Admissible rule

Admissible rule

In logic, a rule of inference is admissible in a formal system if the set of theorems of the system is closed under the rule. The concept of an admissible rule was introduced by Paul Lorenzen (1955).


The concept of admissibility, as introduced above, can be applied generally to formal systems which do not resemble any kind of logic, see this example. However, 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 (for instance, { o,land,lor,ot} in the case of superintuitionistic logics, or { o,ot,Box} in the case of monomodal logics). Well-formed formulas are built freely using these connectives from a countably infinite set of propositional variables "p""n". A substitution σ is a function from formulas to formulas which commutes with the connectives, i.e.,:sigma f(A_1,dots,A_n)=f(sigma A_1,dots,sigma A_n)for every connective "f", and formulas "A"1, …, "A""n". (We may also apply substitutions to sets Γ of formulas, making nowrap|1=σΓ = {σ"A": "A" ∈ Γ}.) A Tarski-style consequence relation [Blok & Pigozzi (1989), Kracht (2007)] is a relation vdash between sets of formulas, and formulas, such that
#Avdash A,
#if Gammavdash A then Gamma,Deltavdash A,
#if Gammavdash A and Delta,Avdash B then Gamma,Deltavdash B,for all formulas "A", "B", and sets of formulas Γ, Δ. A consequence relation such that

  1. if Gammavdash A then sigmaGammavdashsigma A
for all substitutions σ is called structural. (Note that the term "structural" as used here and below is unrelated to the notion of structural rules in sequent calculi.) A structural consequence relation is called a propositional logic. A formula "A" is a theorem of a logic vdash if varnothingvdash A.

For example, we identify a superintuitionistic logic "L" with its standard consequence relation vdash_L axiomatizable by modus ponens and axioms, and we identify a normal modal logic with its global consequence relation vdash_L axiomatized by modus ponens, necessitation, and axioms.

A structural inference rule [Rybakov (1997), Def. 1.1.3] (or just rule for short) is given by a pair (Γ,"B"), usually written as:frac{A_1,dots,A_n}Bqquad ext{or}qquad A_1,dots,A_n/B,where Γ = {"A"1, …, "A""n"} is a finite set of formulas, and "B" is a formula. An instance of the rule is:sigma A_1,dots,sigma A_n/sigma Bfor a substitution σ. The rule Γ/"B" is derivable in vdash, if Gammavdash B. It is admissible if for every instance of the rule, σ"B" is a theorem whenever all formulas from σΓ are theorems. [Rybakov (1997), Def. 1.7.2] We also write Gamma,|!!!sim B if Γ/"B" is admissible. (Note that |!!!sim is a structural consequence relation on its own.)

Every derivable rule is admissible, but not vice versa in general. A logic is structurally complete if every admissible rule is derivable, i.e., {vdash}={,|!!!sim}. [Rybakov (1997), Def. 1.7.7]

In logics with a well-behaved conjunction connective (such as superintuitionistic or modal logics), a rule A_1,dots,A_n/B is equivalent to A_1landdotsland A_n/B with respect to admissibility and derivability. It is therefore customary to only deal with unary rules "A"/"B".


*Classical propositional calculus ("CPC") is structurally complete. [Chagrov & Zakharyaschev (1997), Thm. 1.25] Indeed, assume that "A"/"B" is non-derivable rule, and fix an assignment "v" such that "v"("A") = 1, and "v"("B") = 0. Define a substitution σ such that for every variable "p", σ"p" = op if "v"("p") = 1, and σ"p" = ot if "v"("p") = 0. Then σ"A" is a theorem, but σ"B" is not (in fact, ¬σ"B" is a theorem). Thus the rule "A"/"B" is not admissible either. (The same argument applies to any multi-valued logic "L" complete with respect to a logical matrix whose all elements have a name in the language of "L".)
*The Kreisel­–Putnam rule (aka Harrop's rule, or independence of premise rule)::(mathit{KPR})qquadfrac{ eg p o qlor r}{( eg p o q)lor( eg p o r)}:is admissible in the intuitionistic propositional calculus ("IPC"). In fact, it is admissible in every superintuitionistic logic. [Prucnal (1979), cf. Iemhoff (2006)] On the other hand, the formula::( eg p o qlor r) o( eg p o q)lor( eg p o r):is not an intuitionistic tautology, hence "KPR" is not derivable in "IPC". In particular, "IPC" is not structurally complete.
*The rule::frac{Box p}p:is admissible in many modal logics, such as "K", "D", "K"4, "S"4, "GL" (see this table for names of modal logics). It is derivable in "S"4, but it is not derivable in "K", "D", "K"4, or "GL".
*The rule::frac{Diamond plandDiamond eg p}ot:is admissible in every normal modal logic. [Rybakov (1997), p. 439] It is derivable in "GL" and "S"4.1, but it is not derivable in "K", "D", "K"4, "S"4, "S"5.
*Löb's rule::(mathit{LR})qquadfrac{Box p o p}p:is admissible (but not derivable) in the basic modal logic "K", and it is derivable in "GL". However, "LR" is not admissible in "K"4. In particular, it is "not" true in general that a rule admissible in a logic "L" must be admissible in its extensions.
*The Gödel–Dummett logic ("LC"), and the modal logic "Grz".3 are structurally complete.Rybakov (1997), Thms. 5.4.4, 5.4.8]

Decidability and reduced rules

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 (i.e., its set of theorems) 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 now that admissibility of rule in a decidable logic is Pi^0_1 (i.e., its complement is recursively enumerable). For instance, it is known that admissibility in the bimodal logics "K""u" and "K"4"u" (the extensions of "K" or "K"4 with the universal modality) is undecidable. [ Wolter & Zakharyaschev (2008)] 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. [Rybakov (1997), §3.9] A modal rule in variables "p"0, …, "p""k" is called reduced if it has the form:frac{igvee_{i=0}^nigl(igwedge_{j=0}^k eg_{i,j}^0p_jlandigwedge_{j=0}^k eg_{i,j}^1Box p_jigr)}{p_0},where each eg_{i,j}^u is either blank, or negation eg. For each rule "r", we can effectively construct a reduced rule "s" (called the reduced form of "r") such that any logic admits (or derives) "r" if and only if it admits (or derives) "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 extstyleigvee_{i=0}^nvarphi_i/p_0 be a reduced rule as above. We identify every conjunction varphi_i with the set { eg_{i,j}^0p_j, eg_{i,j}^1Box p_jmid jle k} of its conjuncts. For any subset "W" of the set {varphi_imid ile n} of all conjunctions, let us define a Kripke model M=langle W,R,{Vdash} angle by:varphi_iVdash p_jiff p_jinvarphi_i,:varphi_i,R,varphi_{i'}iffforall jle k,(Box p_jinvarphi_iRightarrow{p_j,Box p_j}subseteqvarphi_{i'}). Then the following provides an algorithmic criterion for admissibility in "K"4: [Rybakov (1997), Thm. 3.9.3]

Theorem. The rule extstyleigvee_{i=0}^nvarphi_i/p_0 is "not" admissible in "K"4 if and only if there exists a set Wsubseteq{varphi_imid ile n} such that
#varphi_i Vdash p_0 for some ile n,
#varphi_iVdashvarphi_i for every ile n,
#for every subset "D" of "W" there exist elements alpha,etain W such that the equivalences::alphaVdashBox p_j if and only if varphiVdash p_jlandBox p_j for every varphiin D::alphaVdashBox p_j if and only if alphaVdash p_j and varphiVdash p_jlandBox p_j for every varphiin D:hold for all "j".

Similar criteria can be found for the logics "S"4, "GL", and "Grz". [Rybakov (1997), Thms. 3.9.6, 3.9.9, 3.9.12; cf. Chagrov & Zakharyaschev (1997), §16.7] Furthermore, admissibility in intuitionistic logic can be reduced to admissibility in "Grz" using the Gödel–McKinsey–Tarski translation: [Rybakov (1997), Thm. 3.2.2] :A,|!!!sim_{IPC}B if and only if T(A),|!!!sim_{Grz}T(B).

Rybakov (1997) developed much more sophisticated techniques for showing decidability of admissibility, which apply to a robust (infinite) class of transitive (i.e., extending "K"4 or "IPC") modal and superintuitionistic logics, including e.g. "S"4.1, "S"4.2, "S"4.3, "KC", "T""k" (as well as the above mentioned logics "IPC", "K"4, "S"4, "GL", "Grz"). [Rybakov (1997), §3.5]

Despite being decidable, the admissibility problem has relatively high computational complexity, even in simple logics: admissibility of rules in the basic transitive logics "IPC", "K"4, "S"4, "GL", "Grz" is coNEXP-complete. [Jeřábek (2007)] This should be contrasted with the derivability problem (for rules or formulas) in these logics, which is PSPACE-complete. [Chagrov & Zakharyaschev (1997), §18.5]

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 (1999, 2000). In the logical setup, a unifier of a formula "A" in a logic "L" (an "L"-unifier for short) is a substitution σ such that σ"A" is a theorem of "L". (Using this notion, we can rephrase admissibility of a rule "A"/"B" in "L" as "every "L"-unifier of "A" is an "L"-unifier of "B".) An "L"-unifier σ is less general than an "L"-unifier τ, written as σ ≤ τ, if there exists a substitution υ such that:vdash_Lsigma pleftrightarrow upsilon au pfor 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 (mgu) of "A" is a unifier σ such that {σ} 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:Avdash_L Bleftrightarrowsigma Bfor every formula "B". Note that σ is a mgu of "A". In transitive modal and superintuitionistic logics with the finite model property (fmp), one can characterize projective formulas semantically as those whose set of finite "L"-models has the extension property: [Ghilardi (2000), Thm. 2.2] 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", "K"4, "S"4, "GL", "Grz" (and more generally in any transitive logic with the fmp whose set of finite frame satisfies another kind of extension property), we can effectively construct for any formula "A" its projective approximation Π("A"): [Ghilardi (2000), p. 196] a finite set of projective formulas such that
#Pvdash_L A for every PinPi(A),
#every unifier of "A" is a unifier of a formula from Π("A").It follows that the set of mgus of elements of Π("A") is a complete set of unifiers of "A". Furthermore, if "P" is a projective formula, then:P,|!!!sim_L B if and only if Pvdash_L Bfor any formula "B". Thus we obtain the following effective characterization of admissible rules: [Ghilardi (2000), Thm. 3.6] :A,|!!!sim_L B if and only if forall PinPi(A),(Pvdash_L B).

Bases of admissible rules

Let "L" be a logic. A set "R" of "L"-admissible rule is called a basis [Rybakov (1997), Def. 1.4.13] 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 |!!!sim_L is the smallest structural consequence relation which includes vdash_L and "R".

Notice that decidability of admissible rules of a decidable logic is equivalent to the existence of recursive (or recursively enumerable) 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. (In other words, we can decide admissibility of "A"/"B" by the following algorithm: we start in parallel two exhaustive searches, one for a substitution σ which unifies "A" but not "B", and one for a derivation of "A"/"B" from "R" and vdash_L. One of the searches has to eventually come up with an answer.) Apart from decidability, explicit bases of admissible rules are useful for some applications, e.g. in proof complexity. [Mints & Kojevnikov (2004)]

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. [Rybakov (1997), Thm. 4.5.5] Finite bases are relatively rare: even the basic transitive logics "IPC", "K"4, "S"4, "GL", "Grz" do not have a finite basis of admissible rules, [Rybakov (1997), §4.2] though they have independent bases. [Jeřábek (2008)]

Examples of bases

*The empty set is a basis of "L"-admissible rules if and only if "L" is structurally complete.
*Every extension of the modal logic "S"4.3 (including, notably, "S"5) has a finite basis consisting of the single rule [Rybakov (1997), Cor. 4.3.20] ::frac{Diamond plandDiamond eg p}ot.
*Visser's rules::frac{displaystyleBigl(igwedge_{i=1}^n(p_i o q_i) o p_{n+1}lor p_{n+2}Bigr)lor r}{displaystyleigvee_{j=1}^{n+2}Bigl(igwedge_{i=1}^{n}(p_i o q_i) o p_jBigr)lor r},qquad nge 1:are a basis of admissible rules in "IPC" or "KC". [Iemhoff (2001, 2005), Roziere (1992)]
*The rules::frac{displaystyleBoxBigl(Box q oigvee_{i=1}^nBox p_iBigr)lorBox r}{displaystyleigvee_{i=1}^nBox(qlandBox q o p_i)lor r},qquad nge0:are a basis of admissible rules of "GL". [Jeřábek (2005)] (Note that the empty disjunction is defined as ot.)
*The rules::frac{displaystyleBoxBigl(Box(q oBox q) oigvee_{i=1}^nBox p_iBigr)lorBox r}{displaystyleigvee_{i=1}^nBox(Box q o p_i)lor r},qquad nge0:are a basis of admissible rules of "S"4 or "Grz". [Jeřábek (2005,2008)]

emantics for admissible rules

A rule Γ/"B" is valid in a modal or intuitionistic Kripke frame F=langle W,R angle, if the following is true for every valuation Vdash in "F"::if forall xin W,(xVdash A) for all AinGamma, then forall xin W,(xVdash B).(The definition readily generalizes to general frames, if needed.)

Let "X" be a subset of "W", and "t" a point in "W". We say that "t" is
*a reflexive tight predecessor of "X", if for every "y" in "W": "t R y" if and only if "t" = "y" or "x" = "y" or "x R y" for some "x" in "X",
*an irreflexive tight predecessor of "X", if for every "y" in "W": "t R y" if and only if "x" = "y" or "x R y" for some "x" in "X".We say that a frame "F" has reflexive (irreflexive) tight predecessors, if for every "finite" subset "X" of "W", there exists a reflexive (irreflexive) tight predecessor of "X" in "W".

We have: [Iemhoff (2001), Jeřábek (2005)]
*a rule is admissible in "IPC" if and only if it is valid in all intuitionistic frames which have reflexive tight predecessors,
*a rule is admissible in "K"4 if and only if it is valid in all transitive frames which have reflexive and irreflexive tight predecessors,
*a rule is admissible in "S"4 if and only if it is valid in all transitive reflexive frames which have reflexive tight predecessors,
*a rule is admissible in "GL" if and only if it is valid in all transitive converse well-founded frames which have irreflexive tight predecessors.

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.

tructural 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. [Rybakov (1997), Thms. 5.5.6, 5.5.9] On the other hand, the Mints rule:frac{(p o q) o plor r}{((p o q) o p)lor((p o q) o r)}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 every its 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 by Tsitkin 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 "K"4 is hereditarily structurally complete if and only if it is not valid in any of certain twenty Kripke frames (including the five intuitionistic frames above).

There exist structurally complete logics that are not hereditarily structurally complete: for example, Medvedev's logic is structurally complete, [Prucnal (1976)] but it is included in the structurally incomplete logic "KC".


A rule with parameters is a rule of the form:frac{A(p_1,dots,p_n,s_1,dots,s_k)}{B(p_1,dots,p_n,s_1,dots,s_k)},whose variables are divided into the "regular" variables "p""i", and the parameters "s""i". The rule is "L"-admissible if every "L"-unifier σ of "A" such that σ"s""i" = "s""i" for each "i" is also a unifier of "B". The basic decidability results for admissible rules also carry to rules with parameters. [Rybakov (1997), §6.1]

A multiple-conclusion rule is a pair (Γ,Δ) of two finite sets of formulas, written as:frac{A_1,dots,A_n}{B_1,dots,B_m}qquad ext{or}qquad A_1,dots,A_n/B_1,dots,B_m.Such a rule is admissible if every unifier of Γ is also a unifier of some formula from Δ. [Jeřábek (2005); cf. Kracht (2007), §7] For example, a logic "L" is consistent iff it admits the rule:frac{;ot;}{},and a superintuitionistic logic has the disjunction property iff it admits the rule:frac{plor q}{p,q}.Again, basic results on admissible rules generalize smoothly to multiple-conclusion rules. [Jeřábek (2005, 2007, 2008)] 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 "S"4 the rule above is equivalent to:frac{A_1,dots,A_n}{Box B_1lordotslorBox B_m}.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:frac{Gammavdash A,DeltaqquadPi,AvdashLambda}{Gamma,DeltavdashPi,Lambda}.(By abuse of language, it is also sometimes said that the (full) sequent calculus admits cut, meaning its cut-free version does.) However, admissibility in sequent calculi is usually only a notational variant for admissibility in the corresponding logic: any complete calculus for (say) intuitionistic logic admits a sequent rule if and only if "IPC" admits the formula rule which we obtain by translating each sequent GammavdashDelta to its characteristic formula igwedgeGamma oigveeDelta.



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*S. Ghilardi, "Best solving modal equations", Annals of Pure and Applied Logic 102 (2000), no. 3, pp. 183–198. doi|10.1016/S0168-0072(99)00032-9
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*R. Iemhoff, "Intermediate logics and Visser's rules", Notre Dame Journal of Formal Logic 46 (2005), no. 1, pp. 65–81. doi|10.1305/ndjfl/1107220674
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*G. Mints and A. Kojevnikov, "Intuitionistic Frege systems are polynomially equivalent", Zapiski Nauchnyh Seminarov POMI 316 (2004), pp. 129–146. [ gzipped PS]
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