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Almost always blue trees and bar induction

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Let us do constructive temporal logic on infinite binary trees whose nodes are colored red or blue. It turns out that there are more positive notions of an almost always blue tree than one might first think and that the relationships between these are subtle.

We organize five almost always blueness predicates into a pentagon-shaped diagram. Along the five arcs, the predicates entail each other intuitionistically. Four of out of the five converse implications hold under specific assumptions: the fan theorem (FAN), bar induction (BI), the lesser principle of omniscience (LPO) and weak continuity for numbers (WCN); the fifth is contradictory. On a very simple example tree, one of the predicates is undecided intuitionistically, but true resp. false in two consistent but mutually contradictory extensions (LPO and WCN ).

(Joint work with Marc Bezem, Keiko Nakata.)

This talk is part of the Logic and Semantics Seminar (Computer Laboratory) series.

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