Multiplicative-Additive Proof Equivalence is Logspace-complete, via Binary Decision Trees

Given a logic presented in a sequent calculus, a natural question is that of equivalence of proofs: to determine whether two given proofs are equated by any denotational semantics, ie any categorical interpretation of the logic compatible with its cut-elimination procedure. This notion can usually be captured syntactically by a set of rule permutations. Very generally, proofnets can be defined as combinatorial objects which provide canonical representatives of equivalence classes of proofs. In particular, the existence of proof nets for a logic provides a solution to the equivalence problem of this logic. In certain fragments of linear logic, it is possible to give a notion of proofnet with good computational properties, making it a suitable representation of proofs for studying the cut-elimination procedure, among other things. It has recently been proved that there cannot be such a notion of proofnets for the multiplicative (with units) fragment of linear logic, due to the equivalence problem for this logic being Pspace-complete. We investigate the multiplicative-additive (without unit) fragment of linear logic and show it is closely related to binary decision trees: we build a representation of proofs based on binary decision trees, reducing proof equivalence to decision tree equivalence, and give a converse encoding of binary decision trees as proofs. We get as our main result that the complexity of the proof equivalence problem of the studied fragment is Logspace-complete.Comment: arXiv admin note: text overlap with arXiv:1502.0199

Involutions on the Algebra of Physical Observables From Reality Conditions

Some aspects of the algebraic quantization programme proposed by Ashtekar are revisited in this article. It is proved that, for systems with first-class constraints, the involution introduced on the algebra of quantum operators via reality conditions can never be projected unambiguously to the algebra of physical observables, ie, of quantum observables modulo constraints. It is nevertheless shown that, under sufficiently general assumptions, one can still induce an involution on the algebra of physical observables from reality conditions, though the involution obtained depends on the choice of particular representatives for the equivalence classes of quantum observables and this implies an additional ambiguity in the quantization procedure suggested by Ashtekar.Comment: 19 pages, latex, no figure

Problematic ideologies in teacher education

A critical problem facing educationiuls is the problematic quality of many teacher training courses. Vie major source o f the problem seems to be the irrational ideological foundations on which these courses are often based. The research theme revolves around a particular type of problematic ideology, viz. lecturing course teams in teacher training whose members refuse to adopt an integrated, muiually-compatible approach when teaching students how to teach. Such course teams in.s'ist on transferring contradictory, subjective views o f teaching to student teachers. It is estimated that a high percentage of students annually qualifying as teachers in South Africa are, from a professional point of view, incompetent to teach. The argument is outlined in three parts: statement of problem, theoretical argimients being forwarded to justify problematic ideologies, and possible solutions. Bearing in mind the far-reaching implications  f the situation, the solution could be to appoint a prescriptive controlling body/inspect orate whose members' main task would be to raise teacher training standards in South Africa. Experience in more developed countries suggests that such a controlling body would only function effectively if composed of representatives from other social sciences, schools, future employers (of pupils), educationists, and the government