First-Order Linear Temporal Logic
5.4 Primed Individual Constants
As in the case of a semi-formal axiomatization briefly mentioned in the main text above, the induction rule (ind) of temporal logic plays the crucial role in an approach to an arith- metically complete formal system. One essential part of the modification ofΣFOLTLcould be to replace (ind) by the rule
(ar-ind) Ay(0)→B,Ay(SUCC(y))→ cA ∀yA→2B
in whichyis a variable fromXNatandBdoes not containy. This rule describes just an- other inductive argumentation (“over the natural numbers”) which is easy to understand informally. It obviously corresponds to the basic semantical fact that the states in a state sequenceW= (η0, η1, η2, . . .)are indexed by the natural numbers. (Examining the con- siderations of this section, it is easy to see that this fact is, on the other hand, essentially responsible for the incompleteness of FOLTL.) Interestingly, we will encounter a similar line of argumentation (for another purpose) in Sect. 5.5.
Observe finally that the rule (ind) is just a trivial case of (ar-ind): ifAdoes not contain the variableythen (ar-ind) reduces to
A→B,A→ cA A→2B which is in fact (ind).
We writea (instead of the direct transcription ea) for such new syntactic entities and call them primed (flexible) individual constants. With this extension the sample phrase is simply expressible by
a=a+ 1.
Note that the next time operator is not really transferred in its full power. We allow only “one priming”, so there is no analogy to eev. Our approach will be sufficient for the usual applications.
Formally we extend FOLTL to a logic FOLTL the languageLFOLTL of which results fromLFOLTLby adding the prime symbolto the alphabet and the clause
• Ifa∈Xthenais a term of the same sort asa to the syntactical definition of terms.
For defining the semantics ofLFOLTLwe slightly modify our technical apparatus.
Up to now terms and atomic formulas were evaluated in states (and with respect to some variable valuationξ) whereas general formulas were interpreted over state sequences. This conceptual difference is emphasized by the different “interpretation functions” S(ξ,ηi) andK(ξ)i , respectively. Primed individual constants and, hence, terms and atomic formulas ofLFOLTL contain a temporal aspect as well referring not only to one but also to the next state in a state sequence. Accordingly, we omit here the separate mappingS(ξ,ηi)and useK(ξ)i instead from the very beginning of the inductive definition.
So, given a temporal structureK= (S,W)for the underlying temporal signature TSIG = ((S,F,P),X,V), a variable valuationξfor the set X of variables, and i∈N, we defineK(ξ)i (t)∈ |S|for termstinductively by the clauses
1. K(ξ)i (x) =ξ(x) forx ∈ X. 2. K(ξ)i (a) =ηi(a) fora∈X.
3. K(ξ)i (a) =ηi+1(a) fora∈X.
4. K(ξ)i (f(t1, . . . ,tn)) =fS(K(ξ)i (t1), . . . ,K(ξ)i (tn)) forf ∈F.
For atomic formulasA,K(ξ)i (A)∈ {ff,tt}is defined by 1. K(ξ)i (v) =ηi(v) forv ∈V.
2. K(ξ)i (p(t1, . . . ,tn)) =pS(K(ξ)i (t1), . . . ,K(ξ)i (tn)) forp∈P.
3. K(ξ)i (t1=t2) =tt ⇔ K(ξ)i (t1)andK(ξ)i (t2)are equal values in|S|.
Finally, the additional clauses definingK(ξ)i (F)for general formulasF and the no- tions of validity and consequence are adopted from FOLTL.
It is evident that for formulasF without primed individual constants,K(ξ)i (F) according to this definition coincides withK(ξ)i (F)whenF is viewed as a formula of FOLTL and evaluated as before.
Example. Fora ∈Xandx ∈ X,A≡a =a+ 1andB ≡x ∗a <a+aare formulas ofLFOLTL (with an obvious signature). AssumingNto be the underlying structure, we get
K(ξ)i (A) =tt ⇔ ηi+1(a) =ηi(a) + 1,
K(ξ)i (B) =tt ⇔ ξ(x)∗ηi(a)< ηi(a) +ηi+1(a).
So, ifξ(x) = 3andKis given by η0 η1 η2 η3 . . .
a 2 5 6 3 . . . then we obtain
K(ξ)0 (A) =ff,K(ξ)1 (A) =tt,K(ξ)2 (A) =ff,
K(ξ)0 (B) =tt,K(ξ)1 (B) =ff,K(ξ)2 (B) =ff.
Above we discussed already that the new formulaa =a+ 1can also be ex- pressed in FOLTL by∃x(e(a=x)∧x =a+1). Actually it turns out quite generally that FOLTLdoes not really produce more expressibility than FOLTL.
Theorem 5.4.1. In anyLFOLTL, for every formula A there is a formula A∗ such thatAandA∗are logically equivalent andA∗does not contain primed individual constants.
Proof. a) We defineA∗inductively according to the syntactic structure ofA.
1. Ais atomic: Then A ≡ p(t1, . . . ,tn)orA ≡ t1 = t2. IfAdoes not contain primed individual constants thenA∗≡A. Otherwise, leta1, . . . ,am ,m≥1, be the primed individual constants occurring in t1, . . . ,tn (ort1,t2, respectively) andx1, . . . ,xmbe variables not occurring inA. Then
A∗ ≡ ∃x1. . .∃xm(e(a1=x1∧. . .∧am =xm)∧A) whereAresults fromAby replacingaibyxifor1≤i≤m.
2. A≡false: ThenA∗≡false.
3. A≡B →C orA≡ eBorA≡2B: ThenA∗≡B∗→C∗orA∗≡ eB∗or A∗ ≡2B∗, respectively, whereB∗andC∗are the results of this construction forBandC.
4. A ≡ ∃xB: ThenA∗ ≡ ∃xB∗ whereB∗is the constructed formula forB(and this construction does not use the variablex in step 1).
Obviously,A∗does not contain primed individual constants.
b) Let now K = (S,W), W = (η0, η1, η2, . . .), be a temporal structure, ξ a variable valuation, andi∈N. For the formulaA∗defined in a) we show by the same induction that
K(ξ)i (A∗) =K(ξ)i (A)
from which the assertion of the theorem follows immediately.
1. Ais atomic: We treat only the caseA ≡p(t1, . . . ,tn). The caseA ≡t1 =t2 runs in quite the same way. IfAdoes not contain primed individual constants then the assertion is trivial. Otherwise, A in the above construction is of the formp(t1∗, . . . ,tn∗)where, for1 ≤i ≤n,tiresults fromtiby the replacement of thea1, . . . ,am byx1, . . . ,xm. So, abbreviatingB ≡a1=x1∧. . .∧am=xm we have
K(ξ)i (A∗) =tt ⇔ there is a ξwith ξ∼x1 ξand K(ξi )(∃x2. . .∃xm(eB∧A)) =tt
⇔ there are ξ, ξwith ξ∼x1ξ,ξ ∼x2 ξand K(ξi )(∃x3. . .∃xm(eB∧A)) =tt
...
⇔ there are ξ, ξ, . . . , ξ(m)with
ξ∼x1 ξ,ξ∼x2ξ,. . .,ξ(m−1)∼xm ξ(m)and K(ξi (m))(eB∧A) =tt
⇔ there are ξ, ξ, . . . , ξ(m)with
ξ∼x1 ξ,ξ∼x2ξ,. . .,ξ(m−1)∼xm ξ(m)and K(ξ
(m))
i (p(t1∗, . . . ,tn∗)) =tt and
ξ(m)(xj) =ηi+1(aj) =ηi(aj)for1≤j ≤m
⇔ there are ξ, ξ, . . . , ξ(m)with
ξ∼x1 ξ,ξ∼x2ξ,. . .,ξ(m−1)∼xm ξ(m)and pS(K(ξ)i (t1), . . . ,K(ξ)i (tn)) =tt
⇔ K(ξ)i (A) =tt.
2. A≡false: In this case the assertion is trivial.
3. A≡B →C,A≡ eB, orA≡2B: Using the respective induction hypothesis in each case, we have
K(ξ)i (A∗) =tt ⇔ K(ξ)i (B∗) =ff or K(ξ)i (C∗) =tt
⇔ K(ξ)i (B) =ff or K(ξ)i (C) =tt
⇔ K(ξ)i (A) =tt forA≡B→C,
K(ξ)i (A∗) = K(ξ)i+1(B∗) = K(ξ)i+1(B) = K(ξ)i (A)
forA≡ eB, and
K(ξ)i (A∗) =tt ⇔ K(ξ)j (B∗) =tt for everyj ≥i
⇔ K(ξ)j (B) =tt for everyj ≥i
⇔ K(ξ)i (A) =tt
forA≡2B.
4. A≡ ∃xB: Using the induction hypothesis, we have:
K(ξ)i (A∗) =tt ⇔ there is a ξwith ξ∼xξandK(ξi )(B∗) =tt
⇔ there is a ξwith ξ∼xξandK(ξi )(B) =tt
⇔ K(ξ)i (A) =tt.
Example. LetA≡a =a+ 1andB ≡y∗a <a+abe the formulas from the previous example. The construction of Theorem 5.4.1 yields
A∗≡ ∃x(e(a=x)∧x =a+ 1)
which is just the formula from the beginning of our discussion and B∗≡ ∃x(e(a=x)∧y∗a<a+x).
Note that the result of the general construction can often be simplified. For example, forC ≡2(a >1)we obtain
C∗≡ ∃x(e(a=x)∧2(x >1)),
and this is logically equivalent to 2 e(a>1).
If we now transfer the expressivity notions from Sect. 4.1 to the present logics then an immediate corollary of Theorem 5.4.1 is that FOLTL is not really more expressive than FOLTL.
Theorem 5.4.2. FOLTL and FOLTLare equally expressive.
Proof. FOLTL≤FOLTL is trivial since FOLTL⊆FOLTL (defined analogously as in Sect 4.1). Theorem 5.4.1 shows that FOLTL≤FOLTL; so together we obtain
FOLTL=FOLTL.
This fact means that we can view FOLTLas “the same” as FOLTL and formu- lasAwith primed individual constants as abbreviations for the correspondingA∗. Formal derivations with such “primed” formulas can use
A↔A∗
as an additional axiom. Consider, for example, the formula A ≡ a=y∧a=a→a=y.
Forgetting for a moment that we deal with temporal logic,A looks like a simple FOL formula (with variables or individual constantsy,a, anda) which should be derivable withinΣFOL. Now
A∗ ≡ a=y∧ ∃x(e(x =a)∧x =a)→ ∃x(e(x =a)∧x =y)
and therefore a derivation ofAas a FOLTLformula could consist of the two steps (1) a=y∧ ∃x(e(x =a)∧x =a)→
∃x(e(x =a)∧x =y) (pred)
(2) a=y∧a =a→a =y axiomA↔A∗,(prop),(1) To summarize, we will freely use in the following primed individual constants within FOLTL without explicitly considering this as a change of the logic (from FOLTL to FOLTL) and we will directly use formulas like the above-derivedAin derivations as applications of (pred). Clearly, non-temporal formulas of a respective language do not contain primed individual constants.
Furthermore, for sake of uniformity we will frequently extend the priming nota- tion to flexible propositional constantsv ∈Vusingvas a synonym for ev:
v ≡ ev.