Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)

This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs, additional examples & figures, additional comparison between classical LTL/schemata algorithms up to the provided translations, and an example of how to do model checking with schemata; 36 pages, 8 figure

Lattice initial segments of the hyperdegrees

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $\omega _{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$

Exploring the episodic structure of algebra story problem solving

This paper analyzes the quantitative and situational structure of algebra story problems, uses these materials to propose an interpretive framework for written problem-solving protocols, and then presents an exploratory study of the episodic structure of algebra story problem solving in a sizable group of mathematically competent subjects. Analyses of written protocols compare the strategic, tactical, and conceptual content of solution attampts, looking within these attempts at the interplay between problem comprehension and solution. Comprehension and solution of algebra story problems are complimentary activities, giving rise to a succession of problem solving episodes. While direct algebraic problem solving is sometimes effective, results suggest that the algebraic formalism may be of little help in comprehending the quantitative constraints posed in a problem text. Instead, competent problem solvers often reason within the situational context presented by a story problem, using various forms of "model-based reasoning" to identify, pursue, and verify quantitative constraints required for solution. The paper concludes by discussing the implications of these findings for acquiring mathematical concepts (e.g., related linear functions) and for supporting their acquisition through instruction