Capturing k-ary existential second order logic with k-ary inclusion–exclusion logic

Abstract

In this paper we analyze k-ary inclusion-exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX [k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO [k].We also introduce several useful operators that can be expressed in INEX [k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.Keywords: inclusion logic, exclusion logic, dependence logic, team semantics, IF-logic, existential second order logic, expressive power. * This paper is an extended version of [18] with additional technical details.Next we generalize these notations for tuples of variables. Let s be an assignment, x := x 1 . . . x k a tuple of variables and a := (a 1 , . . . , a k ) ∈ M k . We use the notation s[ a/ x ] := s[a 1 /x 1 , . . . , a k /x k ]. For a team X, a set A ⊆ M k and a function F : X → P(M k ) we writeLet M be an L-model, s an assignment and t ∈ T L s.t. Vr(t) ⊆ dom(s). The interpretation of t with respect to M and s, t M s , is denoted simply by s(t). For a team X and t ∈ T L s.t. Vr(t) ⊆ dom(X) we write X(t) := {s(t) | s ∈ X}. Let t := t 1 . . . t k be a tuple of L-terms and let X be a team s.t. Vr( t ) ⊆ dom(X). We writeNote that s( t ) is a vector in M and X( t ) is a k-ary relation in M. We use the notation P * (A) for the power set of A excluding the empty set (that is P * (A) := P(A) \ {∅}). We are now ready to define team semantics for FO. Definition 2.2. Let M be an L-model, ϕ ∈ FO L and X a team such that Fr(ϕ) ⊆ dom(X). We define the truth of ϕ in M and X, denoted by M X ϕ:• M X t 1 = t 2 iff s(t 1 ) = s(t 2 ) for all s ∈ X.Definition 2.4. If t 1 , t 2 are k-tuples of L-terms, t 1 ⊆ t 2 is a k-ary inclusion atom. We define Fr( t 1 ⊆ t 2 ) = Vr( t 1 ) ∪ Vr( t 2 ). The language INC L is defined by adding the following condition to the definition of FO L (Definition 2.1).• If t 1 , t 2 are tuples of L-terms of the same length, then t 1 ⊆ t 2 ∈ S.Note that we do not allow negation to appear in front of inclusion atoms. For literals, connectives and quantifiers we use the same semantics as for FO with team semantics. Inclusion atoms have the following truth condition:Definition 2.5. Let M be a model and X a team s.t. Vr( t 1 t 2 ) ⊆ dom(X). We define the truth of t 1 ⊆ t 2 in the model M and the team X:This truth condition can be written equivalently as follows:Example 2.1. Let t 1 , . . . , t m be k-tuples of L-terms and x a k-tuple of fresh variables. Now the following holds for all nonempty teams X:In particular, for t ∈ T...

Safe Dependency Atoms and Possibility Operators in Team Semantics

“…This, in brief, is due to the second order existential quantifications implicit in the Team Semantics rules for disjunction and existential quantification. Thus, exploring the properties of fragments of such logics (as done for instance in [2,3,10,12,21]) provides an interesting avenue to the study of the properties and relations between fragments of Second Order Logic.…”

The Expressive Power of k-ary Exclusion Logic

“…In this paper we will show that the relationship between these two logics becomes nontrivial when we consider their bounded arity fragments. This also leads to results on the relation between inclusion and exclusion logics, which is interesting because they can be seen as duals to each other, as we have argued in [19].…”

“…In an earlier work by the author [19] it was shown that both INC[k]-and EXC[k]-formulas could be translated into k-ary ESO, ESO [k], which gives us an upper bound for the expressive power of EXC [k]. In [19] it was also shown that conversely ESO[k]-formulas with at most k-ary free relation variables can be expressed in k-ary inclusion-exclusion logic, INEX[k], and consequently INEX [k] captures ESO[k] on the level of sentences.…”

“…We will use the observation above to modify our translation (in [19]) from ESO [k] to INEX[k]. If we only consider sentences of exclusion logic, we can quantify the needed complementary values, and then replace inclusion atoms in the translation with the corresponding exclusion atoms.…”

“…If we only consider sentences of exclusion logic, we can quantify the needed complementary values, and then replace inclusion atoms in the translation with the corresponding exclusion atoms. The remaining problem is that in our translation we also needed a new connective called term value preserving disjunction ( [19]) to avoid the loss of information on the values of certain variables when evaluating disjunctions. This operator can be defined by using both inclusion and exclusion atoms ( [19]), but it is undefinable in exclusion logic since it is not closed downwards.…”