A Journey
through the Possible Worlds of Modal Logic
Lecture 3.1: Introduction to temporal reasoning
and logics
Valentin Goranko
Department of Philosophy, Stockholm University
ESSLLI’2016, Bolzano, August 24, 2016
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Time and temporal reasoning
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What is time?
What is temporal logic?
What then is time?
If no one asks me, I know what it is.
If I wish to explain it to him who asks, I do not know.
Saint Augustine
What is time? Two answers:
Einstein: ”Time is what clocks measure.”
Ray Cummings:
”Time is what prevents everything from happening all at once.”
None is really satisfactory. Understanding time takes time...
What about temporal logic?
Classical logic reasons about a snapshot of the universe.
Temporal logic reasons about the dynamics of the universe.
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Reasoning about time: brief historic remarks
Some historical landmarks:
• Aristotle: the ’There will be a sea-battle tomorrow’ argument.
• Diodorus Cronus’ Master Argument.
•
•
•
•
•
possible: : “what is or will ever be”;
necessary: “what is and will always be”.
To discuss next lecture.
Middle ages: historical necessity and determinism vs free will.
William of Ockham: the idea of different possible futures.
Next lecture.
19th century origins of formal logic of time: G. Boole, CS Peirce.
Early modern approaches, fist half of 20th century:
McTaggart, Russell, Lukasiewicz, Quine, ...
Reichenbach: temporality and tenses in natural language.
Arthur Prior, 1950’s: temporal logic as modal logic.
Amir Pnuelli, 1977 paper ”The temporal logic of programs”:
launch of applications of temporal logic in CS.
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The nature of time and temporal ontologies
Temporal reasoning is subject to various philosophical dilemmas
regarding the nature of time:
•
•
•
•
•
•
linear or branching? (forward? backward?)
discrete or dense?
instant-based or interval-based?
bounded or unbounded?
with or without beginning/end?, etc.
or, is it not circular?
Each of these allows different ontological commitments for the nature
and structure of time, leading to different temporal logical systems: for
linear and for branching time, for discrete and for dense time,
point-based and interval-based logics, etc.
Here I will only discuss instant-based temporal models and reasoning.
Interval-based reasoning: a whole different story.
See an overview paper linked on the webpage.
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Instant-based models of time
Primitive temporal entities: time instants.
The basic relationships between them: equality and precedence ≺.
Models of time flow: hT , ≺i, or hT , i
(where x y is an abbreviation of x ≺ y ∨ x = y ).
Two main types of such models:
linear orderings and forward branching partial orderings.
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Temporal reasoning: an example
Socrates is teaching now.
Therefore, tomorrow he will have taught the day before,
and always in the future he will have been teaching sometime in the past,
and always in the past he would be teaching sometime in the future.
Furthermore, Socrates has been teaching since some time in the past
and he will be teaching until sometime in the future.
Moreover, Socrates will either be teaching forever
or after some future moment he will not be teaching anymore.
Can logic help formalizing, analyzing and doing temporal reasoning?
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Instant-based temporal reasoning: FO logic approach
Some natural properties of models of time flow, expressed with FO sentences:
• reflexivity: ∀x(x ≺ x); respectively irreflexivity: ∀x¬(x ≺ x);
• transitivity: ∀x∀y ∀z(x ≺ y ∧ y ≺ z → x ≺ z);
• anti-symmetry: ∀x∀y (x ≺ y ∧ y ≺ x → x = y );
• trichotomy (connectedness): ∀x∀y (x = y ∨ x ≺ y ∨ y ≺ x);
• density (between every two precedence-related instants there is an
instant): ∀x∀y (x ≺ y → ∃z(x ≺ z ∧ z ≺ y ));
• no beginning: ∀x∃y (y ≺ x); no end: ∀x∃y (x ≺ y );
• every instant has an immediate successor:
∀x∃y (x ≺ y ∧ ∀z(x ≺ z → y z)) and, likewise, every instant has an
immediate predecessor: ∀x∃y (y ≺ x ∧ ∀z(z ≺ x → z y ));
• forward discreteness (every instant with a successor has an immediate
successor): ∀x∃y (x ≺ y → ∃y (x ≺ y ∧ ∀z(x ≺ z → y z)));
• backward discreteness (every instant with a predecessor has an immediate
predecessor): ∀x∃y (y ≺ x → ∃y (y ≺ x ∧ ∀z(z ≺ x → z y ))).
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FO logic based temporal reasoning: pros and cons
FO logic based temporal reasoning: logical deduction in FO theories of
time. Common in AI and partly in philosophy.
Pros:
• Well expressive languages
• Complete deductive systems
• Tools for automated reasoning
Cons:
• Some important properties not expressible,
e.g. well-orderness, Dedekind completeness
• Involved syntax and grammar
• Church’s theorem: undecidability of validity
Arthur Prior’s alternative approach: using modal logic instead of FOL.
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Prior’s tense logic and some extensions
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Prior’s basic tense operators
Arthur Prior introduced temporal logic as a brand of modal logic in the
1950s-1960s. He also introduced possible worlds semantics in the
context of temporal logics before Kripke.
Pϕ
“It has at some (past) time been the case that ϕ”
Fϕ
“It will at some (future) time be the case that ϕ”
Hϕ
“It has always (in the past) been the case that ϕ”
Gϕ
“It will always (in the future) be the case that ϕ”
Thus, e.g., GP (Prior invents Tense Logic) translates as
“It will always be the case
that it has some time been the case
that Prior invents Tense Logic”.
P and F: weak temporal operators; H and G: strong temporal operators.
The two pairs are interdefinable:
Pϕ ≡ ¬H¬ϕ, Hϕ ≡ ¬P¬ϕ; Fϕ ≡ ¬G¬ϕ, Gϕ ≡ ¬F¬ϕ
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Prior’s basic tense logic TL: syntax
Basic tense language LTL : a propositional bimodal language containing
propositional connectives ⊥, ¬, ∧, a set of atomic propositions
AP = {p0 , p1 , ...}, and the two strong tense operators.
Formulae of LTL :
ϕ = p | ⊥ | ¬ϕ | ϕ1 ∧ ϕ2 | Gϕ | Hϕ
The other propositional connectives are defined as usual.
In particular: Fϕ := ¬G¬ϕ, Pϕ := ¬H¬ϕ.
Besides, we define Aϕ = Hϕ ∧ ϕ ∧ Gϕ, and Eϕ = Pϕ ∨ ϕ ∨ Fϕ.
On linear time flows these mean resp. ‘always’ and ‘ sometime’.
Other tenses can be expressed, too:
• PPϕ: “It had been the case that ϕ”
• FPϕ: “It will have been the case that ϕ”
• PFϕ: “It was going to be the case that ϕ”
• PFPϕ: “ϕ would have been the case”
Not suitable for progressive tenses.
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Prior’s basic tense logic TL: Kripke semantics
Tense frame: pair T = (T , ≺), where T is a set of time instants and ≺
is a precedence relation on T .
Valuation in T : mapping V : AP → P(T ). Tense model: (T , ≺, V ).
Truth of a formula at an instant in a tense model M = (T , ≺, V ): as in
modal logic, assuming ≺ associated with G, and its converse with H:
• M, t Gϕ if M, s ϕ for every s ∈ T such that t ≺ s
• M, t Hϕ if M, s ϕ for every s ∈ T such that t s, that is,
M, t Hϕ if M, s ϕ for every s ∈ T such that s ≺ t.
The respective clauses for F and P are:
• M, t Fϕ if M, s ϕ for some s ∈ T such that t ≺ s.
• M, t Pϕ if M, s ϕ for some s ∈ T such that s ≺ t.
Validity of tense formulae in models and frames and logical validity
are defined as in modal logic.
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TL vs first-order logic and McTaggart’s time series
McTaggart’s two alternative approaches to modelling time:
A-series: characterise events as Past, Present, or Future.
Presumes some particular present moment.
B-series: characterise events as relatively “Earlier” or “Later”.
No present instant is needed.
Affinity between the A-series and the modal approach of TL and between
the B-series and the first-order logic approach.
Thus, Prior’s TL is a rival to FOL for logical reasoning about time.
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Addendum: Standard translation of TL into FOL
The language and semantics of TL can be translated to classical FOL as follows.
Let L1 be a FO language with =, a binary predicate symbol R, and a countable
set of unary predicate symbols P = {P0 , P1 , ...}, corresponding to the set of
atomic propositions AP = {p0 , p1 , ...}.
The standard translation ST of TL into L1 is now defined as follows:
• ST (pi ) = Pi (x);
• ST (⊥) = ⊥;
• ST (ϕ ∨ ψ) = ST (ϕ) ∨ ST (ψ);
• ST (Gϕ) = ∀y (xRy → ST (ϕ)[y /x]), where y is a fresh variable;
• ST (Hϕ) = ∀y (yRx → ST (ϕ)[y /x]), where y is a fresh variable.
Consequently:
ST (Fϕ) = ∃y (xRy ∧ ST (ϕ)[y /x]) and ST (Pϕ) = ∃y (yRx ∧ ST (ϕ)[y /x]).
Example: ST (Gp1 ∨ FHp2 ) = ∀y (xRy → P1 y ) ∨ ∃y (xRy ∧ ∀z(zRy → P2 z)).V Goranko
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Addendum: Standard translation of TL into FOL:
semantic correspondence and definability
Every tense model M = hT , ≺, V i can be regarded as an L1 -model by
interpreting R as ≺ and each Pi as V (pi ). Then:
M, t |= ϕ iff M |= ST (ϕ)[x := t],
M |= ϕ iff M |= ∀xST (ϕ).
The semantic correspondence on a level of Kripke frames:
T |= ϕ iff T |= ∀P∀xST (ϕ).
|= ϕ iff |= ∀P∀xST (ϕ).
Thus, validity of a temporal formula in a model is a first-order property, while
validity in a frame is a (monadic) second-order property.
A TL formula ϕ defines the class of tense frames in which it is valid.
Likewise, a FO sentence defines the class of tense frames in which it is true.
For instance, each of Gp → GGp and ∀x∀y ∀z(x ≺ y ∧ y ≺ z → x ≺ z)
defines the class of tense frames with transitive precedence relation ≺.
Thus, a correspondence theory between temporal and classical logic arises.
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Prior’s basic tense logic TL: the axiomatic system Kt
Kt consists of the following axioms schemes, added to the classical prop. logic:
(KG ) G(ϕ → ψ) → (Gϕ → Gψ),
“Whatever will always follow from what always will be, always will be”
(KH ) H(ϕ → ψ) → (Hϕ → Hψ),
“Whatever has always followed from what always has been, always has been”
(GP) ϕ → GPϕ “What is, will always have been”
(HF) ϕ → HFϕ “What is, has always been going to be”
and the following rules of inference:
(NECG ) If ` ϕ then ` Gϕ (If ϕ has been derived, then derive Gϕ.)
(NECH ) If ` ϕ then ` Hϕ (If ϕ has been derived, then derive Hϕ.)
added to the classical Modus Ponens: if ` ϕ and ` ϕ → ψ then ` ψ.
The axioms (GP) and (HF) technically say that the temporal operators H and G
correspond to mutually inverse temporal relations.
Exercise: show validity of each axiom scheme of Kt in every tense frame.
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Soundness and completeness of the axiomatic system Kt
Some valid principles of TL
Theorem The axiomatic system Kt is sound and complete for the
logical validity on the class of all tense frames.
Each of the following formulae is a valid principle of TL,
in the sense of being valid in every tense frame,
and in the sense of being derivable in Kt :
• G(ϕ → ψ) → (Fϕ → Fψ)
• H(ϕ → ψ) → (Pϕ → Pψ)
• G(ϕ ∧ ψ) ↔ (Gϕ ∧ Gψ)
H(ϕ ∧ ψ) ↔ (Hϕ ∧ Hψ)
• F(ϕ ∨ ψ) ↔ (Fϕ ∨ Fψ)
P(ϕ ∨ ψ) ↔ (Pϕ ∨ Pψ)
• PGϕ → ϕ
FHϕ → ϕ
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Reflexivity and transitivity axioms
Time precedence is usually assumed transitive, and often reflexive, too.
Claim. Each of the following axiom schemes, added to TL, axiomatises
the validities in the class of transitive tense frames:
Gϕ → GGϕ, Hϕ → HHϕ, FFϕ → Fϕ, PPϕ → Pϕ
Exercises. Show that:
1) each of these is valid in a tense frame if and inly if it is transitive.
2) Adding any of these axiom schemes to TL makes the others derivable.
Claim. Each of the following axiom schemes, added to TL, axiomatises
the validities in the class of reflexive tense frames:
Gϕ → ϕ, Hϕ → ϕ, ϕ → Fϕ, ϕ → Pϕ
Exercises. Show that:
1) each of these is valid in a tense frame if and inly if it is reflexive.
2) Adding any of these axiom schemes to TL makes the others derivable.
Hereafter transitive tense frames will be called temporal frames, and
the logic TL restricted to temporal frames, the basic temporal logic.
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Some axioms expressing properties of temporal frames
Most, but not all, of the correspondences below assume transitivity of ≺.
(Recall that Ep = Pp ∨ p ∨ Fp and Ap = Hp ∧ p ∧ Gp.)
(REF): G ϕ → ϕ
(TRAN): G ϕ → GG ϕ
(BEG): H⊥ ∨ PH⊥
(NOBEG): P> or Hp → Pp
(reflexivity of ≺)
(transitivity of ≺)
(the time has a beginning)
(the time has no beginning)
(END): G⊥ ∨ FG⊥
(the time has an end)
(NOEND): F> or Gp → Fp
(the time has no end)
(LIN-F): PFp → Ep
(linearity in the future)
(LIN-P): FPp → Ep
(linearity in the past)
(LIN): (FPp ∨ PFp) → Ep
(DENSE): GGp → Gp or Fp → FFp
(linearity)
(density)
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Some axioms expressing non FO-definable properties
1. A linear ordering is called well-ordered if every non-empty subset of it
has a least element.
Example: N; non-examples: Z, Q, R.
(WELLORD): H(Hϕ → ϕ) → Hϕ
(Well-ordering)
2. Induction principles in linearly ordered temporal frames:
(INDG ): Fp ∧ G(p → Fp) → GFp
(INDH ): Pp ∧ H(p → Pp) → HPp
(forward induction)
(backward induction)
3. A linear ordering is called Dedekind complete if every non-empty and
bounded above subset has a least upper bound.
Examples: N, Z, R; a non-example: Q.
(COMP): A(Hϕ → FHϕ) → (Hϕ → Gϕ)
(Dedekind completeness)
Neither of these is definable in the FO language L1
(with ≺ and equality).
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Basic temporal logics of partially ordered flows of time
Claim: The following axiomatic systems are sound and complete for the
temporal logics of the indicated classes of frames:
K4t = Kt +(TRAN): transitive frames.
K4t is also complete for the class of strict partial orders.
S4t = K4t +(REF): reflexive and transitive frames.
S4t is also complete for the class of (non-strict) partial orders.
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Linear time temporal logics
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Some temporal logics of linear flows of time
Theorem: The following axiomatic systems are sound and complete for
the temporal logics of the indicated classes of frames:
K4t = Kt +(TRAN): transitive frames.
Lt = Kt +(TRAN)+(LIN): strict linear orderings.
Nt = Lt +(NOEND)+ (INDG ) +(WELLORD): (N, <).
Zt = Lt +(NOBEG)+(NOEND)+ (INDG )+(INDH ): (Z, <).
Qt = Lt +(NOBEG)+(NOEND)+(DENSE): (Q, <).
Rt = Lt +(NOBEG)+(NOEND)+(DENSE)+(COMPL): (R, <).
The basic language of TL is not very expressive.
Some natural additional operators for linear time can be added.
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Addendum: Tenses and equivalences of tenses
A tense (in a formal tense logic sense) is any formula X1 , . . . Xn q where
q is a (fixed) propositional variable and X1 , . . . Xn is a (possibly empty)
string of temporal operators from {F , P, G, H}.
Examples: q, Fq, GGFPq, FGFGHHPGFq, etc.
Some of these are equivalent in the given temporal logic.
E.g. Gq ≡ GGq in S4t .
Hamblin (1958); Hamblin and Prior (1965): There are exactly 15
different tenses in the temporal logic of the time flow of the rational
numbers (Q, ≤) (i.e., reflexive and dense linear order without endpoints).
Exercise: identify these 15 tenses and prove Hamblin’s theorem.
How about the temporal logic of all linear orders? Or, S4t ?
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Nexttime operator
Let T = hT , ≺i be a linear temporal frame and s, t ∈ T .
s is called an immediate successor of t, denoted t C s if
(T , ≺) |= t ≺ s ∧ ¬∃y (t ≺ y ∧ y ≺ s).
Forward discreteness means that every instant has an immediate
successor. Assuming also linearity, it is unique. Then, define a new
operator X: ’at every immediate successor’ or, neXttime:
M, t Xϕ iff M, s ϕ for (the only)s such that t C s.
Xϕ: nexttime ϕ
Xϕ
ϕ
···
The past analogue Y of X, can be defined likewise.
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Some valid properties of Nexttime
In any linear and forward discrete frame T :
• (KX )
T |= X(ϕ → ψ) → (Xϕ → Xψ),
• (FUNC) T |= X¬ϕ ↔ ¬Xϕ.
• (FPG )
T |= Gϕ ↔ (ϕ ∧ XGϕ) if ≺ is assumed reflexive.
• (FPirG )
T |= Gϕ ↔ X(ϕ ∧ Gϕ) if ≺ is assumed irreflexive.
• (IND)
(N, <) |= ϕ ∧ G(ϕ → Xϕ) → Gϕ
(Induction principle)
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Some logics with Nexttime
In the language with G, H and X:
• Lt (X) = Lt + (KX ) + (FUNC) + (FPG ) axiomatizes completely
the temporal logic of unbounded, forward-discrete linear orderings.
• Nt (X) = Nt + (KX ) + (FUNC) + (FPG ) + (IND) axiomatizes
completely the temporal logic of hN, s, <i, where s(n) = n + 1.
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Since and Until
Hans Kamp proposed in his 1968’ PhD thesis the binary temporal
operators S (Since) and U (Until).
The formula ϕSψ states intuitively that
ϕ has been true since a past moment when ψ was true.
Likewise, ϕUψ states that ϕ will be true until ψ becomes true.
More precisely: ψ will become true at some future instant, and ϕ will
hold true in the meantime.
ϕUψ, ϕ
ϕ
ϕ
ϕ
ψ
···
Formally: M, t |= ϕUψ iff M, s |= ψ for some s t and
M, u |= ϕ for every u, such that t u ≺ s.
Likewise: M, t |= ϕSψ iff M, s |= ψ for some s t and
M, u |= ϕ for every u, such that s ≺ u t.
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Expressing other temporal operators with Since and Until
F is a special case of U: Fϕ ≡ > U ϕ. Likewise: Pψ ≡ >Sψ.
”ϕ holds eventually always” G∞ ϕ ≡ FGϕ
”ϕ holds infinitely often” F∞ ϕ ≡ GFϕ
(assuming no end of time)
Weak until: Intuitively, still expresses “ϕ until ψ”, but without the
inevitable occurrence of ψ; if ψ never occurs, then ϕ must remain true
forever. Thus:
ϕWψ ≡ Gϕ ∨ (ϕUψ).
Before: Defined as dual of U on the first argument:
ϕBψ ≡ ¬(¬ϕUψ).
Meaning: before every occurrence of ψ there is an occurrence of ϕ.
Likewise, ‘Weak since’ and ‘After’ can be defined using S.
Theorem: (Kamp’68): Every first order definable operator on any
Dedekind complete linear ordering (e.g., in N) in expressible there in
terms of X, S and U.
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Expressing temporal statements with Since and Until
“I will keep trying until I succeed, unless I die meanwhile”
can be formalised using Until and the special atomic propositions
try, succeed, die as:
try U (succeed ∨ die)
“Whenever smoke is detected, the alarm is activated at the next moment
and remains activated until the fire brigade arrives” is formalised by
G(smoke → X(alarm ∧ (alarm U fire brigade)))
“Ever since Mia left home, Joe has been unhappy and has been drinking
until passing out over and over again.” is formalised as:
(Joe unhappy ∧ (Joe drinking U(Joe passes out))) S (Mia leaves home)
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Complete axiomatic system for the logic of Since and Until
on the class of all linear time flows
The following system of axioms is originally due to Burgess (1982),
further simplified by Xu (1988). It extends the classical propositional
logic with the following axiom schemata (for the reflexive versions) and
their duals, with S and U swapped:
• Gϕ → ϕ,
• G(ϕ → ψ) → ϕUχ → ψUχ,
• G(ϕ → ψ) → χUϕ → χUψ,
• ϕ ∧ χUψ → χU(ψ ∧ χSϕ),
• ϕUψ → (ϕ ∧ ϕUψ)Uψ,
• ϕU(ϕ ∧ ϕUψ) → ϕUψ;
• ϕUψ ∧ χUθ → (ϕ ∧ χ)U(ψ ∧ θ) ∨ (ϕ ∧ χ)U(ψ ∧ χ) ∨ (ϕ ∧ χ)U(ϕ ∧ θ).
and the inference rules NECH and NECG .
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Some complete axiomatic extensions
of the the logic of Since and Until
The Burgess-Xu axiomatic system, translated for the strict versions S s
and Us , was extended by Venema (1993) to complete axiomatic systems
for S s and Us for:
• all discrete linear orderings,
by adding Fs > → ⊥Us > and its dual Ps > → ⊥S s >.
• all well-orderings,
by further adding Hs ⊥ ∨ Ps Hs ⊥ and Fs ϕ → (¬ϕ)Us ϕ,
• hN, <i, by adding Fs > to the previous system.
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The linear time temporal logic LTL
Using temporal logics for specification and verification of concurrent and
reactive systems was first suggested by Pnueli in his landmark 1977
paper ”The Temporal Logic of Programs”.
(For this and related work, he received the Turing Award in 1996.)
The linear time temporal logic LTL was first introduced and studied by
Gabbay, Pnueli, Shelah and Stavi in 1980.
LTL extends classical propositional logic with future temporal operators
over discrete linear time, essentially over (N, <).
Used to reason about computations in transition systems.
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Syntax and semantics of LTL
LTL formulae:
propositional logic
temporal extension
}|
{
z
}|
{
z
ϕ ::= ⊥ | p | ¬ϕ | ϕ ∧ ψ | Xϕ | Fϕ | Gϕ | ϕUψ
where p ranges over a set of atomic propositions AP = {p0 , p1 , ...}.
F and G can be regarded as definable in terms of U.
Semantics: LTL formulae are interpreted over infinite computations –
sequences of labels (sets of atomic propositions) of systems states.
Abstractly, these are Kripke models over (N, <).
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Expressing properties with LTL: examples
“Every time when a message is sent, it will not be marked as ’sent’
before an acknowledgment of receipt is returned.”
G(Sent → (¬MarkedSent U AckReturned)).
“Between every two green signals there must be a red signal”
G(Green → X(¬Green U Red))?
or:
G(Green → X(¬Green W Red))?
or:
G(Green ∧ FGreen → X(¬Green U Red))?
or
G(Green → X(Red B Green))?
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Complete axiomatic system for LTL
CL: Axioms for the classical propositional logic CL.
KG : G (ϕ → ψ) → (G ϕ → G ψ)
KX : X (ϕ → ψ) → (X ϕ → X ψ)
FUNC: X ¬ϕ ↔ ¬X ϕ
FPG : G ϕ ↔ (ϕ ∧ XG ϕ)
GFPG : ψ ∧ G (ψ → (ϕ ∧ X ψ)) → G ϕ
FPU ϕUψ ↔ (ψ ∨ (ϕ ∧ X (ϕUψ)))
LFPU : G ((ψ ∨ (ϕ ∧ X θ)) → θ) → (ϕUψ → θ)
Inference rules: MP and NECG .
The axioms FPG , GFPG , FPU and LFPG characterise G and U as the greatest,
reps. the least, fixed point certain operators.
GFPG generalises the Induction axiom: ϕ ∧ G (ϕ → X ϕ) → G ϕ.
In fact, GFPG can be replaced by the following Induction rule:
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If ` ψ → ϕ ∧ X ψ then ` ψ → G ϕ. Likewise for LFPU , see exercises.
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Addendum: Decidability and deductive systems for LTL
For completeness proof of an axiomatic system for LTL, see e.g.
[Goldblatt, 1992].
[Clarke, Sistla’85] Validity/satisfiability in LTL is decidable (PSpace).
Small model property : every satisfiable LTL formula is satisfiable in an
ultimately periodic model, with lengths of prefix and period exponentially
bounded by the size of the input formula.
A tableaux-based decision method for LTL was developed in [Wolper’83].
Natural deduction, sequent calculi and resolution-based deductive
systems for LTL have also been developed.
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