Count-Invariance Including Exponentials

We deﬁne inﬁnitary count-invariance for categorial logic, extending count-invariance for multiplicatives (van Benthem, 1991) and additives and bracket modalities (Valent´ın et al., 2013) to include exponentials. This provides an e ﬀ ective tool for pruning proof search in categorial parsing / theorem-proving.


Introduction
In logical grammar, which dates back to (Ajdukiewicz, 1935), grammar is reduced to logic: an expression is grammatical if and only if an associated logical statement is a theorem of a calculus.

Sharing
In standard logic information does not have multiplicity. Thus where + is the notion of addition of information and ≤ is the notion of inclusion we have x+x ≤ x and x ≤ x+x; both these two properties together amount to idempotency: x+x = x. These properties are expressed by the rules of inference of Contraction and Expansion: (1) a.
Linguistic resources do not freely have these properties: grammaticality is not generally preserved under addition or removal of copies of words or expressions. However, there are some constructions manifesting something similiar. Parasitic gaps involve a kind of Contraction. Parasitic gaps cannot occur anywhere, thus: (2) *the slave that i John sold e i to e i Rather, we assume here that as the term 'parasitic' suggests, a parasitic gap must fall within an island. Extraction from weak islands can become fully acceptable when accompanied by a cobound non-island extraction: ( And iterated coordination allows a kind of Expansion: (4) John likes, Mary dislikes and Bill loves London.
That is, in logical grammar a controlled use of idempotency, or sharing, is motivated. Girard (1987) introduced exponentials for such control.

Count-Invariance
Count-invariance for multiplicatives in (sub)linear logic is introduced in van Benthem (1991). This involves simply checking the number of positive and negative occurrences of each atom in a sequent. Thus where #(Σ) is a count of the sequent Σ we have: I.e. the numbers of positive and negative occurrences of each atom must exactly balance for the sequent to be a theorem. This provides a necessary, but of course not sufficient, criterion for theoremhood, and can be checked rapidly. It can be used as a filter in proof search: if backward chaining proof search generates a goal which does not satisfy the count-invariant, the goal can be discarded. This notion of count for multiplicatives was included in the categorial parser/theoremprover CatLog (Morrill, 2012). In Valentín et al. (2013) the idea is extended to additives (and bracket modalities). Instead of a single count for each atom of a sequent Σ we have a minimum count # min (Σ) and a maximum count # max (Σ) and for a sequent to be a theorem it must satisfy two inequations: I.e. the count functions # min and # max define an interval which must include the point of balance 0; for the multiplicatives, # min = # max = # and (6) reduces to the special case (5). This generalised notion of count is included in the categorial parser/theorem-prover CatLog2.
The structure of the continuation of the paper is as follows. In Section 2 we present the infinitary count algebra which we employ, we define the fragment of categorial logic for which we illustrate count invariance, and we define the (infinitary) count functions for this fragment. In Section 3 we state and prove our count-invariance theorem. In Section 4 we evaluate the introduction of exponential count invariance experimentally in relation to CatLog parsing/theorem-proving.

Infinitary Count Algebra
We consider terms built over constants 0, 1, ⊥ (−∞: minus infinity), and (+∞: plus infinity) by binary operations of plus (+), minus (−), minimum (min) and maximum (max), and the infinitary step functions X and Y as follows where i and j are integers (* indicates undefined):

The count functions
The count function, or count functions, are functions from types and sequents into values in the count algebra such that if sequents are provable their images under the count functions fall within a certain range. It follows that if their images do not fall within the required range then the sequents are not provable; we give examples after defining the count functions, in the next subsection. This provides an efficient filter on parsing/theoremproving, as we show in the last section.
Let us assume primitive types P. For Q ∈ P∪{[]}, m ∈ {min, max} and min = max and max = min we define where # • and # • are as below. We define the enrichment LAb! b ? of the Lambek calculus (Lambek, 1958) with types Tp as follows: Where P ∈ P, p ∈ {•, •}, and • = • and • = • we define the count functions: Proof. By induction as in Figure 1; justifications refer to the Proposition (7).
To present sequents we define configurations Config and tree terms TreeTerm in terms of types Tp as follows, where Λ is the empty string: The rules for LAb! b ? are shown in Figure 2. Note that !C is of a generalised form necessary to prove Cut-elimination in the presence of !R. Note also that ?L is an infinitary rule; it is not used in linguistic applications. We include it here for the sake of showing technical completeness of the count invariance. For tree terms and configurations, counts are: Lemma 8 extends to configurations.

Examples
Relativisation including medial and parasitic extraction is obtained by assigning a relative pronoun a type (CN\CN)/(!N\S ) whereby the body of a relative clause is analysed as !N\S . By way of example of count-invariance, we show how it discards N, N\S ⇒ !N\S corresponding to the ungrammaticality of a relative clause without a gap: *paper that John walks. We have the max N-count: Iterated sentential coordination is obtained by assigning a coordinator the type (?S \S )/S . By way of a second example we show how count-invariance discards N, N, N\S ⇒ ?S corresponding to the ungrammaticality of unequilibrated coordination: *John Mary walks and Suzy talks.

Theorem and Proof
Our main theorem is: Proof. The proof is by induction on the length of derivations. For the base case P ⇒ P we have The inductive cases are as follows, where we use: • a − (b − c) = (a − b) + c (Where we write #(∆) with ∆ a context we should more precisely understand that ∆ is a configuration with a hole where the count of a hole is always zero.)

Multiplicatives
For every atom or bracket, The induction hypothesis (i.h.) tells us that # min (Γ ⇒ A) ≤ 0 and # min ( For every atom or bracket, For every atom or bracket, For every atom or bracket, Therefore by i.h., For every atom or bracket, For every atom or bracket, For atoms: I.e. the property for the conclusion follows from the induccion hypothesis for the premise since brackets and bracket modalities are transparent to atom count.
For brackets: For atoms: Since brackets and bracket modalities are transparant to atom count.
For brackets: since brackets and bracket modalities are transparent to atom count.
For brackets, since brackets and bracket modalities are transparent to atom count.
For brackets, Therefore by i.h.: For brackets, For atoms and brackets, And for atoms and brackets, For atoms and brackets, For atoms and brackets,