A Propositional Linear Time Logic with Time Flow Isomorphic to \omega^2
Abstract
Primarily guided with the idea to express zerotime transitions by means of temporal propositional language, we have developed a temporal logic where the time flow is isomorphic to ordinal $\omega^2$ (concatenation of $\omega$ copies of $\omega$). If we think of $\omega^2$ as lexicographically ordered $\omega\times \omega$, then any particular zerotime transition can be represented by states whose indices are all elements of some $\{n\}\times\omega$. In order to express noninfinitesimal transitions, we have introduced a new unary temporal operator $[\omega] $ ($\omega$jump), whose effect on the time flow is the same as the effect of $\alpha\mapsto \alpha+\omega$ in $\omega^2$. In terms of lexicographically ordered $\omega\times \omega$, $[\omega] \phi$ is satisfied in $\ < i,j\ >$th time instant iff $\phi$ is satisfied in $\ < i+1,0\ >$th time instant. Moreover, in order to formally capture the natural semantics of the until operator $\mathtt U$, we have introduced a local variant $\mathtt u$ of the until operator. More precisely, $\phi\,\mathtt u \psi$ is satisfied in $\ < i,j\ >$th time instant iff $\psi$ is satisfied in $\ < i,j+k\ >$th time instant for some nonnegative integer $k$, and $\phi$ is satisfied in $\ < i,j+l\ >$th time instant for all $0\leqslant l<k$. As in many of our previous publications, the leitmotif is the usage of infinitary inference rules in order to achieve the strong completeness.
 Publication:

arXiv eprints
 Pub Date:
 September 2013
 arXiv:
 arXiv:1309.0829
 Bibcode:
 2013arXiv1309.0829M
 Keywords:

 Mathematics  Logic;
 Computer Science  Logic in Computer Science
 EPrint:
 Journal of Applied Logic 12 (2). p. 208  229, 2014