Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA

Modal formulae express monadic second-order properties on Kripke frames, but in many important cases these have first-order equivalents. Computing such equivalents is important for both logical and computational reasons. On the other hand, canonicity of modal formulae is important, too, because it implies frame-completeness of logics axiomatized with canonical formulae. Computing a first-order equivalent of a modal formula amounts to elimination of second-order quantifiers. Two algorithms have been developed for second-order quantifier elimination: SCAN, based on constraint resolution, and DLS, based on a logical equivalence established by Ackermann. In this paper we introduce a new algorithm, SQEMA, for computing first-order equivalents (using a modal version of Ackermann's lemma) and, moreover, for proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on modal formulae, thus avoiding Skolemization and the subsequent problem of unskolemization. We present the core algorithm and illustrate it with some examples. We then prove its correctness and the canonicity of all formulae on which the algorithm succeeds. We show that it succeeds not only on all Sahlqvist formulae, but also on the larger class of inductive formulae, introduced in our earlier papers. Thus, we develop a purely algorithmic approach to proving canonical completeness in modal logic and, in particular, establish one of the most general completeness results in modal logic so far.Comment: 26 pages, no figures, to appear in the Logical Methods in Computer Scienc

FINITE H-DIMENSION DOES NOT IMPLY EXPRESSIVE COMPLETENESS

Accepted versio

Andreev-Tunneling, Coulomb Blockade, and Resonant Transport of Non-Local Spin-Entangled Electrons

We propose and analyze a spin-entangler for electrons based on an s-wave superconductor coupled to two quantum dots each of which is tunnel-coupled to normal Fermi leads. We show that in the presence of a voltage bias and in the Coulomb blockade regime two correlated electrons provided by the Andreev process can coherently tunnel from the superconductor via different dots into different leads. The spin-singlet coming from the Cooper pair remains preserved in this process, and the setup provides a source of mobile and nonlocal spin-entangled electrons. The transport current is calculated and shown to be dominated by a two-particle Breit-Wigner resonance which allows the injection of two spin-entangled electrons into different leads at exactly the same orbital energy, which is a crucial requirement for the detection of spin entanglement via noise measurements. The coherent tunneling of both electrons into the same lead is suppressed by the on-site Coulomb repulsion and/or the superconducting gap, while the tunneling into different leads is suppressed through the initial separation of the tunneling electrons. In the regime of interest the particle-hole excitations of the leads are shown to be negligible. The Aharonov-Bohm oscillations in the current are shown to contain single- and two-electron periods with amplitudes that both vanish with increasing Coulomb repulsion albeit differently fast.Comment: 11 double-column pages, 2 figures, REVTeX, minor revision

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity

Effects of Compression and Collective Expansion on Particle Emission from Central Heavy-Ion Reactions

Conditions under which compression occurs and collective expansion develops in energetic reactions of heavy nuclei, are analyzed, together with their effects on emitted light baryons and pions. Within transport simulations, it is shown that shock fronts perpendicular to beam axis form in head-on reactions. The fronts separate hot compressed matter from normal. As impact parameter increases, the angle of inclination of the fronts relative to beam axis decreases, and in-between the fronts a weak tangential discontinuity develops. Hot matter exposed to the vacuum in directions perpendicular to shock motion (and parallel to fronts), starts to expand sideways, early within reactions. Expansion in the direction of shock motion follows after the shocks propagate through nuclei, but due to the delay does not acquire same strength. Expansion affects angular distributions, mean-energy components, shapes of spectra and mean energies of different particles emitted into any one direction, and further particle yields. Both the expansion and a collective motion associated with the weak discontinuity, affect the magnitude of sideward flow within reaction plane. Differences in mean particle energy components in and out of the reaction plane in semicentral collisions, depend sensitively on the relative magnitude of shock speed in normal matter and speed of sound in hot matter.Comment: 71 pages, 33 figures (available on request), report MSUCL-94

Universality of electron correlations in conducting carbon nanotubes

Effective low-energy Hamiltonian of interacting electrons in conducting single-wall carbon nanotubes with arbitrary chirality is derived from the microscopic lattice model. The parameters of the Hamiltonian show very weak dependence on the chiral angle, which makes the low energy properties of conducting chiral nanotubes universal. The strongest Mott-like electron instability at half filling is investigated within the self-consistent harmonic approximation. The energy gaps occur in all modes of elementary excitations and estimate at $0.01-0.1$ eV.Comment: 4 pages, 2 figure

Controlled release from zein matrices: Interplay of drug hydrophobicity and pH

Purpose: In earlier studies, the corn protein zein is found to be suitable as a sustained release agent, yet the range of drugs for which zein has been studied remains small. Here, zein is used as a sole excipient for drugs differing in hydrophobicity and isoelectric point: indomethacin, paracetamol and ranitidine. Methods: Caplets were prepared by hot-melt extrusion (HME) and injection moulding (IM). Each of the three model drugs were tested on two drug loadings in various dissolution media. The physical state of the drug, microstructure and hydration behaviour were investigated to build up understanding for the release behaviour from zein based matrix for drug delivery. Results: Drug crystallinity of the caplets increases with drug hydrophobicity. For ranitidine and indomethacin, swelling rates, swelling capacity and release rates were pH dependent as a consequence of the presence of charged groups on the drug molecules. Both hydration rates and release rates could be approached by existing models. Conclusion: Both the drug state as pH dependant electrostatic interactions are hypothesised to influence release kinetics. Both factors can potentially be used factors influencing release kinetics release, thereby broadening the horizon for zein as a tuneable release agent

Measurement of mechanical vibrations excited in aluminium resonators by 0.6 GeV electrons

We present measurements of mechanical vibrations induced by 0.6 GeV electrons impinging on cylindrical and spherical aluminium resonators. To monitor the amplitude of the resonator's vibrational modes we used piezoelectric ceramic sensors, calibrated by standard accelerometers. Calculations using the thermo-acoustic conversion model, agree well with the experimental data, as demonstrated by the specific variation of the excitation strengths with the absorbed energy, and with the traversing particles' track positions. For the first longitudinal mode of the cylindrical resonator we measured a conversion factor of 7.4 +- 1.4 nm/J, confirming the model value of 10 nm/J. Also, for the spherical resonator, we found the model values for the L=2 and L=1 mode amplitudes to be consistent with our measurement. We thus have confirmed the applicability of the model, and we note that calculations based on the model have shown that next generation resonant mass gravitational wave detectors can only be expected to reach their intended ultra high sensitivity if they will be shielded by an appreciable amount of rock, where a veto detector can reduce the background of remaining impinging cosmic rays effectively.Comment: Tex-Article with epsfile, 34 pages including 13 figures and 5 tables. To be published in Rev. Scient. Instr., May 200