LSM is not generated by binary functions

For all arities k ≥ 0 define B k to be the set of functions {0, 1} k → R ≥0 and define B = k B k . A function F ∈ B k is log-supermodular (lsm) if F (x ∨ y)F (x ∧ y) ≥ F (x)F (y) for all x, y ∈ {0, 1} k where x ∨ y and x ∧ y denote elementwise maximum and minimum respectively. Denote the set of all lsm functions by LSM. Define IMP ∈ B 2 by IMP(x, y) = 0 if (x, y) = (1, 0) 1 otherwise Bulatov et al. [1] defined the operation of (efficient) pps ω -definability in order to study the computational complexity of certain approximate counting problems. They asked ( [1], Section 4) whether all functions in LSM can be defined by IMP and B 1 in this sense. We give a negative answer to this question. Živný et al. [4] proved a negative result for the analogous optimisation problem.

We will construct a set C containing binary lsm functions and closed under a minimalist set of operations related to T 2 -constructibility [3]. Later, in section 3, we will relate this to pps ω -definability. For concision we will use vector notation such as

For all k ≥ 0 and all F ∈ B k and all permutations π of {1, • • • , k} define the permutation F π ∈ B k by

This work was supported by an EPSRC doctoral training grant.

For all k ≥ 0, all F ∈ B k , and all w ∈ {0, 1} k , define B(F, w) by

For all k ≥ 1 and all F ∈ B k and all 1 ≤ i ≤ k and all v ∈ {0, 1}, define the primitive pinning F i →v by

We say that F ′ ∈ B is a pinning of F if F ′ can be obtained from F by a (possibly empty) sequence of primitive pinnings. Define

Lemma 2. C contains all lsm functions of arity at most 2.

Proof. Let F ′ be an lsm function of arity at most 2. Let F be a pinning of F ′ of arity k and let w ∈ {0, 1} k . We would like to show that B(F, w) ≥ 0. By Lemma 1 we may assume

Proof. We argue by induction on the number of primitive pinnings used to construct H. The first statement is trivial if

In both cases H is of the required form. The second statement is trivial if H = tr 1,2 F . Otherwise, by induction H = (tr 1,2 F ′ ) i →v for some pinning F ′ of F , so H = tr 1,2 (F ′ (i+2) →v ) as required.

Lemma 4. C is closed under tensor products, primitive contractions, and permutations.

Proof. Let F ′ , G ′ ∈ C and let H be a pinning of F ′ ⊗ G ′ . By Lemma 3 there exist pinnings F of F ′ and G of G ′ such that H = F ⊗ G.

Let k and ℓ be the arities of F and G respectively. For all

Let H be a pinning of tr 1,2 F ′ . By Lemma 3 there exists a pinning F of F ′ such that H = tr 1,2 F . Let k be the arity of F . For all w ∈ {0, 1} k-2 , B(tr 1,2 F, w)

The definitions of B and C are clearly invariant under permutations.

Lemma 5. Let F be a subset of B closed under tensor products, primitive contractions, and permutations. Let F be a function of the form

where each φ j is a function application G j (x i j,1 , • • • , x i j,a(j) ) with G j ∈ F , such that for 1 ≤ i ≤ n exactly one pair (j, k) satisfies i j,k = i, and for n + 1 ≤ i ≤ n + m exactly two pairs (j, k) satisfy i j,k = i. Then F , and hence all permutations of F , are in F .

Proof. We will prove the statement by induction on m.

Proof. Let F be a pinning of EQ 3 . Let k be the arity of F . Let w ∈ {0, 1} k . We would like to show that B(F, w) ≥ 0. By Lemma 1 we may assume i w i = 2. If F = EQ 3 we have B(EQ 3 , w) = (-1) 0 + (-1) 2 ≥ 0. Otherwise F has arity 2 hence F = (EQ 3 ) i →v for some i ∈ {1, 2, 3} and v ∈ {0, 1}. By symmetry we may assume i = 3. But for all x 1 , x 2 we have EQ 3 (x 1 , x 2 , v)EQ 3 (1-x 1 , 1-x 2 , v) = 0 which implies B((EQ 3 ) 3 →v , (1, 1)) = 0.

We will use the following definitions of pps-formulas, -,ω , pps ω -definability, and F φ for pps-formulas φ. These definitions are taken from [1] (Section 2), except that we specialise to functions in B and we rename EQ to EQ 2 .

Suppose F ⊆ B is some collection of functions, V = {v 1 , . . . , v n } is a set of variables and x : {v 1 , . . . , v n } → {0, 1} is an assignment to those variables. An atomic formula has the form φ = G(v i 1 , . . . , v ia ) where G ∈ F , a = a(G) is the arity of G, and (v i 1 , v i 2 , . . . , v ia ) ∈ V a is a scope. Note that repeated variables are allowed. The function

where from now on we write x j = x(v j ). A pps-formula (“primitive product summation formula”) is a summation of a product of atomic formulas. A pps-formula ψ over F in variables

where φ j are all atomic formulas over F in the variables V ′ . (The variables V are free, and the others, V ′ \ V , are bound.) The formula ψ specifies a function F ψ : {0, 1} n → R ≥0 in the following way:

(2)

where x and y are assignments x : {v 1 , . . . , v n } → {0, 1} and y : {v n+1 , . . . , v n+m } → {0, 1}. The functional clone F generated by F is the set of all functions in B that can be represented by a pps-formula over F ∪ {EQ 2 } where EQ 2 is the binary equality function defined by EQ 2 (x, x) = 1 and EQ 2 (x, y) = 0 for x = y.

Then we say that an a-ary function F is pps ω -definable over F if there exists a finite subset S F of

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