Switching Lemmas and Proof Complexity Paul Beame University of Washington

AC0 circuits and restrictions Unbounded fan-in AND, OR gates, NOT gates Constant depth Restrictions Partial assignment ρ on X : 𝝆∈ 𝟎,𝟏,∗ 𝑿 𝝆 = 𝝆 −𝟏 𝟎,𝟏 Vars(𝝆) = 𝝆 −𝟏 ∗ Given 𝑭:𝑿→{𝟎,𝟏} define 𝑭 ​ 𝝆 : Vars(𝝆) →{𝟎,𝟏} by 𝑭 ​ 𝝆 𝒙′ =𝑭 𝒙 where 𝒙 𝒊 = 𝒙 𝒊 ′ 𝒊∈Vars(𝝆) 𝝆 𝒊 𝒊∉Vars(𝝆)

Circuit lower bounds via restrictions To show lower bound for function 𝑭 choose family/distribution on restrictions 𝑹 𝑭 and show that: For any small, shallow circuit 𝑪, Pr 𝝆∼ 𝑹 𝑭 𝑪 ​ 𝝆 is not 𝑠𝑖𝑚𝑝𝑙𝑒 is small P r 𝝆∼ 𝑹 𝑭 𝑭 ​ 𝝆 is not 𝑠𝑖𝑚𝑝𝑙𝑒 is large 𝑠𝑖𝑚𝑝𝑙𝑒 = computed by small height decision tree Write 𝑫 𝑭 for the smallest height of any decision tree computing 𝐹.

Decision Tree for 𝑷𝒂𝒓𝒊𝒕𝒚(𝒙𝟏, 𝒙𝟐, 𝒙𝟑, 𝒙𝟒) 1 𝒙 𝟏 𝟎 𝟏 𝒙 𝟐 𝒙 𝟑 𝒙 𝟒

Lower bound for Parity For every restriction 𝝆, 𝑫 𝐏𝐚𝐫𝐢𝐭𝐲 ​ 𝝆 =|Vars 𝝆 | Let 𝑹 𝒏 ℓ be the set of restrictions on 𝟎,𝟏 𝒏 with Vars 𝝆 =ℓ. Theorem (Håstad): If 𝑪 is an AC0 circuit of size 𝑺 and depth 𝒅 then for ℓ=𝒏/ (𝟏𝟔 log 𝑺) 𝒅−𝟏 there is a 𝝆∈ 𝑹 𝒏 ℓ s.t. 𝑫 𝑪 ​ 𝝆 ≤𝟏𝟔 log 𝑺 Corollary: Depth d Parity circuits require 𝟐 𝒏 𝟏/𝒅 𝟏𝟔 size.

Håstad’s Switching Lemma Lemma (Håstad): Let 𝑭 be a 𝒓-DNF formula on 𝟎,𝟏 𝒏 . For ℓ=𝒑𝒏 we have 𝐏𝐫 𝝆∼ 𝑹 𝒏 ℓ 𝑫 𝑭 ​ 𝝆 ≥𝒔 ≤ 𝟖𝒑𝒓 𝒔 Note: If 𝑭′ has a decision tree 𝑻 of height 𝒉 then both 𝑭’ and 𝑭 ′ are expressible as 𝒉-DNF formulas Circuit lower bound follows by repeatedly setting 𝒓= log 𝑺 , 𝒑= 𝟏 𝟏𝟔 log 𝑺 , and 𝒔=𝒓+𝟏.

Håstad’s Switching Lemma Lemma (Håstad): Let 𝑭 be a 𝒓-DNF formula on 𝟎,𝟏 𝒏 . For ℓ=𝒑𝒏 we have 𝐏𝐫 𝝆∼ 𝑹 𝒏 ℓ 𝑫 𝑭 ​ 𝝆 ≥𝒔 ≤ 𝟖𝒑𝒓 𝒔 Proof idea (Razborov): Consider a canonical way of converting 𝑭 ​ 𝝆 to a decision tree. Use existence of a path of length ≥𝒔 in canonical decision tree for 𝑭 (plus knowledge of 𝑭) to find a more efficient description of ρ. Conclude that restrictions with this property are rare.

Applying a restriction to a DNF For a DNF 𝑭 and restriction 𝝆, 𝑭 ​ 𝝆 is a DNF that contains a reduced form of each term of 𝑭 not falsifed 𝝆 by removing all literals set to 1, keeps the same order among all the terms that survive, and has no other terms.

Canonical decision tree for a DNF Given an ordered set of variables 𝑽= 𝒙 𝟏 , 𝒙 𝟐 ,…, 𝒙 𝒌 the full tree on 𝑽 is: A full binary tree of height 𝒌 that queries the 𝒊th variable in 𝑽 at every node on the 𝒊th level. with a branch for each total assignment 𝝅 to 𝑽. 𝒙 𝟏 𝒙 𝟐 𝒙 𝟑 𝟎 𝟏

Canonical decision tree for a DNF Let DNF 𝑭=𝑻∨𝑭′ where 𝑻 is the first term of 𝑭. The canonical decision tree for 𝑭 consists of: the full tree 𝑫 𝑻 for Vars(𝑻) at the root. for the unique assignment 𝝈 to Vars(𝑻) that sets 𝑻 to 1, the branch of 𝑫 𝑻 labelled 𝝈 has leaf label 1 for every other assignment 𝝅 to Vars 𝑻 , the branch of 𝑫 𝑻 labelled 𝝅 has the canonical decision tree for 𝑭′ ​ 𝝅 at its leaf.

Switching Lemma Proof Suppose ∃ a path of length 𝒔 in canonical decision tree for 𝑭 ​ 𝝆 Pick leftmost such path 𝝅= 𝝅 𝟏 𝝅 𝟐 … 𝝅 𝒕 Say that 1st term of 𝑭 that survives in 𝑭 ​ 𝝆 is 𝑻 𝒊 𝟏 𝝅 𝟏 assigns to Vars( 𝑻 𝒊 𝟏 ​ 𝝆 ) 𝝈 𝟏 sets (𝑻 𝒊 𝟏 ​ 𝝆 ) ​ 𝝈 𝟏 =𝟏 Repeat for 𝒋=𝟐,…,𝒕 Canonical decision tree for 𝑭 ​ 𝝆 𝝅 𝟏 𝝅 𝟐 𝝅 𝟑 𝝅 𝟒 𝝅 𝟓 𝝈 𝟏 𝝈 𝟐 𝝈 𝟑 𝝈 𝟒 𝝈 𝟓 1

Switching Lemma Proof Say that 1st term of 𝑭 that survives in 𝑭 ​ 𝝆 𝝅 𝟏 is 𝑻 𝒊 𝟐 𝝅 𝟐 assigns to Vars( 𝑻 𝒊 𝟐 ​ 𝝆 𝝅 𝟏 ) 𝝈 𝟐 sets (𝑻 𝒊 𝟐 ​ 𝝆 𝝅 𝟏 ) ​ 𝝈 𝟐 =𝟏 … Obtain 𝝈 𝟏 ,𝝈 𝟐 ,…, 𝝈 𝒕 and indices of the terms in 𝑭, 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 𝝈 𝟏 𝝅 𝟏 1 𝝈 𝟐 𝝅 𝟐 𝝅 𝟑 1 𝝈 𝟑 𝝅 𝟒 1 𝝈 𝟒 𝝅 𝟓 1 𝝈 𝟓 1 Canonical decision tree for 𝑭 ​ 𝝆

Switching Lemma Proof Obtain 𝝈 𝟏 ,𝝈 𝟐 ,…, 𝝈 𝒕 and indices of the terms in 𝑭, 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒕 Specify where the 𝒔 vars set by 𝝅, 𝝈 are in 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify values set by 𝝅 Claim: 𝑭 plus 1, 2, & 3 let us reconstruct 𝝆 Canonical decision tree for 𝑭 ​ 𝝆 𝝅 𝟏 𝝅 𝟐 𝝅 𝟑 𝝅 𝟒 𝝅 𝟓 𝝈 𝟏 𝝈 𝟐 𝝈 𝟑 𝝈 𝟒 𝝈 𝟓 1

Switching Lemma Proof Specify 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒕 Specify where the 𝒔 vars set by 𝝈, 𝝅 are in 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify values set by 𝝅 Claim: 𝑭 plus 1, 2, & 3 let us reconstruct 𝝆 1. 𝝆 ′ ∈ 𝑹 𝒏 ℓ−𝒔 so | 𝑹 𝒏 ℓ−𝒔 | possibilities Bit vector of length 𝒔−𝟏 for partition of [𝒔] into 𝝈 𝟏 , 𝝈 𝟐 ,…, 𝝈 𝒕 plus elt of [𝒓] 𝒔 to indicate positions of each var in its term. Total 𝟐 𝒔−𝟏 𝒓 𝒔 Bit vector of length 𝒔. Total 𝟐 𝒔 So: less than 𝟒𝒓 𝒔 | 𝑹 𝒏 ℓ−𝒔 | of the 𝝆 have such a bad path

Switching Lemma Proof So: 𝐏𝐫 𝝆∼ 𝑹 𝒏 ℓ 𝑫 𝑭 ​ 𝝆 ≥𝒔 ≤ 𝟒𝒓 𝒔 𝑹 𝒏 ℓ−𝒔 𝑹 𝒏 ℓ = 𝟒𝒓 𝒔 𝒏 ℓ−𝒔 𝟐 𝒏−ℓ+𝒔 𝒏 ℓ 𝟐 𝒏−ℓ = 𝟖𝒓 𝒔 ℓ…(ℓ−𝒔+𝟏) 𝒏−ℓ+𝒔 …(𝒏−ℓ+𝟏) ≤ 𝟖𝒓 𝒔 𝒑 𝟏−𝒑 𝒔 Can do better with a sharper analysis since ℓ=𝒑𝒏

Switching Lemma Proof Specify 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒕 Specify where the 𝒔 vars set by 𝝈, 𝝅 are in 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify values set by 𝝅 Claim: 𝑭 plus 1, 2, & 3 let us reconstruct 𝝆 Define 𝝆 𝒌 =𝝆 𝝅 𝟏 … 𝝅 𝒌−𝟏 𝝈 𝒌 … 𝝈 𝒕 𝝆 𝟏 = 𝝆 ′ 𝑻 𝒊 𝒌 is 1st term of 𝑭 surviving in 𝑭 ​ 𝝆 𝝅 𝟏 … 𝝅 𝒌−𝟏 and 𝑻 𝒊 𝒌 ​ 𝝆 𝒌 =𝟏 Can build 𝝆 𝒌+𝟏 from 𝒊 𝒌 and 𝑭 plus 𝝆 𝒌 , 2, & 3 Once all 𝝈 𝟏 … 𝝈 𝒕 are known they can be removed to get 𝝆 from 𝝆′

Proof Complexity Switching Lemmas

Pigeonhole formulas and matchings For PH P 𝒏 : variables 𝒙 𝒊𝒋 for 𝒊∈ 𝒏+𝟏 , 𝒋∈ 𝒏 ¬( 𝒙 𝒊𝟏 ∨…∨ 𝒙 𝒊𝒏 ) for 𝒊∈[𝒏+𝟏] ¬( 𝒙 𝒊𝒌 ∨ 𝒙 𝒋𝒌 ) for 𝒊,𝒋∈ 𝒏+𝟏 , 𝒌∈[𝒏] Take disjunction, add dual terms, get bijective PHP tautology Restrictions that do not simplify PH P 𝒏 too much are matching restrictions. For 𝝆 ∃ partial bipartite matching 𝑴(𝝆) s.t.: 𝝆 sets 𝒙 𝒊𝒋 to 1 for all 𝒊,𝒋 ∈𝑴(𝝆) 𝝆 sets 𝒙 𝒊𝒋 to 0 for all 𝒊,𝒋 matched in 𝑴(𝝆) but not to each other 𝝆 leaves all other variables unset Define 𝝆 = 𝑴 𝝆 Vars 𝝆 ={ 𝒙 𝒊𝒋 :𝒊,𝒋 both not matched in 𝑴(𝝆)} For matching restriction 𝝆, PH P 𝒏 ​ 𝝆 ≡PH P 𝒏− 𝝆

Matching decision trees Each path or branch 𝝅 is a matching restriction. Each node queries a currently unmatched vertex on the left or right and asks which vertex it is matched to One child node for each possible match Queried vertex will always be matched Leaf labels 0 or 1 Unlike Boolean decision trees, not all assignments have branches consistent with them

A matching decision tree on 4 ×[3] ?𝟐 𝟐? 𝟐𝟐 𝟐𝟑 𝟏𝟐 𝟑𝟐 1 ?𝟏 𝟏𝟏 𝟒𝟏 𝟐𝟏 𝟑𝟏 𝟒𝟐 𝟏? 𝟏𝟑 𝟒? 𝟒𝟑 ?𝟑 𝟑𝟑 Associated DNF (𝒙 𝟏𝟏 𝒙 𝟐𝟐 ∨ 𝒙 𝟏𝟏 𝒙 𝟑𝟐 𝒙 𝟒𝟑 ∨ 𝒙 𝟏𝟏 𝒙 𝟒𝟐 ∨ 𝒙 𝟐𝟏 𝒙 𝟏𝟑 ∨ 𝒙 𝟑𝟏 𝒙 𝟐𝟐 ∨ 𝒙 𝟑𝟏 𝒙 𝟐𝟑 ∨ 𝒙 𝟒𝟏 𝒙 𝟏𝟐 ∨ 𝒙 𝟒𝟏 𝒙 𝟐𝟐 𝒙 𝟏𝟑 )

Matching disjunctions A 𝒌-matching disjunction is a monotone 𝒌-DNF each of whose terms is a (bipartite) matching For 𝑻 a matching decision tree Let B r 𝒃 𝑻 = set of branches of 𝑻 with leaf label 𝒃. Let 𝐷𝑁𝐹 𝑻 = 𝝅∈B r 𝟏 (𝑻) 𝒊,𝒋 ∈𝑴(𝝅) 𝒙 𝒊𝒋 If 𝑻 has height 𝒌 then 𝐷𝑁𝐹(𝑻) is a 𝒌-matching disjunction Let 𝑻 𝒄 be the same decision tree as 𝑻 but with leaf labels 𝟎 and 𝟏 swapped. A matching decision tree 𝑻 represents a matching disjunction 𝑭 iff every branch 𝝅 in 𝑻 has 𝑭 ​ 𝝅 =𝒃 iff 𝝅 has leaf label 𝒃

Full matching decision trees Suppose that 𝑫 and 𝑹 are ordered sets. For a subset 𝑽⊂𝑫∪𝑹, the full matching tree 𝑻 𝑽 for 𝑽 on 𝑫×𝑹 is given by the following: If 𝑽=∅ then 𝑻 𝑽 consists of a single root node. If 𝑽∩𝑫≠∅, let 𝒊 be 1st node of 𝑫 in 𝑽; 𝑻 𝑽 has root with query 𝒊? and edges for all possible 𝒋∈𝑹; edge labelled 𝒊𝒋 leads to root of 𝑻 𝑽− 𝒊,𝒋 . Else 𝑽∩𝑹≠∅ so let 𝒋 be 1st node of 𝑹 in 𝑽; then 𝑻 𝑽 has root with query ?𝒋 and edges for all possible 𝒊∈𝑫; edge labelled 𝒊𝒋 leads to root of 𝑻 𝑽− 𝒊,𝒋 If 𝑴 is a matching, write 𝑻 𝑴 for full matching tree for set 𝑽 of nodes matched by 𝑴

Full matching tree on 1,2 ⊂ 4 ×[3] 1? 𝟏𝟏 𝟏𝟑 𝟏𝟐 ?𝟐 ?𝟐 𝟐𝟐 𝟒𝟐 𝟑𝟐 𝟐𝟐 𝟒𝟐 𝟑𝟐

Full matching tree on 1,2 ,(2,1) ⊂ 4 ×[3] 1? 𝟏𝟏 𝟏𝟑 𝟏𝟐 𝟐? 𝟐𝟐 𝟐𝟑 𝟐? 𝟐𝟏 𝟐𝟑 𝟐? 𝟐𝟏 𝟐𝟐 ?𝟏 𝟒𝟏 𝟑𝟏 𝟒𝟐 𝟑𝟐 ?𝟐 𝟒𝟏 𝟑𝟏 ?𝟏 ?𝟐 𝟒𝟐 𝟑𝟐

Canonical matching decision tree Given a matching disjunction 𝑭=𝑻∨𝑭′ where 𝑻= 𝒊,𝒋 ∈𝑴 𝒙 𝒊𝒋 is the 1st term of 𝑭, the canonical matching decision tree for 𝑭 consists of: the full matching tree 𝑻 𝑴 for matching 𝑴 at the root for the matching restriction 𝝈 with 𝑴 𝝆 =𝑴, the branch of 𝑻 𝑴 labelled 𝝈 has leaf label 1 for every other matching restriction 𝝅 labelling a branch of 𝑻 𝑴 , the branch labelled 𝝅 has the canonical matching decision tree for 𝑭′ ​ 𝝅 at its leaf. Note: The canonical matching decision tree for 𝑭 represents 𝑭

Intuition for PH P 𝒏 lower bound Assume wlog proof connectives are unbounded ∨,¬ Associate a height 𝒌 matching decision tree 𝑻 𝑨 for each subformula 𝑨 appearing in the proof s.t.: 𝑻 𝟎 = 𝑻 𝟏 = 𝑻 𝒙 𝒊𝒋 is the full matching tree on 𝒊,𝒋 with leaf label 1 at level 1 (𝒊𝒋) and leaf label 0 at level 2. 𝑻 ¬𝑨 = 𝑻 𝑨 𝒄 If 𝑨= ℓ 𝑨 ℓ and each 𝑨 ℓ does not have an ∨ at its root then represents ℓ 𝐷𝑁𝐹( 𝑻 𝑨 ℓ ) “𝒌-evaluation”. Any tree with all leaves labelled 1 approximates true; all leaves labelled 0 ≈ false If inference rule size ≤𝒄 and 𝒌< 𝒏/𝒄 then all lines are ≈ true but PH P 𝒏 is ≈ false 1

Matching Switching Lemma Let 𝑴 𝒏 ℓ denote the set of matching restrictions 𝝆 on 𝒏+𝟏 × 𝒏 with 𝝆 =𝒏−ℓ. i.e., Vars 𝝆 =𝑫×𝑹 where 𝑹 =ℓ For any matching disjunction 𝑭, let 𝑫 𝑴 (𝑭) be the minimum height of any matching decision tree representing 𝑭. Lemma (BIKPPW): Let 𝑭 be any 𝒓-matching disjunction on [𝒏+𝟏]×[𝒏]. Then for 𝟒𝒓ℓ ℓ+𝟏 ≤𝒏−ℓ Pr 𝝆∼ 𝑴 𝒏 ℓ 𝑫 𝑴 𝑭 ​ 𝝆 ≥𝒔 ≤ 𝟐𝒓 ℓ 𝟑 (ℓ+𝟏) 𝒏−ℓ 𝒔/𝟐

Matching Switching Lemma Proof Suppose ∃ a path of length 𝒔 in canonical matching decision tree for 𝑭 ​ 𝝆 Pick leftmost such path 𝝅= 𝝅 𝟏 𝝅 𝟐 … 𝝅 𝒕 Say that 1st term of 𝑭 that survives in 𝑭 ​ 𝝆 is 𝑻 𝒊 𝟏 𝝅 𝟏 assigns to all vertices of matching in 𝑻 𝒊 𝟏 ​ 𝝆 𝝈 𝟏 sets (𝑻 𝒊 𝟏 ​ 𝝆 ) ​ 𝝈 𝟏 =𝟏 Repeat for 𝒋=𝟐,…,𝒕 𝝈 𝟏 𝝅 𝟏 1 𝝈 𝟐 𝝅 𝟐 𝝅 𝟑 1 𝝈 𝟑 𝝅 𝟒 1 𝝈 𝟒 𝝅 𝟓 1 𝝈 𝟓 1 Canonical matching decision tree for 𝑭 ​ 𝝆

Matching Switching Lemma Proof Obtain 𝝈 𝟏 ,𝝈 𝟐 ,…, 𝝈 𝒕 and indices of the terms in 𝑭, 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒕 Specify where the ≤𝒔 vars set by 𝝈 are in 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify values set by 𝝅 Claim: 𝑭 plus 1, 2, & 3 let us reconstruct 𝝆 Canonical matching decision tree for 𝑭 ​ 𝝆 𝝅 𝟏 𝝅 𝟐 𝝅 𝟑 𝝅 𝟒 𝝅 𝟓 𝝈 𝟏 𝝈 𝟐 𝝈 𝟑 𝝈 𝟒 𝝈 𝟓 1

Matching Switching Lemma Proof Specify 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒕 Specify where the ≤𝒔 vars set by 𝝈 are in 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify values set by 𝝅 Claim: 𝑭 plus 1, 2, & 3 let us reconstruct 𝝆 Define 𝝆 𝒌 =𝝆 𝝅 𝟏 … 𝝅 𝒌−𝟏 𝝈 𝒌 … 𝝈 𝒕 𝝆 𝟏 = 𝝆 ′ 𝑻 𝒊 𝒌 is 1st term of 𝑭 surviving in 𝑭 ​ 𝝆 𝝅 𝟏 … 𝝅 𝒌−𝟏 and 𝑻 𝒊 𝒌 ​ 𝝆 𝒌 =𝟏 Can build 𝝆 𝒌+𝟏 from 𝒊 𝒌 and 𝑭 plus 𝝆 𝒌 , 2, & 3 Once all 𝝈 𝟏 … 𝝈 𝒕 are known they can be removed to get 𝝆 from 𝝆′

Matching Switching Lemma Proof Specify 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒕 Specify where the 𝒔′≤𝒔 vars set by 𝝈 are in 𝑻 𝒊 𝟏 ,𝑻 𝒊 𝟐 ,…, 𝑻 𝒊 𝒕 Specify values set by 𝝅 Claim: 𝑭 plus 1, 2, & 3 let us reconstruct 𝝆 1. # of vars 𝒔 ′ set by 𝝈 satisfies 𝒔 𝟐 ≤ 𝒔 ′ ≤𝒔 and 𝝆 ′ ∈ 𝑴 𝒏 ℓ− 𝒔 ′ so 𝒔/𝟐≤ 𝒔 ′ ≤𝒔 | 𝑴 𝒏 ℓ− 𝒔 ′ | possibilities Bit vector of length 𝒔′−𝟏 for partition of [𝒔′] into 𝝈 𝟏 , 𝝈 𝟐 ,…, 𝝈 𝒕 plus elt of [𝒓] 𝒔 ′ to indicate positions of each var in its term. Total ≤𝟐 𝒔 ′ −𝟏 𝒓 𝒔 ′ Matching of length 𝒔 touching all vertices of 𝝈. Total ℓ 𝒔 So: less than 𝟐 −𝟏 ℓ 𝒔 𝒔/𝟐≤ 𝒔 ′ ≤𝒔 (𝟐𝒓) 𝒔 ′ | 𝑴 𝒏 ℓ− 𝒔 ′ | of the 𝝆 have such a bad path

Matching Switching Lemma Proof | 𝑴 𝒏 ℓ |= 𝒏 ℓ 𝒏+𝟏 …(ℓ+𝟐) 𝑴 𝒏 ℓ− 𝒔 ′ 𝑴 𝒏 ℓ = ℓ… ℓ− 𝒔 ′ +𝟏 𝒏−ℓ+ 𝒔 ′ … 𝒏−ℓ+𝟏 ℓ+𝟏 …(ℓ− 𝒔 ′ +𝟐) ≤ ℓ(ℓ+𝟏) 𝒏−ℓ 𝒔 ′ So since 𝟒𝒓ℓ ℓ+𝟏 ≤𝒏−ℓ the probability of height at least 𝒔 is at most 𝟐 −𝟏 ℓ 𝒔 𝒔/𝟐≤ 𝒔 ′ ≤𝒔 𝟐𝒓 𝒔 ′ | 𝑴 𝒏 ℓ− 𝒔 ′ | 𝑴 𝒏 ℓ ≤ ℓ 𝒔 𝟐𝒓ℓ(ℓ+𝟏) 𝒏−ℓ 𝒔/𝟐 ≤ 𝟐𝒓 ℓ 𝟑 (ℓ+𝟏) 𝒏−ℓ 𝒔/𝟐

Beyond Bipartite Matching Similar switching lemmas for Matching restrictions on the variables of the complete graph 𝑲 𝟐𝒏+𝟏 [B-Pitassi 1996] 𝒌-hypergraph matching restrictions on complete 𝒌-regular hypergraphs over [𝒌𝒏+𝒋] for 𝟏≤𝒋<𝒌. [B-Riis 1998] Switching lemmas for 𝑹𝒆𝒔(𝒌): [Buss, Impagliazzo, Segerlind 2004] Set a polynomially-small fraction of inputs. [Razborov 2017] Switching lemmas for Tseitin formulas: [Pitassi, Rossman, Servedio, Tan 2016] [Håstad 2017]

Multi-Switching Lemma

Using Hastad’s Switching Lemma Lemma (Håstad): Let 𝑭 be a 𝒓-DNF formula on 𝟎,𝟏 𝒏 . For ℓ=𝒑𝒏 we have 𝐏𝐫 𝝆∼ 𝑹 𝒏 ℓ 𝑫 𝑭 ​ 𝝆 ≥𝒔 ≤ 𝟖𝒑𝒓 𝒔 Use it for 𝒌 different 𝒓-DNFs at once by setting 𝒑=𝟏/(𝟏𝟔𝒓) and 𝒔=log⁡𝒌+𝟏 to get failure probability at most 𝒌 𝟖𝒑𝒓 𝒔 ≤𝟏/𝟐. To get better failure probability need larger 𝒔. Alternative: A multiswitching lemma.

What multiswitching produces A single decision tree 𝑻 of height 𝒕 is a common 𝒔-partial decision tree for 𝒓-DNF formulas 𝑭 𝟏 , 𝑭 𝟐 ,…, 𝑭 𝒌 iff for every branch 𝝉 of 𝑻, for every 𝒊∈ 𝒌 , 𝑫 𝑭 𝒊 ​ 𝝉 <𝒔. Define 𝑷( 𝑭 𝟏 ,…, 𝑭 𝒌 , 𝒕,𝒔) to be the property that no such tree exists. Multiswitching Lemma (Håstad 2014*): Let 𝑭 𝟏 , 𝑭 𝟐 ,…, 𝑭 𝒌 be 𝒓-DNF formulas on 𝟎,𝟏 𝒏 . For ℓ=𝒑𝒏, integer 𝒕, and 𝒔≥ 𝐥𝐨𝐠 𝒌 we have 𝐏𝐫 𝝆∼ 𝑹 𝒏 ℓ [ 𝑷(𝑭 𝟏 ​ 𝝆 , …, 𝑭 𝒌 ​ 𝝆 ,𝒕,𝒔)]≤𝒌 (𝟐𝟒𝒑𝒓) 𝒕 . *related lemma shown in [Impagliazzo, Matthews, Paturi 2012]

What multiswitching produces Multiswitching Lemma (Håstad 2014): Let 𝑭 𝟏 , 𝑭 𝟐 ,…, 𝑭 𝒌 be 𝒓-DNF formulas on 𝟎,𝟏 𝒏 . For ℓ=𝒑𝒏, integer 𝒕, and 𝒔≥ 𝐥𝐨𝐠 𝒌 we have 𝐏𝐫 𝝆∼ 𝑹 𝒏 ℓ [ 𝑷(𝑭 𝟏 ​ 𝝆 , …, 𝑭 𝒌 ​ 𝝆 ,𝒕,𝒔)]≤𝒌 (𝟐𝟒𝒑𝒓) 𝒕 . Corollary: Let 𝑭 𝟏 , 𝑭 𝟐 ,…, 𝑭 𝒌 be 𝒓-DNF formulas on 𝟎,𝟏 𝒏 . For ℓ=𝒑𝒏 and 𝒔≥ 𝐥𝐨𝐠 𝒌 there is a distribution on restrictions 𝒟 depending on 𝑭 𝟏 , 𝑭 𝟐 ,…, 𝑭 𝒌 such that: ∀𝝆∈𝒟. |Vars(𝝆)|=ℓ 𝐏𝐫 𝝆∼𝒟 ∃𝒊∈ 𝒌 .𝑫(𝑭 𝒊 ​ 𝝆 ≥𝒔]≤𝒌 (𝟒𝟖𝒑𝒓) ℓ .

Proof Multiswitching Lemma The original proof used conditional probability using arbitrary downward-closed conditions and random restrictions with independent probabilities over the bits. We give a bit simpler proof using similar ideas to the proof of the basic lemma. Suppose that 𝑷(𝑭 𝟏 ​ 𝝆 , …, 𝑭 𝒌 ​ 𝝆 ,𝒕,𝒔) holds. We will give a specific construction for a decision tree 𝑻 and bound that probability that it fails to be a height-𝒕 𝒔-partial decision tree for 𝑭 𝟏 ​ 𝝆 , …, 𝑭 𝒌 ​ 𝝆 .

Multiswitching Lemma Proof Given 𝝆,𝑭 𝟏 , …, 𝑭 𝒌 let 𝒋 𝟏 be the first index such that |𝑫 𝑻 𝑭 𝒋 𝟏 ​ 𝝆 |≥𝒔 Let 𝝅 𝟏 be the lexicographically first path in the canonical decision tree for 𝑭 𝒋 𝟏 ​ 𝝆 of length 𝒔. Define 𝝈 𝟏 = 𝝈 𝟏 𝟏 … 𝝈 𝒖 𝟏 as before, except make sure that 𝝈 𝒖 𝟏 reaches the 1-leaf. Then 𝒔 ′ = 𝝈 𝟏 ≥𝒔 and 𝒔 ′ <𝒔+𝒓. 𝝅 𝟏 is enough to determine 𝝈 𝟏 . Begin building tree 𝑻 by with complete tree of height 𝒔′ on the variables set by 𝝈 𝟏 For every path 𝝉 𝟏 path of length 𝒔’ in 𝑻 continue as follows:

Multiswitching Lemma Proof Given 𝝆, 𝝉 𝟏 ,𝑭 𝟏 , …, 𝑭 𝒌 let 𝒋 𝟐 be the first index such that |𝑫 𝑻 𝑭 𝒋 𝟐 ​ 𝝆 𝝉 𝟏 |≥𝒔 We have 𝒋 𝟐 ≥ 𝒋 𝟏 Let 𝝅 𝟐 be the lexicographically first path in the canonical decision tree for 𝑭 𝒋 𝟐 ​ 𝝆 𝝉 𝟏 of length 𝒔. Define 𝝈 𝟐 = 𝝈 𝟏 𝟐 … 𝝈 𝒖 ′ 𝟐 as before, except make sure that 𝝈 𝒖 ′ 𝟐 reaches the 1-leaf. Then 𝒔 ′′ = 𝝈 𝟐 ≥𝒔 and 𝒔 ′′ <𝒔+𝒓. 𝝅 𝟐 is enough to determine 𝝈 𝟐 . Continue tree 𝑻 at leaf 𝝉 𝟏 with complete tree of height 𝒔′′ on the variables set by 𝝈 𝟐 For every path 𝝉 𝟏 𝝉 𝟐 in 𝑻 continue and repeat until no more indices.

Multiswitching Lemma Proof Failure occurs iff there is a path of length ≥𝒕 in 𝑻 Let 𝝉 be lex first such path Let 𝝈 𝟏 ,…, 𝝈 𝒗 correspond to 𝝉 𝟏 ,…, 𝝉 𝒗 . Define 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒗 ∈ 𝑹 𝒏 ℓ−𝒕 As before we include info to decode 𝝆 from 𝝆′: 𝒋 𝟏 ,…, 𝒋 𝒗 𝒔 vars within terms of 𝑭 𝒋 𝒊 set by 𝝅 𝒊 for each 𝒊∈[𝒗] Bit vector of values set by 𝝅 𝟏 … 𝝅 𝒗 Bit vector of values set by 𝝉 𝟏 … 𝝉 𝒗 𝝈 𝟏 𝝉 𝟏 𝝈 𝟐 𝝉 𝟐 𝝉 𝟑 𝝈 𝟑 𝝉 𝟒 𝝈 𝟒 𝝉 𝟓 𝝈 𝟓 Tree 𝑻

Multiswitching Lemma Proof Define 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒗 ∈ 𝑹 𝒏 ℓ−𝒕 . Info to decode 𝝆 from 𝝆′: 𝒋 𝟏 ,…, 𝒋 𝒗 ∈[𝒌] 𝒔 vars within terms of 𝑭 𝒋 𝒊 set by 𝝅 𝒊 for each 𝒊∈[𝒗] Bit vector of values set by 𝝅 𝟏 … 𝝅 𝒗 Bit vector of values set by 𝝉 𝟏 … 𝝉 𝒗 Claim: this is enough to decode given 𝑭 𝟏 ,…, 𝑭 𝒌 ,𝒔,𝒕: Look at first term 𝑻 𝒊 𝟏 set to 1 in 𝑭 𝒋 𝟏 ​ 𝝆 ′ (previous terms will be set to 0). This yields 𝝈 𝟏 𝟏 . Use the information from 2 to determine which vars of 𝑻 𝒊 𝟏 are unset by 𝝆 and hence set by 𝝅 𝟏 𝟏 . Use 3 to determine 𝝅 𝟏 𝟏 . Continue as in the ordinary switching lemma decoding by replacing 𝝈 𝟏 𝟏 by 𝝅 𝟏 𝟏 in 𝝆 ′ to get 𝝆 ′′ . Then look at 1st term 𝑻 𝒊 𝟐 set to 1 in 𝑭 𝒋 𝟏 ​ 𝝆 ′′ etc. Continue replacing until get all of 𝝈 𝟏 . Now go back and replace 𝝈 𝟏 by 𝝉 𝟏 in 𝝆′ to get 𝝆 𝝉 𝟏 𝝈 𝟐 … 𝝈 𝒗 , using 4.

Multiswitching Lemma Proof Define 𝝆 ′ =𝝆 𝝈 𝟏 … 𝝈 𝒗 ∈ 𝑹 𝒏 ℓ−𝒕 . Info to decode 𝝆 from 𝝆′: 𝒋 𝟏 ,…, 𝒋 𝒗 ∈[𝒌] 𝒔 vars within terms of 𝑭 𝒋 𝒊 set by 𝝅 𝒊 for each 𝒊∈[𝒗] Bit vector of values set by 𝝅 𝟏 … 𝝅 𝒗 Bit vector of values set by 𝝉 𝟏 … 𝝉 𝒗 Counts: 𝒌 𝒗 ≤ 𝟐 𝒗−𝟏 𝒔 𝒌≤𝒌⋅ 𝟐 𝒕 since 𝒔≥ 𝐥𝐨𝐠 𝒌 and 𝒔 𝒗−𝟏 ≤𝒕 𝟐 𝒕 ′ −𝟏 𝒓 𝒕 ′ where 𝒕 ′ ≤𝒕 is the total length of 𝝅 𝟏 … 𝝅 𝒗 Total ≤ 𝟐 𝒕 𝒓 𝒕 𝟐 𝒕 ′ 𝟐 𝒕 Total fraction of restrictions where 𝑷( 𝑭 𝟏 ,…, 𝑭 𝒌 , 𝒕,𝒔) holds is at most: 𝟏𝟔𝒓 𝒕 |𝑹 𝒏 ℓ−𝒕 | 𝑹 𝒏 ℓ =𝒌 𝟏𝟔𝒓 𝒕 𝒏 ℓ−𝒕 𝟐 𝒏−ℓ+𝒕 𝒏 ℓ 𝟐 𝒏−ℓ = 𝒌 𝟑𝟐𝒓 𝒕 ℓ…(ℓ−𝒕+𝟏) 𝒏−ℓ+𝒕 …(𝒏−ℓ+𝟏) ≤𝒌 𝟑𝟐𝒑𝒓 𝟏−𝒑 𝒕