterizations and First-Order Non-definability

A walk $W=w_1w_2\dots w_k$, $k\geq 2$, is called a toll walk if $w_1\neq w_k$ and $w_2$ and $w_{k-1}$ are the only neighbors of $w_1$ and $w_k$, respectively, on $W$ in a graph $G$. A toll walk interval $T(u,v)$, $u,v\in V(G)$, contains all the vertices that belong to a toll walk between $u$ and $v$. The toll walk intervals yield a toll walk transit function $T:V(G)\times V(G)\rightarrow 2^{V(G)}$. We represent several axioms that characterize the toll walk transit function among chordal graphs, trees, asteroidal triple-free graphs, Ptolemaic graphs, and distance hereditary graphs. We also show that the toll walk transit function can not be described in the language of first-order logic for an arbitrary graph.Comment: 31 pages, 4 figures, 25 reference

A first-order axiomatization of the theory of finite trees

We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes

To Be Announced

In this survey we review dynamic epistemic logics with modalities for quantification over information change. Of such logics we present complete axiomatizations, focussing on axioms involving the interaction between knowledge and such quantifiers, we report on their relative expressivity, on decidability and on the complexity of model checking and satisfiability, and on applications. We focus on open problems and new directions for research

A complete axiomatization of a theory with feature and arity constraints

CFT is a recent constraint system providing records as a logical data structure for logic programming and for natural language processing. It combines the rational tree system as defined for logic programming with the feature tree system as used in natural language processing. The formulae considered in this paper are all first-order-logic formulae over a signature of binary and unary predicates called features and arities, respectively. We establish the theory CFT by means of seven axiom schemes and show its completeness. Our completeness proof exhibits a terminating simplification system deciding validity and satisfiability of possibly quantified record descriptions

A Process Model of Non-Relativistic Quantum Mechanics

A process model of quantum mechanics utilizes a combinatorial game to generate a discrete and finite causal space upon which can be defined a self-consistent quantum mechanics. An emergent space-time and continuous wave function arise through a uniform interpolation process. Standard non-relativistic quantum mechanics (at least for integer spin particles) emerges under the limit of infinite information (the causal space grows to infinity) and infinitesimal scale (the separation between points goes to zero). This model is quasi-local, discontinuous, and quasi-non-contextual. The bridge between process and wave function is through the process covering map, which reveals that the standard wave function formalism lacks important dynamical information related to the generation of the causal space. Reformulating several classical conundrums such as wave particle duality, Schrodinger's cat, hidden variable results, the model offers potential resolutions to all, while retaining a high degree of locality and contextuality at the local level, yet nonlocality and contextuality at the emergent level. The model remains computationally powerful