Expressivity of extensions of dynamic first-order logic

Dynamic predicate logic (DPL), presented in [5] as a formalism for representing anaphoric linking in natural language, can be viewed as a fragment of a well known formalism for reasoning about imperative programming [6]. An interesting difference from other forms of dynamic logic is that the distinction between formulas and programs gets dropped: DPL formulas can be viewed as programs. In this paper we show that DPL is in fact the basis of a hierarchy of formulas-as-programs languages. 1 Formulas-as-Programs in DPL and its Extensions In this paper we investigate the landscape of extensions of DPL with the six operations ∪, ,̆ σ, σ̆, ∩, ∃∃. All 64 combinations are classified with respect to their expressive power. Extensions of DPL with ∩ (relation intersection) and ∃∃ (local variable declaration) are studied in [8], while in [1], an extension of DPL with ∪ (relation union) and σ (simultaneous substitution) is axiomatized, and ω-completeness is proved for the extension of DPL with ∪, σ and ∗ (Kleene star). In a dynamic setting, left-to-right substitutions σ have right-to-left counterparts σ̆ (converse substitutions). For preand postcondition reasoning with extensions of DPL, converse substitution and relation converse ̆ are attractive. Some of these operators have been studied from a linguistic perspective. In particular, [4] discusses potential linguistic applications of the ∪ and ∩ operator. The relevance of ∩ operator is mentioned there for the analysis of plurals, and it is argued that ∪ is needed for a proper analysis of sequences such as: “A professor or an assistant professor will attend the meeting of the university board. She will report to the faculty.” Unlike [5], we allow function symbols, so DPL terms are given by t ::= x | c | f(t1, . . . , tn). DPL formulas are given by φ ::= ∃x | Rt1 . . . tn | t1 ≈ t2 | ¬φ | φ1;φ2. Terms are interpreted as usual. We use t for the interpretation of term t in model M under valuation g. A substitution is a finite set of bindings x := t, with the usual conditions that no binding is trivial (of the form x := x) and that every x in the set has at most one binding (substitutions are functional). Examples are {x := f(x)} (“set new x equal to f -value of old x”), {x := y, y := x} (“swap values of x and y”). If a substitution contains just a single binding we omit the curly brackets and write just the assignment statement x := t. A converse substitution is a finite set of converse bindings (x := t)̆ , with the same conditions as those for substitutions. An example is (x := f(x))̆ (“set old x equal to f -value of new x”, or “look at all inputs g that differ from the output h only in x, and that satisfy f(g(x)) = h(x)”). A variable state in a model M = 〈D, I〉 is a member of D, where Var is the set of variables of the language. Letting g, h, k range over variable states, the interpretation of DPL and the extensions that we study, in a model M = 〈D, I〉, is given by the following definition: g[[∃x]]h iff h = g d for some d ∈ D g[[Rt1 . . . tn]] M h iff h = g and 〈t 1 , . . . , t M,g n 〉 ∈ R M g[[t1 ≈ t2]] M h iff h = g and t 1 = t M,g 2 g[[¬φ]]h iff h = g and there is no k such that g[[φ]]k g[[φ;ψ]]h iff g[[φ]]k and k[[ψ]]h for some k ∈ D g[[σ]]h iff h = g1n d1···dn where {x1, . . . , xn} = dom(σ) and di = σ(xi) M,g g[[σ̆]]h iff g = h1n d1···dn where {x1, . . . , xn} = dom(σ) and di = σ(xi) M,h g[[∃∃x(φ)]]h iff (g d )[[φ]] M k and h = k g(x) for some d ∈ D, k ∈ D Var g[[φ ∪ ψ]]h iff g[[φ]]h or g[[ψ]]h g[[φ ∩ ψ]]h iff g[[φ]]h and g[[ψ]]h g[[φ ]̆]h iff h[[φ]]g. 1ILLC, Amsterdam 2CWI and ILLC, Amsterdam, Uil-OTS, Utrecht 3ILLC, Amsterdam With DPL we will denote the basic language. Extensions will be indicated with DPL(X), where X is a set of operators. For instance, DPL(σ, )̆ denotes the extension of DPL with simultaneous substitutions and converse. 2 The Lattices of DPL and DPL(∗) Extensions The following figure represents the lattice of all possible combinations of DPL with operators from {∪,∩, ,̆ σ, σ̆, ∃∃} (union, intersection, converse, simultaneous substitution, converse substitution, hiding). It indicates which operators can be defined in terms of which; the labels on the arrows indicate counterexamples to equal expressivity, i.e., formulas from the lower language that don’t have a counterpart in the upper language.