Book of Abstracts - IRIT

Positive Logic Is 2-Exptime Hard
Schubert et al
2. a selected flag can be raised at a given ancestor node of w;
3. the machine can check if a selected flag is up at a given ancestor node of w;
4. a new child of w can be added to the tree T .
In case 3, if the flag in question is down, the computation gets stuck; one cannot make negative
tests. The same happens in case 1 if we attempt to move up when w is a root or to move down
when w has no children. In case 2, raising a flag which is already 1 has no effect (and remains
unnoticed). In all cases but 4, machine instructions also specify the next state. However,
when a new node is created at level l the automaton has no choice and must enter state q0,
l
a designated initial state for that level.
The behaviour of Eden automata somehow resembles tree-walking automata [1], but there
are essential differences. First of all, our automata build their own trees rather than recognise
them. Also the node branching is potentially unbounded. We also notice that tree automata
with additional memory were introduced by Kaminski and Tan [4].
Our main result is obtained in two steps. First, we show that the halting problem for Eden
automata reduces to the decision problem for positive formulas. In fact, a Logspace reduction
from automata to formulae is also possible at the cost of a slight complication in the definition
of automata. This way we obtain a machine model exactly equivalent in complexity to the
decision problem for our formulas. But for the lack of space we do not pursue this issue here.
In the second step we prove the 2-Exptime hardness for automata, using a direct Turing
Machine simulation. It is worth noting that the main difficulty here is to ensure the unequivocality
of representation, namely that a single code represents a single entity, but not any other,
which is the result of the situation that Eden automata are earthly entities with too limited
memory to deal with the complexity of Eden citizens such as trees of knowledge. Putting
together the two reductions we obtain the doubly-exponential lower bound for provability.
References
[1] A.V. Aho and J.D. Ullman. Translations on a context-free grammar. Inf. Control, 19(5):439–475,
1971.
[2] Gilles Dowek and Ying Jiang. Eigenvariables, bracketing and the decidability of positive minimal
predicate logic. Theor. Comput. Sci., 360(1–3):193–208, 2006.
[3] Gilles Dowek and Ying Jiang. Enumerating proofs of positive formulae. Comput. J., 52(7):799–807,
2009.
[4] Michael Kaminski and Tony Tan. Tree automata over infinite alphabets. In Arnon Avron, Nachum
Dershowitz, and Alexander Rabinovich, editors, Pillars of Computer Science, volume 4800 of LNCS,
pages 386–423. Springer-Verlag, Berlin, Heidelberg, 2008.
[5] G.E. Mints. Solvability of the problem of deducibility in LJ for a class of formulas not containing
negative occurrences of quantifiers. Steklov Inst., 98:135–145, 1968.
[6] Ivar Rummelhoff. Polymorphic Π1 Types and a Simple Approach to Propositions, Types and Sets.
PhD thesis, University of Oslo, 2007.
73