After the last chapter we append an extensive list of formal laws of the various temporal logics studied in this book. Thus, a special effect of the last relation is that the set of valid formulas of (any language of) PL can be "mechanically generated" (in technical terms: it is recursively enumerable).
Classical First-Order Logic
Furthermore, we can define S(ξ)(A)∈ {ff,tt}for each atomic formula:. where is this kind of often1andt2). The formula∀x◦(NEL,x) =x is valid inZ, but not universally valid: consider the structure that differs from Z by defining NELS= 1.
Theories and Models
We have introduced the concept of theories in the framework of classical first-order logic. Of course, it can be defined in the same way for any other logic.
Extensions of Logics
A second-order theory (LSOL,A) consists of a second-order language LSOL and a set A of non-logical axioms (formulas of LSOL). A proper second-order theory for the natural numbers takes the signature SIGNat and the first six axioms of the first-order theory together with the new induction axiom.
Bibliographical Notes
This intention implicitly assumes that in a formal system the set of axioms can be determined and for any finite sequence A1,. To summarize, a logical LOG with a consequence relation is called incomplete if there is no formal systemΣ for LOG in this sense such that.
Basic Propositional Linear Temporal Logic
The Basic Language and Its Normal Semantics
Phrases in next state, in all states (and so on) are at the heart of the temporal logic approach. As we will see later, many interesting valid formulas will be of the syntactic form A↔B.
Temporal Logical Laws
By Theorems 2.2.1 and 2.2.2 we know the logical laws in LTL, which derive from the "classical basis" of the new logic. Finally, (T20) and (T21) assert that “infinitely often” distributes over ∨ and “almost always” distributes over ∧. These formulas claim that at least "some direction" of the further distributions 2, 3, 23, and 32 holds.
Axiomatization
However, we are not concerned here with how tautologically valid formulas can be derived (which would be done entirely within the 'classical part' of the formal system). The subtraction theorem can be used to abbreviate derivations, as illustrated by the following example: to derive the valid formula.
Completeness
This is a contradiction; thus (∗) is proved. From Lemma 2.4.6 a), we know that the KP contains only a finite number of distinct vertices. Now we actually have all the means to prove the theorem, which is a rather trivial rewrite of the desired perfection result.
Decidability
To apply an ω rule, the derivation of their infinitely many premises needs an argument "outside" the system, usually an inductive argument. So the table root is also not closed and the table is successful;.
Initial Validity Semantics
Altogether, we find that the tableau method can be implemented in time exponentially in the size of P. Despite all these differences between LTL and LTL0, it is however remarkable that the two (universal) notions of validity still coincide: .. The assertion follows directly from Theorem 2.6.2 b), by choosing F=∅wat. If A is an axiom of ΣLTL, then 2A is an axiom of ΣLTL0 and thus derivable in the latter.
Extensions of LTL
Binary Temporal Operators
It is worth noting that the recursive equivalences for the strong and weak versions of an operator have the same form. Let us now summarize our discussion for extending the basic language LLTL with binary operators. The semantics of the new operators of LbLTL has already been defined above, and the proof theory for the extended logic (which we will denote by LTL+b) can be given quite uniformly.
Fixpoint Operators
Similarly, it can be shown that false and 2A are the smallest and largest fix points of the equivalence (i) above, and that Auntil B and Aunless B are the smallest and largest fix points of (iii). We want to introduce new (unary) logical operators that, when applied to a formula F (which generally contains a variable u), give formulas that are the smallest and largest fix points of the equivalence. From this completed proof of Lemma 3.2.2 and Theorem 3.2.1, we have shown that μuAΞK defines the smallest fixpoint of the mappingΥA as intended.
Propositional Quantification
As in the similar situation in the previous section, we assume a corresponding language for which the semantic clause defining Ki(v1unlv2) in LTL+b is transferred to K(Ξ)i (v1unlv2) for each time structureK,i ∈N, and arbitraryΞ). The semantic definitions for LTL+μ and LTL+q have a "global" flavor in the sense that the valuation Ξ is used in its entirety for the definition of K(Ξ)i (A), and not just its suffix Ξi = (ξi , ξi+1, Nevertheless, a natural generalization of Lemma 2.1.5 holds for these logics, which we now show for the logic LTL+q.
Past Operators
We will see in the next section that this is related to the initial validity discussed in Sect. Ki(B) = ttwhat the statement proves. A close look at this proof shows that the if part of the LTL relation is still valid in LTL+p.). We conclude with the obvious remark that the subtraction theorem does not hold in LTL+p in the form of theorem 2.3.3, but must be modified.
Syntactic Anchoring
The informal meaning of init is to behave like efalse in LTL+p, ie. to keep exactly in the initial state of a temporal structure. An axiomatization of LTL+i is given by expanding the formal systemΣLTL with the two characteristic laws introduced above, i.e. the additional axiom. iltl) e¬init and the addition rule (init) init→2A A. According to the semantic considerations above, this no longer applies in LTL+i in the general form of LTL.
Combinations of Extensions
They can be derived in the formal system ΣLTLp supplemented by one of the above axioms. An extended version adds the binary tense operators until and since, i.e. similar formulas. 2.3 (and now colloquially understood as a series of "points of time" and as the relation "earlier than") the semantics of the tense operators are given by the clauses.
Expressiveness
LTL and Its Extensions
In fact, an immediate transfer of the proof of the equality2AΞK = νu(A∧ eu)ΞK to an example in Sect. We now show that LTL+μ≤LTL+q and leave the proof of the other direction for Sect. From the assumptions on the polarity ofuinA phenomena, Lemma 3.2.2 and Theorem 3.2.1 imply ΥA(μΥA) =μΥA; hencej ∈μΥA=uΞK, which completes the proof.
Temporal Logic and Classical First-Order Logic
If A≡B→C, we define LTL(A,x0) =LTL(B,x0)→LTL(C,x0), and the statement follows by means of the induction hypothesis. Therefore, the only occurrences of x0 in B are of the form x0 < y,x0 = y ory In fact, no random ordering can be found for the left-hand direction dag, and we will now show that a B¨uchiΩ automaton does not accept the time structureKif and only ifdag(Ω,K). Note in passing that each run dag trivially accepts an equal ranking, for example by assigning rank0 to each node.) The "if" part of this theorem is quite obvious. Because "infinitely oftenAnd infinitely often B" is not the same as "infinitely oftenAandB", the construction of a B¨uchi automaton that accepts the intersection of two languages is a bit more complicated than the standard construction of the product, as might already be noted. in proving Theorem 4.3.1. A generalized B¨uchiΩ= (V,Q,Q0, δ,Acc) automaton for a finite set of propositional constants Vof has the same structure as a B¨uchi automaton, except that the acceptance condition is given by a finite setAcc = {See (1), . 4.2, the only terms of the second-order language defined by this subscript are the individual variablesx ∈ X. For the remainder of the proof, recall that the only terms of LSOL(SIGV) are the (individual)x variables. A simple consequence of the results described in this section is that B¨uchi automata can also be encoded in LTL+q logic. When reading a state that satisfies v, the automaton reactivates the original location, because the original formula must also be true in the rest of the temporal structure. Since the rules of the game reflect the definition of a run, player AUTOMATON has a winning strategy if and only if K is accepted by Ω. B¨uchi automata were introduced in [24] for the proof of the decidability of the monadic second-order theory of one successor. Given a temporal signature TSIG = (SIG,X,V), SIG = (S,F,P), let LFOL(SIG+) be a first-order language in the sense of Sect. A temporal structureK = (S,W)together with a variable valuationξwith respect to S (which is a mappingξ: X → |S| as in Section 1.2) defines, for each stateηi of W, mappingsS(ξ,ηi)those valuesS( ξ,ηi)(t)∈ |S|for each term and S(ξ,ηi)(A)∈ {ff,tt}for each atomic formula A. If A is a rigid and non-temporal formula, considering A as a formula of LFOL(SIG), we can also evaluate S(ξ)(A) in the sense of classical FOL, and the respective clauses in Sect. Furthermore, we can include classical first-order reasoning in derivations within ΣFOLTL because of the first-order axioms and the (par) rule. However, a formal definition of the notion "first-order consequence" is not as easy as it was for the "tautological consequence" in (prop). The deduction theorem 2.3.3 of LTL can be transferred to ΣFOLTL with some restrictions, as discussed in Sect. Indeed, the "much larger" step from incompleteness to (full) completeness in FOLTL (but not in SOL) can be achieved in the same way. Interestingly, it is even the sameω rule. appropriate in the LTL case), which works here. Replacing (in) with (ω-in) in ΣFOLTL gives a (sound) semi-formal system which is complete in the sense that. For such new syntactic entities we write a (instead of the direct transcription ea) and call them empty (flexible) individual constants. A there exists a formula A∗ such that A and A∗ are logically equivalent and A∗ does not contain individual prime constants. This fact means that FOLTL can be viewed as "the same" as FOLTL and A formulas with individual prime-number constants as shorthand for the corresponding A*. To recap, in the following we will freely use primed individual constants within FOLTL without explicitly considering this as a change of logic (from FOLTL to FOLTL) and we will use formulas such as the Ain derivations derived above directly as applications of (pred). It should be noted that the premise in (wfr) is a formula of the form C→3D itself;. As a simple application, we want to show that repeated application of the rule (chain) can be coded into one application of (wfr). As can be seen, (wfr) is actually the only proper logical time rule that occurs in this derivation. There we discovered that natural numbers (the standard model Nof) can be characterized in FOLTL. In fact, both 0 and SUCC can be expressed (also with FOLTL formulas) which results in the fact that in FOLTL+q the natural numbers (me0,SUCC,+,∗) can. The desired distinction, that is to say the specification of the system, is carried out - as in classical logic theories - by certain non-logical axioms. We write LTL for any language LFOLTL with or without one or more of the extensions discussed in the preceding chapters. A specification of the circuit (more precisely: aC-LTL theory for the class of the temporal structures for representing all possible runs of the circuit) is given by the non-logical axioms. 5.3,ThN specifies the standard modelN of natural numbers in the sense that there exists a model(N,W)ofThN, and even more: for any model(S,W)ofThN,SandNare. For example, in the "even and odd number" system in the main text above, the axioms for N can only be given by the formulas P1–P8. Note, however, that in the definition of a STSΓ the setsFΓ and PΓ of SIGΓ and the interpretations of their elements in SΓ are not (yet) relevant.Non-deterministic ω-Automata
LTL and B ¨uchi Automata
Weak Alternating Automata
First-Order Linear Temporal Logic
Basic Language and Semantics
A Formal System
Incompleteness
Primed Individual Constants
Well-Founded Relations
Flexible Quantification
State Systems
Temporal Theories and Models
State Transition Systems
