Sometimes measuring the dimensions of a flow gives an insufficiently accurate estimate of its number of paths. This section shows that only certain interactions betweenac↓
and ac↑ nodes generate significant complexity in aKS+-flow. In the absence of these the number of open ai-paths in a flow is still polynomial in its size, regardless of its length.
Definition 6.26. Acontraction loop in a flow is a pair of (ac↑,ac↓) nodes (ν1, ν2) such that there are at least two disjoint (directed) paths betweenν1 and ν2
For example we give the following flow and all its contraction loops,
u v
w
⋆ x
y z
(u, y),(v, z).
whereas every other pair has only one path between them. If the edge ? were broken and there was no path from wtox then there would be no contraction loops at all.
Lemma 6.27. If there are no contraction loops in a KS+-flowφ thenpφq≤ |φ|3. Proof. For an edge inφconsider the following two notions:
• The weight of , denoted w(), is the number of directed paths from to the bottom ofφ, i.e. to anaw↑-node or an edge with lower end pending.
• For an atomic structural ruleρ letN(ρ, ) denote the number ofρ-nodes below that are connected toby a directed path.
We show thatw()≤N(aw↑, ) +N(ac↑, ) + 1 by induction on the distance offrom the bottom ofφ. The inequality is clear for the base cases whenis directly connected to aaw↑node, ǫ , and whenhas lower end pending, . We have two inductive steps:
1. is an upper edge of an ac↓-node,
ǫ
δ . In this case we clearly have that w() =w(δ) and so the inequality follows by the inductive hypothesis.
2. is the upper edge of an ac↑-node, γ δ
ǫ
. Observe that, since there are no contraction loops inφand ac↓is the only node type with in-degree greater than
1, any node below this ac↑-node can be directed-path-connected to at most one of γ or δ. Consequently we have that N(aw↑, ) = N(aw↑, γ) +N(aw↑, δ) and N(ac↑, ) =N(ac↑, δ) +N(ac↑, γ) + 1. Therefore,
w() =w(δ) +w(γ)
≤(N(aw↑, δ) +N(ac↑, δ) + 1) + (N(aw↑, γ) +N(ac↑, γ) + 1)
≤(N(aw↑, δ) +N(aw↑, γ) + (N(ac↑, δ) +N(ac↑, γ) + 1) + 1
≤N(aw↑, ) +N(ac↑, ) + 1
Notice in particular that by this inequality we have thatw()≤ |φ|.
Clearly the number of openai-paths going through an edge with upper end pending is bounded above by its weight, and so by|φ|by the bound above, while the number of open ai-paths going through any ai↓ node is bounded above by the product of the weights of each of its edges, and so by |φ|2. In particular we have that the number of open ai-paths going through any edge at the top of a flow is bounded above by |φ|2, and there are at most|φ|many such edges, whence the bound follows.
Example 6.28. In fact the bound given above is optimal, up to multiplication by a constant. Consider the flow φbelow, where there aren ai↓ nodes:
.. .
. . . . . .
... ..
.
Clearly the flow has size linear innand notice that the weights (as defined in the above proof) of the topmost edges, starting from the left, are 1,2, . . . , n, n, . . . ,1 respectively.
Consequently the number of openai-paths going through the ai↓ nodes, starting from the outside, is 12,22, . . . , n2 respectively, and so taking the sum we obtainpφq= Ω(n3).
Lemma 6.27.
Chapter 7
Complexity of the logical fragment of deep inference
In this chapter we turn to the logical fragment of SKS. The objects of study here are {s,m}-derivations, and we analyse them in the setting of term rewriting. The motivation is to understand the contribution of these rules to the complexity of a derivation; as we will see, {s,m} generally does not contribute superpolynomially to the size of a derivation, but it is expressive enough to encode all the information of a proof, from the point of view of complexity, and consequently determines the complexity of proof search in deep inference.
As we have mentioned before, the normalisation and complexity results of the previ-ous chapter hold independently of the rules of the logical fragment, provided all logical rules are linear and switch and medial are derivable - derivability of both these rules is necessary in order to obtain atomicity for the structural rules. Consequently it is an interesting pursuit to see what effects other linear rules may have on a proof system, and we consider examples of such rules also in this chapter.
One thing to note is that the polynomial bounds we obtain on lengths of linear derivations are specific to{s,m}-derivations; there is nothing to stop other linear rules from contributing superpolynomially to the size of a derivation, again something that we will see in this chapter.
The material in this chapter is based on work that has appeared in [Das13].
7.1 The term rewriting setting
corresponding to rewrite paths. In this chapter we find that many questions about the complexity of the logical fragment of deep inference, or indeed of proofs in general, can be phrased as natural questions in the setting of term rewriting.
On a point of notation, to maintain consistency with the term rewriting literature, we now use formula variablesA, B, etc. as formal symbols occurring in termss, tetc., and we denote ground terms, i.e. formulae, by meta-variablesα, β etc.
As convention, we construe inference rules as term rewriting rules. We no longer assume that = is contained in every system, but rather distinguish between the re-bracketing rules and the unit rules. We denote byAC the equational theory generated by the rebracketing rules, and byUthe equational theory generated by the unit rules.
We associate deep inference derivations with rewrite paths by their translation to CoS form.
We define the systemMS={s,m} and the systemMSU=MS∪U. All systems are assumed to operate modulo AC, although we may indicate an instance ofAC, or any other equational theory we are working modulo, by a ‘fake’ inference step A
....
B.