Admissible rule


In logic, a rule of inference is admissible in a formal system if the set of theorems of the system does not change when that rule is added to the existing rules of the system. In other words, every formula that can be derived using that rule is already derivable without that rule, so, in a sense, it is redundant. The concept of an admissible rule was introduced by Paul Lorenzen.

Definitions

Admissibility has been systematically studied only in the case of structural rules in propositional non-classical logics, which we will describe next.
Let a set of basic propositional connectives be fixed. Well-formed formulas are built freely using these connectives from a countably infinite set of propositional variables p0, p1,.... A substitution σ is a function from formulas to formulas which commutes with the connectives, i.e.,
for every connective f, and formulas A1,..., An. A Tarski-style consequence relation is a relation between sets of formulas, and formulas, such that
for all formulas A, B, and sets of formulas Γ, Δ. A consequence relation such that
for all substitutions σ is called structural. A structural consequence relation is called a propositional logic. A formula A is a theorem of a logic if.
For example, we identify a superintuitionistic logic L with its standard consequence relation axiomatizable by modus ponens and axioms, and we identify a normal modal logic with its global consequence relation axiomatized by modus ponens, necessitation, and axioms.
A structural inference rule is given by a pair, usually written as
where Γ = is a finite set of formulas, and B is a formula. An instance of the rule is
for a substitution σ. The rule Γ/B is derivable in, if. It is admissible if for every instance of the rule, σB is a theorem whenever all formulas from σΓ are theorems. In other words, a rule is admissible if, when added to the logic, does not lead to new theorems. We also write if Γ/B is admissible.
Every derivable rule is admissible, but not vice versa in general. A logic is structurally complete if every admissible rule is derivable, i.e.,.
In logics with a well-behaved conjunction connective, a rule is equivalent to with respect to admissibility and derivability. It is therefore customary to only deal with unary rules A/B.

Examples

The basic question about admissible rules of a given logic is whether the set of all admissible rules is decidable. Note that the problem is nontrivial even if the logic itself is decidable: the definition of admissibility of a rule A/B involves an unbounded universal quantifier over all propositional substitutions, hence a priori we only know that admissibility of rule in a decidable logic is . For instance, it is known that admissibility in the bimodal logics Ku and K4u is undecidable. Remarkably, decidability of admissibility in the basic modal logic K is a major open problem.
Nevertheless, admissibility of rules is known to be decidable in many modal and superintuitionistic logics. The first decision procedures for admissible rules in basic transitive modal logics were constructed by Rybakov, using the reduced form of rules. A modal rule in variables p0,..., pk is called reduced if it has the form
where each is either blank, or negation. For each rule r, we can effectively construct a reduced rule s such that any logic admits r if and only if it admits s, by introducing extension variables for all subformulas in A, and expressing the result in the full disjunctive normal form. It is thus sufficient to construct a decision algorithm for admissibility of reduced rules.
Let be a reduced rule as above. We identify every conjunction with the set of its conjuncts. For any subset W of the set of all conjunctions, let us define a Kripke model by
Then the following provides an algorithmic criterion for admissibility in K4:
Theorem. The rule is not admissible in K4 if and only if there exists a set such that
  1. for some
  2. for every
  3. for every subset D of W there exist elements such that the equivalences
Similar criteria can be found for the logics S4, GL, and Grz. Furthermore, admissibility in intuitionistic logic can be reduced to admissibility in Grz using the Gödel–McKinsey–Tarski translation:
Rybakov developed much more sophisticated techniques for showing decidability of admissibility, which apply to a robust class of transitive modal and superintuitionistic logics, including e.g. S4.1, S4.2, S4.3, KC, Tk.
Despite being decidable, the admissibility problem has relatively high computational complexity, even in simple logics: admissibility of rules in the basic transitive logics IPC, K4, S4, GL, Grz is coNEXP-complete. This should be contrasted with the derivability problem in these logics, which is PSPACE-complete.

Projectivity and unification

Admissibility in propositional logics is closely related to unification in the equational theory of modal or Heyting algebras. The connection was developed by Ghilardi. In the logical setup, a unifier of a formula A in a logic L is a substitution σ such that σA is a theorem of L. An L-unifier σ is less general than an L-unifier τ, written as σ ≤ τ, if there exists a substitution υ such that
for every variable p. A complete set of unifiers of a formula A is a set S of L-unifiers of A such that every L-unifier of A is less general than some unifier from S. A most general unifier of A is a unifier σ such that is a complete set of unifiers of A. It follows that if S is a complete set of unifiers of A, then a rule A/B is L-admissible if and only if every σ in S is an L-unifier of B. Thus we can characterize admissible rules if we can find well-behaved complete sets of unifiers.
An important class of formulas which have a most general unifier are the projective formulas: these are formulas A such that there exists a unifier σ of A such that
for every formula B. Note that σ is a mgu of A. In transitive modal and superintuitionistic logics with the finite model property, one can characterize projective formulas semantically as those whose set of finite L-models has the extension property: if M is a finite Kripke L-model with a root r whose cluster is a singleton, and the formula A holds in all points of M except for r, then we can change the valuation of variables in r so as to make A true in r as well. Moreover, the proof provides an explicit construction of a mgu for a given projective formula A.
In the basic transitive logics IPC, K4, S4, GL, Grz, we can effectively construct for any formula A its projective approximation Π: a finite set of projective formulas such that
  1. for every
  2. every unifier of A is a unifier of a formula from Π.
It follows that the set of mgus of elements of Π is a complete set of unifiers of A. Furthermore, if P is a projective formula, then
for any formula B. Thus we obtain the following effective characterization of admissible rules:

Bases of admissible rules

Let L be a logic. A set R of L-admissible rule is called a basis of admissible rules, if every admissible rule Γ/B can be derived from R and the derivable rules of L, using substitution, composition, and weakening. In other words, R is a basis if and only if is the smallest structural consequence relation which includes and R.
Notice that decidability of admissible rules of a decidable logic is equivalent to the existence of recursive bases: on the one hand, the set of all admissible rule is a recursive basis if admissibility is decidable. On the other hand, the set of admissible rules is always co-r.e., and if we further have an r.e. basis, it is also r.e., hence it is decidable. Apart from decidability, explicit bases of admissible rules are useful for some applications, e.g. in proof complexity.
For a given logic, we can ask whether it has a recursive or finite basis of admissible rules, and to provide an explicit basis. If a logic has no finite basis, it can nevertheless has an independent basis: a basis R such that no proper subset of R is a basis.
In general, very little can be said about existence of bases with desirable properties. For example, while tabular logics are generally well-behaved, and always finitely axiomatizable, there exist tabular modal logics without a finite or independent basis of rules. Finite bases are relatively rare: even the basic transitive logics IPC, K4, S4, GL, Grz do not have a finite basis of admissible rules, though they have independent bases.

Examples of bases

A rule Γ/B is valid in a modal or intuitionistic Kripke frame, if the following is true for every valuation in F:
Let X be a subset of W, and t a point in W. We say that t is
We say that a frame F has reflexive tight predecessors, if for every finite subset X of W, there exists a reflexive tight predecessor of X in W.
We have:
Note that apart from a few trivial cases, frames with tight predecessors must be infinite, hence admissible rules in basic transitive logics do not enjoy the finite model property.

Structural completeness

While a general classification of structurally complete logics is not an easy task, we have a good understanding of some special cases.
Intuitionistic logic itself is not structurally complete, but its fragments may behave differently. Namely, any disjunction-free rule or implication-free rule admissible in a superintuitionistic logic is derivable. On the other hand, the Mints rule
is admissible in intuitionistic logic but not derivable, and contains only implications and disjunctions.
We know the maximal structurally incomplete transitive logics. A logic is called hereditarily structurally complete, if any extension is structurally complete. For example, classical logic, as well as the logics LC and Grz.3 mentioned above, are hereditarily structurally complete. A complete description of hereditarily structurally complete superintuitionistic and transitive modal logics was given respectively by Citkin and Rybakov. Namely, a superintuitionistic logic is hereditarily structurally complete if and only if it is not valid in any of the five Kripke frames
Similarly, an extension of K4 is hereditarily structurally complete if and only if it is not valid in any of certain twenty Kripke frames.
There exist structurally complete logics that are not hereditarily structurally complete: for example, Medvedev's logic is structurally complete, but it is included in the structurally incomplete logic KC.

Variants

A rule with parameters is a rule of the form
whose variables are divided into the "regular" variables pi, and the parameters si. The rule is L-admissible if every L-unifier σ of A such that σsi = si for each i is also a unifier of B. The basic decidability results for admissible rules also carry to rules with parameters.
A multiple-conclusion rule is a pair of two finite sets of formulas, written as
Such a rule is admissible if every unifier of Γ is also a unifier of some formula from Δ. For example, a logic L is consistent iff it admits the rule
and a superintuitionistic logic has the disjunction property iff it admits the rule
Again, basic results on admissible rules generalize smoothly to multiple-conclusion rules. In logics with a variant of the disjunction property, the multiple-conclusion rules have the same expressive power as single-conclusion rules: for example, in S4 the rule above is equivalent to
Nevertheless, multiple-conclusion rules can often be employed to simplify arguments.
In proof theory, admissibility is often considered in the context of sequent calculi, where the basic objects are sequents rather than formulas. For example, one can rephrase the cut-elimination theorem as saying that the cut-free sequent calculus admits the cut rule
However, admissibility in sequent calculi is usually only a notational variant for admissibility in the corresponding logic: any complete calculus for intuitionistic logic admits a sequent rule if and only if IPC admits the formula rule which we obtain by translating each sequent to its characteristic formula.