>> Nikolaj Bjorner: It’s my pleasure to introduce Ori Lahave who’s visiting this week. We’re going to
finish a paper by Friday.
[laughter]
So, but, but not on this talk, the talk isn’t as we can see on the slide, on Analytic Calculi an SAT. So Ori is
post doc at Tel Aviv with Muli who was visiting us earlier this summer. Ori we look forward to your talk.
>> Ori Lahav: Thank you. Okay, thank you for inviting me to talk here. So this is a joint work with Yoni
Zohar from the Tel Aviv University. This is a work in progress. This means that we still don’t have papers
and, but this is our current state exploring this topic. It’s SAT-based Decision Procedure for Analytic
Sequent Calculi.
So a few words for the, about the motivation for this talk, so sequent calculi are a prominent prooftheoretic famework, I guess needless to say that because they provide, as I wrote here algorithmic
presentation of logic. It is, they’re suitable for a variety, a large variety of logics. Many non-classical
logics that will be the main issue in this talk, and our goal in this work was to try to effectively reduce the
derivability problem in a given propositional sequent calculi to SAT. We try, of course this is not possible
for all these variety, huge variety of logics, but only for a small subset of them. I’ll first try to explain this
small family of sequent calculi that we can handle.
Okay, so just, few, some big ground, some notations. So we take sequents to be objects of the form a
Gamma double L Delta. Well Gamma and Delta are finite sets of formulas. This automatically means
that all our systems are fully structural. They have all the usual structural rules as contraction and
exchange, because Gamma and Delta are sets not multi-sets or sequentses.
This is, again the usual thing, the intuition behind a sequents. So we have this sequent, this is a set of
formulas and another set of formulas. You can think about it as this implication the conjunction of the
formulas and the left side implies the disjunction of the formulas of the right side. But this is just
intuition because we are dealing with non-classical objects where we might, don’t have these
connectives or have them with unusual interpretations. So this cannot be a theorem, some
interpretation of theorem in all the systems we study. Yes?
>>: Question, you said because you take this form you automatically get on sub-structural properties.
Why do you get weakening for example?
>> Ori Lahav: We don’t get weakening you’re right. We can agree to add to our systems, but
contraction and exchange, and extension. The opposite of contraction are built in.
>>: Okay.
>> Ori Lahav: Thank you. Please interrupt me whenever because we are a very small group and…
>>: It would also happen in [indiscernible] school.
>> Ori Lahav: Yeah.
[laughter]
So this is the most famous calculus LK, its propositional part and of Gentzen. We have this identity
axiom and a cut rule; Gamma and Delta are Meta variables for sets of formulas in A and B. Meta
variables for formulas and each connective I have here a conjunction, disjunction, implication. Each
connective has two rules, one for introducing on the left side, and one for the right side. I guess this is
well known. This is the first example of sequent calculus.
This next example is for a logic studied, invented here I think. Propositional of Primal Logic, this would
be a running example in this talk. So we have all the usual structural rules, just as before weakening and
cut, and identity, axiom. Logical rules are a bit different for implication, for disjunction. We only have
the right rule. We don’t have the left rule. For implication this is the usual rule for introducing
implication to the left side, but this rule is a bit different.
Let’s go back before we had an LK, we had A, B, this is on the left side and in the right side. From this
you can deduce A implies B on the right side. In the, in primal logic we have only this B here and no A on
the left side. It is not hard to see that this multiple-conclusion calculus that I present here is equivalent
to the sequent-style natural deduction system that appears in the paper of Beklemishev and Gurevich.
>>: Is that, you have to refresh my memory. So this is LJ, so this is an intuitionistic logic because…
>> Ori Lahav: No, here it gives the multiple-conclusion calculus. So it does not…
>>: So it’s not, but you can do a multiple-conclusion intuitionistic version when [indiscernible] certain
operations?
>> Ori Lahav: Yeah, if you take this LK and you change the rule for this junction, implication. You don’t
allow this Delta here…
>>: Right.
>> Ori Lahav: Then you get LJ, something which is equivalent to LJ.
>>: Yes.
>> Ori Lahav: But here I take the multiple-conclusion calculus for primal logic. You’re right saying that
this, I can take single conclusion I will get the same logic. If I omit all these Deltas, in this case, in this
case specific calculus this would not, these would not change anything.
>>: [inaudible] calculus [indiscernible].
>> Ori Lahav: Yeah, yeah.
>>: [inaudible] I show this [indiscernible]. His reaction was what a stupid rule, you already know B.
[laughter]
What’s the point to useful, the rule is useful?
>>: Right.
>> Ori Lahav: Sorry?
>>: So, just, so this is a classical calculus?
>> Ori Lahav: In what sense classical?
>>: In the classical model [indiscernible].
>>: Neither, so it’s not a classical calculus, because…
>> Ori Lahav: With your statement…
>>: The statement was that it’s equivalent to one where you also drop the Deltas.
>> Ori Lahav: Yes, it’s a multiple-conclusion calculus. But it is equivalent to a single-conclusion calculus
without the Deltas.
>>: Okay, [indiscernible].
>> Ori Lahav: I think that primal logic was introduces originally as some sort of intuitionistic logic. But I
must say that I’m not sure that it is intuitionistic logic. For me it is more close to classical logic than
intuitionistic logic, but it is a matter of point of view.
>>: Yes.
>> Ori Lahav: So this is the general family that we study, we call them Pure Sequent Calculi. Pure
sequent calculi are propositional sequent calculi that include all usual structural rules as we saw before,
and any finite set of pure logical rules. This notion is due to Avron in this old paper from ninety-one. So
this rule would be a pure rule. We have Gamma and Delta on both sides. The idea is you allow any
context. If you have this Gamma and Delta in all the premises and the conclusion we call it a pure rule.
But this rule would not be a pure rule because we don’t have Delta on the right side.
Also if you know many rules for modalities they are not pure because you change the context. So we
allow only this kind of simple rules. We call them pure rules and this is the calculi that we can work
with, in this work.
Here is another example just to show that there is some variety of pure calculi. So this is for the logic of
da Costa’s, the logic called C1. This is a, one of the most fundamental Paraconsistent Logics where from
A, a negation of A there’s some cases where you cannot deduce any formula you want. The idea is that
you take the positive fragment of LK, all the positive connective without negation. You add all these
rules that involve negations. So we have the rule from introducing negation on the right but no rule for
introducing it on the left.
This is what gives us the power consistent nature. We have this rule for double negation and all these
special rules for negation and conjunction, and other connectives. So this system appeared in this paper
and I just give it here to show that it is a pure system. You have Gamma and Delta, and Gamma and
Delta in all rules, all premises, and all conclusions. So it falls in our scope.
>>: So you, the left rule is redundant because it’s [indiscernible].
>> Ori Lahav: Which one?
>>: If you said B to not A.
>> Ori Lahav: This one?
>>: On the left, no. If you said B to not A then you get Gamma not, not A, then you have a rule. So I’m
trying to see what, so the only thing you get from the one on the left hand side is that you, you’re by
passing…
>>: Which, which?
>>: The upper left…
>>: Which rule are you on?
>>: The second rule to the left.
>>: That one, this one?
>>: So if you do stensignal pattern matching with B, B not A.
>> Ori Lahav: But it…
>>: You don’t, yeah…
>> Ori Lahav: But I think this the opposite side here and here.
>>: Yeah, so you get not, not A, if you said B to not A’s. You get not, not A.
>> Ori Lahav: But here you have the negation of A and here you have A on the right side.
>>: Ah that you get as well.
>> Ori Lahav: I, I, if I copied it correctly from the paper I don’t believe there is a redundant tool but
maybe it had some mistake in copy paste. But this is the most problematic rule in the system, because
you have this A and not A. It couldn’t be, it shouldn’t be redundant.
>>: [inaudible] the A on the right hand side you’ve got, you’re getting variables, okay.
>> Ori Lahav: Okay, I’m not going into the details because it’s not so impertinent, this work is trying to
be general. This is just a specific example.
>>: Okay, so the point is that this is a, you don’t have, there’s Gamma and Delta…
>> Ori Lahav: Yeah and this is a pure system so you don’t have restrictions on the context anywhere. So
it falls in this family of systems. Now the property of Analyticity of the calculus is also crucial for this
thing to work. This is of course problematic, but because it might be too strong a requirement. But let
me formulate it. So we say that the calculus is analytic if the fact that there is a derivation of Gamma L
or Delta implies that there is a derivation of Gamma L or Delta using only the sub-formulas of Gamma
and Delta.
This is an analytic calculus. In some cases we will allow, use a weaker notion that allows the negations
of the sub-formulas in Gamma, and Gamma and Delta, as well. Not only the sub-formulas also their
negations. It’s easy to see that if a propositional pure calculus is, as I defined before, is analytic
according to this definition, or the weaker definition then it is decidable, simply because you can
enumerate all possible proofs.
I want to say that analytic pure calculi exist for some, not so many but there are some important
propositional logics. So of course classical logic and primal logic as we saw in the examples, these calculi
are analytic. All important three and four valued logics of analytic calculi, and paraconsistent logics, and
I’m sure there are more that I don’t know about. But these are the families of logics that I can recognize
easily that have pure calculi, analytic pure calculi.
>>: So let say something. I can see that you know if a calculus analytic is decidable, but where are you
relying on purity?
>> Ori Lahav: You’re right, this is not, this will be true if, even for a non-purest calculus. But I will only
consider pure calculi, because this is the, these are those that we can do this. The last sentence here
and this is the first observation, that there is a simple reduction of the derivability in analytic pure calculi
to SAT. So in these cases we claim that it is possible to replace proof search by SAT solver.
This is the first observation that led to this work and later I’ll say a few words about it and then go to
several extensions.
>>: [inaudible]
>> Ori Lahav: Sorry.
>>: So the point is that you guess and verify the [indiscernible] because of [indiscernible] property?
>> Ori Lahav: I have a few slides about this, how to do that.
>>: Oh, [indiscernible]
>> Ori Lahav: Okay.
>>: So purity of onset [indiscernible] is important for your particular reduction to SAT.
>> Ori Lahav: Exactly.
>>: Okay.
>> Ori Lahav: If you take an un-pure calculi then you can do almost some, you can do many model logics
and obviously the reduction to SAT would not be simple. Simple is meant to say linear time. I will have
several more restrictions for linear time but we want to have a linear time reduction to SAT, not just a
reduction that you have, will always have in P.
Our reduction goes unlike the, I think the reductions in the papers of primal logic. In most of them our
reduction goes through the semantics. So we’ll have first semantics for every calculus and then use it
for the reduction to SAT. So this is another observation in, I found it in the general sense in a paper of
Beziau, that pure calculi can be characterized by two-valued valuations.
So it doesn’t matter if the calculus is for a three valued logic, four valued logic, or what so ever, you can
always use a two-valued valuations. The idea is that each pure rule is easily translated into some
semantic condition. We just read the rule from a semantic point of view. If you join all the semantic
conditions of the rules in some calculus you, calculus G you get the set of G-legal valuations. This is the
set for which the calculus is sound and complete.
I have an example in the next slide. But this would be the general theorem once you do these two steps
then Gamma L or Delta is probable in G if and only if every G-legal valuation is a model of Gamma L or
Delta. So each calculus G induces a set of G-legal valuations. This is a usual notion of a model. A
valuation is a model of Gamma L or Delta if either there is a formula on the left side which is false or
some formal on the right side which is true. So all our valuations are two-valued and things are
relatively simple. Once you see the reduction you’ll see that it is almost trivial and another reduction,
sorry the translation of rules to semantic conditions.
Here is an example for the calculus for C1. I took just a few rules, not all of them. So for example what
does this rule mean from a semantic point of view? It gives us this condition. The value of, if the value
of A is false then the value of negation of A should be true. This is the semantic reading of this rule. This
rule would say that if the value of A is false then the value of the negation, the negation of A should be
false. Then each rule you can find this easily the semantic condition and you obtain a set of valuations,
the valuations that satisfy these conditions for which the calculus is sound and complete.
This is a crucial observation that we get the semantics which is non-deterministic. This means that the
value of compound formula might be not uniquely determined by the value of its sub-formulas. So if I
know for example that the value of A is true and there is no condition here in general case that will tell
me what is the value of the negation of A. It can be either true or false. By doing, by relying this kind of
weird valuations, non-deterministic valuations we are able to do everything with two-values, only true
and false, and simply read the rules from a semantic point of view. The general theorem that I showed
before tells us that this, for every system this SAT valuation is sound and complete for this calculus.
>>: What is your paper did you non-deterministic that?
>> Ori Lahav: Yes, of course.
>>: [inaudible]
>> Ori Lahav: There is no other option if you want, if you only have two-values. To do three and four
valued logics you use non-deterministic semantics.
>>: So we didn’t discover, ha, ha, we were not the first to use them.
>> Ori Lahav: Yeah I, there, I worked with during my Ph.D. studies and I found through very old works of
for example Shuta and Gerard, all of them use non-deterministic semantics without saying that, so. This
is very useful once you start from an arbitrary proof system and you want to find semantics.
The reduction to SAT would simply be, let me go back one slide. You can easily observe that all these
conditions can be formulated in propositional classical logic. Then the reduction to SAT is very easy.
You associate a variable X A for every sub-formula A of Gamma L or Delta. You generate a set of clauses
for the semantic conditions and you apply the semantic condition, these clauses on every sub-formula
and every X A like that. Then if you want to decide whether Gamma L or Delta is provable you generate
singleton clauses X A for every A in Gamma and negation of X A for every A in Delta, and you get that
Gamma or Delta is provable if and only if the set of clauses is not satisfiable.
>>: Per classical logic this is, would be equivalent to the [indiscernible] coding then?
>> Ori Lahav: Sorry, to what?
>>: For classical logic this would be equivalent to the [indiscernible] encoded of…
>> Ori Lahav: Yes, simply translate everything to this [indiscernible]…
>>: [indiscernible] important that it’s on linear number of sub-formulas.
>> Ori Lahav: Yes, for the, I’ll say a few words about complexity. So of course it’s one of the crucial
things is that there is a linear number of sub-formulas here, so you have a linear number of clauses.
Let’s see an example of primal logic. So this would be the rules for primal logic that I showed before.
These are the semantic reading of the rules. So you see for conjunction we have the usual thing, the
two usual truth table of conjunction actually. For disjunction we only have this rule, this condition. If
the value of A is true or the value of B is true, then the value of the disjunction should be true. You
don’t have the other thing that if both of them are false we cannot know the value of the disjunction. It
can be either true or false.
For implication we have this usual thing for if the value of the first argument is true and the second is
false, then the value of the implication is false. But we, as it seems we only have this rule, so the other
thing that we have is that if the value of B is true then the value of the implication is true. So we don’t
have any rule that tells us, any condition that tells us what to do if both the value of A is false and the
value of B is false. Then it’s just completely free the value of the implication.
Question?
>>: I’m asking if [indiscernible]…
>> Ori Lahav: Okay, just let me know. Now we do the reduction. So Gamma L or Delta is provable in
the system of primal logic if and only if the following set of clauses is not satisfiable. So we have, I have
only conjunction and implications. So I do have a shorter slide so I omitted this junction. So this would
be the three clauses that we have for conjunction. We have them for every formula of the form A
conjunction B in, that occurs in Gamma L or Delta. So this would give us this number, one part of this
number one. This will give us the second part and this is equivalent to the second condition here. A
implies B, we, you have these conditions, this is for three and this is for four. As I said we have a
singleton clauses X A for every A in Gamma and negation of X A for every A in Delta. Gamma L or Delta
is provably primal logic if and only if this set of clauses is unsatisfied.
In this particular case I should mention that we obtain essentially the same reduction that appears in the
paper of Beklemishev and Gurevich, from this general point of view of reduction of pure calculi to SAT.
You can also note that all these clauses in this case are Horn clauses. For the case of primal logic this is
what makes it so efficient. So not only did our reduction is linear also we will have a linear time
algorithm to solve this SAT instance using a HORN-SAT solver.
>>: So if you add a quantifier…
>> Ori Lahav: If I add to primal logic…
>>: Yes, but just in a careful way [indiscernible]. Then wouldn’t the same reduction just go through?
>> Ori Lahav: I don’t know. I have to think about it, but…
>>: Can you translate the question, I didn’t hear.
[laughter]
>>: So you, so for primal logic with universal rules…
>>: Oh quantifiers.
>>: Quantifiers.
>>: I see.
>>: The reduction is data log. So if you do the semantic embedding [indiscernible] against the system.
>> Ori Lahav: I guess it will work and get the reduction to data log as what?
>>: You just add [indiscernible] so these become rules and then…
>> Ori Lahav: We can then check it but I guess…
>>: The problem is the [indiscernible] with quotations.
>> Ori Lahav: I’ll get to quotations in a moment.
>>: Oh.
>> Ori Lahav: Okay, I think I have quantifiers in the further work at least. But if not I can, should add it.
Let me say a few words why does it work, all these idea, this reduction? The idea is that we have
semantic and analyticity. This is the property which is equivalent to the analyticity of the calculus. The
semantic property which is equivalent to the analyticity of the calculus is stated in this theorem. If G is
analytic and we assume this is something not so fair to do. But we assume that the input calculus is
analytic, if G is analytic then every G-legal partial valuation. A partial valuation is a valuation, is defined
only on a subset of formulas. Then I also assumed its domain is closed on under sub-formulas, if every
G-legal partial valuation can be extended to a full G-legal valuation.
So this would be the semantic meaning of analyticity. If we know that G is analytic then we have this
property. This is something that we proved and this is a property which is essential for the correctness
of the reduction. Because once you find a satisfying assignment it only gives you a partial valuation. It
doesn’t give you a full valuation which would be a counter model. Then you have to use the theorem to
show that this partial valuation can be extended to a full valuation.
This is crucial here and for this reason we need a calculus to be analytic otherwise it won’t work. I
mentioned here that something that I like very much is the other direction of this theorem works as
well. So if you know that every partial valuation can be extended to full valuation, this would mean that
the calculus is analytic from a syntactic point of view. That proofs can be, a second, proofs can be
confined to the sub-formulas. Yes?
>>: Sorry, I’m not quite sure if I [indiscernible]. So suppose you did the translation and now the SAT
solver says SAT not UNSAT. Are you saying that the model that the SAT solver produces has to be
changed?
>> Ori Lahav: If we go back we have these clauses only for the sub-formulas occurring in Gamma L or
Delta.
>>: Yes.
>> Ori Lahav: So we don’t have anything for the formulas which are outside. If you want, and the
soundness and completeness general theorem talks about full valuations, you have to give…
>>: That’s okay.
>> Ori Lahav: Okay and you have to show that if you find a small model you can extend it to a full one.
>>: Oh, okay.
>>: Can you say what exactly is the other direction?
>> Ori Lahav: That other direction says that if every G-legal partial valuation can be extended to a full
one…
>>: But…
>> Ori Lahav: Then G is analytic.
>>: Should allow two variant valuations?
>> Ori Lahav: Yeah, we, every pure calculus has a two-valued semantics, according to the…
>>: Now you say if G is pure.
>> Ori Lahav: If G is pure of course. I only speak about pure systems.
>>: Oh [indiscernible].
>> Ori Lahav: Yeah and this gives you semantic method to prove an analyticity. In many cases it is very
easy; this is not the main issue in this work. But I think it’s nice to mention it. For example for LK an
analyticity of LK is usually a consequence of cut elimination. Then you have the sub-formula property.
But this gives you a different method. You only have to prove something that there is nothing to prove
here that every partial classic valuation whose domain is closed under sub-formulas can be extended to
all formulas. This is trivial; people don’t even bother to mention it usually that partial classical
valuations are extendable. If you have this property then the calculus is analytic. This is just a reminder
what I mean by calculus is analytic. I should have begun with that.
About the complexity of the reduction, so now we have this assumption we are trying to weaken it. But
suppose that the rules in the calculus G have the following natural structure. That every rule contains a
main formula and all other formulas are sub-formulas of the main formula. This is what we had for
classical logic, for primal logic. For the paraconsistent logics we allow also negation of sub-formulas, but
it doesn’t change much in this idea. In this case the reduction that I mentioned requires only linear
time. This, we did not implement it but implemented it but we think that it is possible to use the same
data structure that is used in these papers.
>>: So we call this maternal formula.
>> Ori Lahav: Sorry?
>>: What you call main formula I think it didn’t appear in our papers but [indiscernible]…
>>: Call it maternal.
>>: Maternal.
[laughter]
>>: Formal, so it’s kind of model formula for this.
>> Ori Lahav: Yes.
>>: There could be more than one.
>> Ori Lahav: Yeah we are working on weakening this condition because for some rules it is too strong.
I’ll have an example in the last slide. We don’t, we want to say something like there should be a formula
that includes all the atomic formulas of the rule and not that all other formulas are sub-formulas of it.
These are the small details. Here we use the same parse tree with harmonily there’s all the stuff that
you had to do we think the same can be used to do this reduction.
Okay, last thing about this general reduction without quotations. The next part of the talks would be
adding quotations, but as we, as I noted before for propositional primal logic we get only Horn clauses.
In this case this logic can be decided in linear time and this is well known. This is different from the
original linear-time algorithm that appeared in the paper of Gurevich and [indiscernible] Neeman.
If you want to be again general we can recognize a family of calculi, we call them Horn pure calculi for
which we will get Horn clauses. We can decide all of them in linear time using a HORN-SAT solver, if
they’re analytic of course. This would be the condition for every rule in R, you should have this
property. The number of premises for every rule R in G, the number of premises of R whose left side is
not empty, this number plus the number of formulas on the right side of the conclusion should be less
than or equal to one.
>>: Can you give an example of this because…
>> Ori Lahav: Unfortunately I don’t have many examples because primal logic, you can see here that
every rule the number of premises whose left side is not empty, plus the number of formulas on the
right side on the conclusion you forget about Gamma and Delta, is always less than or equal to one. This
is just a general characterization of, in the cases in which you get Horn clauses. But…
>>: So let’s see, we take the disjunction rule, what is hash L?
>> Ori Lahav: It is zero. It’s the [indiscernible] mistake…
>>: Hash what?
>> Ori Lahav: Because there are no premises whose left side is not empty. The hash R is one because
we have one formula here. So if you take the usual rule for implication, if you have this A here…
>>: Yeah.
>> Ori Lahav: Then I will have the hash L, L would be one because I have one premise with non-empty
left side.
>>: Okay.
>> Ori Lahav: The hash R will also be one because I have one formula on the right side of the conclusion.
So together it will be more than one and [indiscernible].
>>: Okay.
>> Ori Lahav: But it is correct to state that we don’t other calculi, useful calculi that fall in this family.
We, what we can do I think is to go into the dual Horn fragment and then you can do dual primal logic.
You can have some connective which is dual to implication. Then it is also solvable in linear time but I’m
not sure about its use in practice.
>>: What is dual Horn?
>> Ori Lahav: Dual Horn is at most one negative. It is also decide, it can be decided in linear time.
>>: Which is the prime negation over everything [indiscernible].
>> Ori Lahav: Yeah, now quotations. This was the main difficulty of course for DKAL primal logic enough
does not suffice and you have to use quotations. If you have things like this P said A, Q said P said A
implies B. We tried to adjust our reductions to quotations and to see what we get, what is the
semantics? The same thing but we add quotations.
Quotations are simply unary modalities that we’ll denote in like this box one, box two, and so on. For
primal logic you can have two options how to add quotations. The first option would be to decorate all
the rules with these prefixes here. So in all active formulas everything except for the context we add
this prefix, which this is a Meta viable for some concatenation of these unary modalities. We add them
to all premises and to the conclusion you do it in all rules. You get a system which is equivalent to the
Hilbert system in this paper for primal logic with quotations.
>>: Bertillon.
>> Ori Lahav: You have a Bertillon system…
[laughter]
And we prefer the Gentzen system so we work with this one. But there is another option, alternatively
you can take the same propositional calculus, don’t change anything and add this one additional rule.
The name is not so good there KD exclamation mark. Let’s call it KD!, KD! rule that allows you to take,
this is not a pure rule but we still allow this rule. It will allow you to add a prefix and it should be the
same prefix of course when it, this rule is applied to all the formulas on the left side and all the formulas
on the right side.
This name comes from a functional Kripke frames. This is the rule that is used for functional Kripke
framing. This rule is used for box and diamond, box and diamond actually the same when the Kripke
frame is function, if the Kripke model is functional. Then if you want to have a rule for a box you will
take this rule. This is the same rule that is used for the operator next in LTL. So somehow quotations
behave exactly like next of the temporal logic LTL, from this point of view.
This is an easy to prove proposition that for every pure calculus if you take this rule and you add it, or
alternatively you go into this approach and prefix all the premises and all the conclusion with this thing
you get the same logic. The same things are derivable so these two options are equivalent. We will go
in the general study with this rule.
>>: It’s not pure.
>> Ori Lahav: This is not pure?
>>: Yeah.
>> Ori Lahav: Yeah but we will define a very simple definition that a pure calculus with quotations is a
proposition of pure calculus like you had before, augmented with this special rule which is not pure, KD!
rule. This would be a general pure calculus with quotations. I’m not sure that the word quotations is
the right word in general, but we don’t have a better, these are the modalities of a very restricted sort.
People would say that they are unusable just this special implication, but…
>>: Where does this notation come from, KD!?
>>: DeBank.
>> Ori Lahav: DeBank is the name of the model, the axiom for model logic when you have, you require
the accessibility relation to be a function, a full function. Usually people take the same name of the
axiom case as model logic and use it for the rule, but we can change it, because we didn’t find it
anywhere else.
>>: So may I ask intuition about this proposition. So the rule that adds the modality to each noncontext formula seems like that’s going to let you derive formulas that have sort of a different quotation
that’s [indiscernible] for each term in Delta. Whereas this one is adding it uniformly across, why are
they equivalent? Maybe it’s some simple…
>> Ori Lahav: Again here it’s, for example here it is. This rule should be the same.
>>: Right.
>> Ori Lahav: So you’re right saying that A might include more quotations inside. But I can apply for
example if we don’t have these quotations here I can, it’s not a good example. I am, I can try to…
>>: It doesn’t say it’s the same quotation.
>> Ori Lahav: Sorry?
>>: It’s the same box. Is the box the same for each term?
>>: No the same box but a [indiscernible] Gamma, some [indiscernible] this quotation.
>> Ori Lahav: Yeah.
>>: I’m saying Delta it does not.
>>: But these agree to add the quotations to the front, right?
>>: Yes.
>>: So that means uniformly they would all share a same quotation frequently.
>>: No, no, suppose Gamma’s one formula in Delta.
>>: Yeah.
>>: And suppose Gamma starts this quotation.
>>: Yeah.
>>: But Delta does not.
>>: Okay.
>>: Then you get different in the depth.
>>: That I can see. But if you have multiple terms in Delta then all of them are going to start with the
same quotation prefix.
>>: Not necessarily.
>>: [indiscernible] conclusion maybe the conclusion but…
>>: But you have weakening.
>> Ori Lahav: Yeah I can try to recover this argument. It was really simple I can try to think about it
later. Which direction bothers you? You think that this might be weaker than the other?
>>: Yes, yes.
>> Ori Lahav: I can try to see one of the directions was easy and…
>>: [inaudible]
>> Ori Lahav: But to this we proved, from a syntactic point of view just trying that this rule is admissible
there and the [indiscernible].
>>: Well primal logic will also have that.
>> Ori Lahav: Yeah, you have that in single conclusion usually…
>>: So it…
>>: Right.
>> Ori Lahav: But the same thing works with multiple conclusions.
>>: [indiscernible]
>>: No [indiscernible]
>> Ori Lahav: So this is definition of a pure calculus with quotations. Take a propositional pure calculus
and add the special KD! rule. This is the approach that we chose. This is a nice property that we proved
that this always preserves analyticity of the calculus. So if the calculus was analytic as we assumed and
you add this special rule it will still be analytic. In particular if you take the primal logic and you add this
KD! rule you get an analytic system, which is a bit different from the first calculus. This one with all the
prefixes inside the rules, this calculus is what we call locally analytic. If you have a proof of Gamma L or
Delta then you have a proof using the local formulas of Gamma and Delta. Not the sub-formulas, local
formulas are defined here and we will need them later as well. If the formula is local to itself, if you
have some connective diamond and some prefix then prefix A I is for every I is local to this formula. It’s
not a sub-formula but it’s local to it. You take the transity of closures. So if A is local to B and B is local
to C then A is local to C.
So we, this is, the main reason we prefer this rule because we are more close to the original tradition in
sequent calculi. We want them to be analytic. There are some other properties that you can use. But if
you can’t, prefer to use the usual notion of sub-formula which is a less complicated, I think, I don’t
know.
>>: They still were analytic. Did you invent it or it was in use?
>> Ori Lahav: Avron uses it all the time. So of course I did not invent it. But I don’t know if he invented
it, I don’t know.
>>: Somehow…
>>: This is different than the standard definition of analytic use here?
>>: I think it’s the same, right?
>> Ori Lahav: Right.
>>: Analytic is that you only use sub-formulas?
>>: You only use sub-formulas in a proof.
>> Ori Lahav: Yeah.
>>: Okay so if I understand in like in some logics you need to extend the notion of sub-formula a little
bit…
>> Ori Lahav: Yeah.
>>: Or what you call sub-formulas.
>> Ori Lahav: So we have some generized analyticity, property, negations, some things like that. What
would be the, we want to do the reduction and again our reduction is based on the semantics. We are
not doing proof search. So what would be the general semantics of pure calculi with quotations? The
equivalent thing to the two-valued valuations here would be two-valued functional Kripke models. Its
definition is straightforward. We have a triple W, W is a set of possible worlds. We have this
calligraphic R which assigns a function, this would be the accessibility relation, it is a function, R box
from W to W to every quotation. So we have functions as many as we have quotations. We have this V
here that assigns a valuation V W, this is a valuation like we had before, a function from the set of all
formulas of the language to false and true. So we have V W for every world W in our set of worlds.
The interpretation of box of the quotation is the usual thing. For every world quotation and box in
formula A the value of box A in the world W. So usually model logic it would be it should be true if and
only if all the accessible worlds the value of A is true. Here we have only one accessible world so we can
use this function. So it’s the value in the one accessible world in the unique one of the formula A.
The idea is exactly as we had before. In G-legal Kripke models the semantic conditions are imposed on
each function V W. So the same semantic conditions that we had before that comes from, that come
from the logical rules of the system. We impose them on each function, on each valuation V W. So in
each world we have these local conditions between the values of compound formulas and their
components.
We have the soundness and completeness that the Gamma L or Delta is provable in G, G is a pure
system with quotations if and only if every G-legal, this is the notion of G-legal Kripke model is a model
of Gamma L or Delta. I have an example very similar to what we had before. Take the same sequent
calculus for the power consistent logic C1 but add this rule, this quotation rule. So we have exactly the
same four conditions that we had before but they are applied on each V W. For every world we have
this local requirements from the valuation. In addition we have this special condition for the
interpretation of box, which is the usual thing in model logic.
Now using these semantics we are able to extend the reduction for pure calculi with quotations and still
do it in linear time. The idea is that now we have a valuable X prefix A for every formula prefix A that is
local to Gamma L or Delta. Instead of sub-formula of local formulas and again the number of local
formulas is only linear and this is crucial for the linear time of the reduction. We will generate the same
clauses as we had before. But now we apply them on the local sub-formulas, local formulas of Gamma L
or Delta, not the sub-formulas.
I have another example in the next slide and this can be done in linear time and correctness is now
proved by showing that if we have satisfying assignment, of course there is nothing else to prove here.
Then you can construct for them a Kripke counter-model. Here we use the fact that the underlying
propositional calculus is analytic. Again for the thing is that you have to show that partial Kripke frame,
partial valuation can be extended to a full one and you use this syntactic analyticity which we assumed
on the calculus.
Yeah?
>>: When we prove linear time we use powerful tools like suffix arrays.
>> Ori Lahav: I don’t know about suffix arrays, but we need the same data structure of the three that
represent the formula with the harmonily there’s all this. So we imported all of these ideas to, in order
to show that this can be done in linear time.
>>: U-huh.
>>: For the translation just to make the, because of this analytic sub-formula [inaudible]…
>> Ori Lahav: Local formulas.
>>: Local formulas.
>> Ori Lahav: Yeah.
>>: So the next line has [indiscernible] Kripke [indiscernible]?
>>: So are all the Kripke models essentially linear or do you have models that are sort of forest?
>> Ori Lahav: So if you have just one quotation it would be like, just like LTL.
>>: It has to be linear.
>> Ori Lahav: But if you have more of them it’s like a tree, it’s like a suffix tree.
>>: What’s not clear yet is how you search over Kripke models. The Kripke command is [indiscernible]
metric [indiscernible] construction. [indiscernible] so when you [indiscernible] to SAT are you going to
explain that or?
>> Ori Lahav: So I don’t have here these details but what we, this is the current translation to SAT. This
does not, if you get a satisfying assignment, let’s look at example, so this is what we’d get, what would
you get for a primal logic. So for every formula prefix A and B that is local to Gamma L or Delta you will
have these three clauses. This does not give you a Kripke model. Just, if you get a satisfying assignment
for this you can have some valuation. It tells you whether this is true or this, or things like that. From
this we have to show that we can construct a Kripke model.
The idea again imported from your work is that the prefixes would be the worlds of the Kripke model.
This, then the values from this assignment would be used in the Kripke model.
>>: To answer his question, you don’t search the Kripke models.
>>: Yeah, yeah, that was it.
>>: So my confusion was I thought that you were going to search through the Kripke models, but that’s
not what you’re doing.
>> Ori Lahav: No.
>>: Indeed you take the rules of the Gentzen system and write down their semantic conditions.
>> Ori Lahav: Yeah and the…
>>: And that suffices. Now for the KD! [indiscernible] what happens to it?
>> Ori Lahav: So it’s the same the KD! [indiscernible] and the other sort of system are equivalent. So…
>>: Then there’s no…
>> Ori Lahav: What is, what we have to prove and I guess this is the most…
>>: Then is transparent.
>> Ori Lahav: Yeah, it gives you this requirement, the semantics of box.
>>: Yeah.
>> Ori Lahav: But it doesn’t affect the reduction, you’re right. It just, it’s not so long but it’s not so
short, so proof that if you have a satisfying assignment for this you can construct from that a Kripke
model, yet we’ll respect all the conditions and with the usual interpretation of the box, of the quotation.
>>: So would it be…
>> Ori Lahav: I don’t have the details in the slides, so.
>>: Okay, so would it be correct to say that in your Kripke model every world corresponds to a path of
quotations from outer to inner, or something from your original formula…
>> Ori Lahav: Exactly. What we had to prove is that again the thing is that you get a partial model and
some formulas are defined in some worlds. You have to show that you can, prove, get a full model from
that. I think that in this case, specific case we don’t have variables. But if you restrict the reduction to
data log from these papers then you get actually the same, the same result. I mean, so I don’t think this
is, it should be new for primal logic. Then the new thing is the generality in the other point of view that
we have here a semantic one.
I think I skip this but I guess it’s obvious that it means that analytic calculi with quotations can be
decided by a SAT solver. That if you have a Horn pure calculi, analytic one with quotations then you can
use a HORN-SAT solver just as we had without quotations.
This would be the last part, general part of the talk. I have two slides of small extensions for primal logic
that we can do using this general framework. But this is, would be very specific so each of these was
tailored. The idea is that all these things here are not provable in primal logic. The usual, the original
one with quotations, so we cannot prove that A implies A because you don’t have the rule for that. Also
from a semantic point of view if A is false then you cannot know what would be the true value of A
implies A.
So all the other things, but all of them are classically true and we would like to recover them, to have
them inside a primal logic. So I don’t know about application whether these kind of formulas really
appear in practice. There are, this is just a small variety but if they have importance then from these we
can handle them. If we do this it will get a bit closer to classical logic.
The only thing that you have to verify using the general result before is that each of them preserves the
analyticity of the calculus. So if you add this axiom as a pure rule you have this Gamma and Delta and
no premises, only just one conclusion. If the calculus is still analytic for primal logic without quotations
then the calculus with quotations would be analytic and this whole procedure for reduction to a HORNSAT can work.
>>: So is the suggestion to add truth valuation, the valuation function for each of these axioms to treat
the…
>> Ori Lahav: Then, I’m not sure…
>>: To treat the formulas and the axioms as atomic at the bat, so B of A and B implies A equals T?
>> Ori Lahav: Yeah, this would be the semantic condition...
>>: [indiscernible]
>> Ori Lahav: You add this…
>>: You add [indiscernible] SAT solver.
>> Ori Lahav: Yeah add it for every formula of this form that you have in your assumption so
[indiscernible] in Gamma and Delta.
>>: So A and B implies A has to be present in the formula?
>> Ori Lahav: A and B implies?
>>: So the third, the third formula has to be present.
>> Ori Lahav: Yeah, if it doesn’t…
>>: It’s not that you [indiscernible] pairs of A and B but you loop over all sub-formulas that are this kind
of implication of this [indiscernible].
>> Ori Lahav: Exactly.
>>: So, right [indiscernible], if you have proof of un-SAT from a SAT solver, can you recover the
[indiscernible] proof?
>> Ori Lahav: I don’t know I didn’t think about it. Are you interested in the proof or?
>>: No I was just curious because you know you were saying we had these analytic proof systems. So
we know if it’s true it has an analytic proof.
>> Ori Lahav: Yeah, this…
>>: So can we recover it after the SAT solver’s…
>> Ori Lahav: I think this is what you asked me before I begin to talk…
>>: [indiscernible]
>>: The answer is yes and non-trivial.
>>: [indiscernible] what exactly what he knows. I had an intern here, Esteban, and what we discover we
used this translation that you mentioned to get the [indiscernible]. Then we used the Z three reduced to
SAT instead of using kind of data log mentioned because SAT was here. Then was the problem because
you SAT, ha, ha, kind of proof and so what should we do?
>> Ori Lahav: Right.
>>: So Esteban worked on this so it’s not exactly the same answer but pretty clear that the answer is
yes, but we need to do some work some more.
>> Ori Lahav: Yeah, of course.
>>: But in general I think it’s very nice because we knew some of these formulas, or maybe all of them it
was clear that it can be done. But if you can get them for free from [indiscernible].
>> Ori Lahav: Yeah, this…
>>: But I think, isn’t one of the extensions like a transitivity…
>> Ori Lahav: Transitivity counter is this flexivity here.
>>: Yes, the flexivity counters but then you need to do it quadratically, don’t you, I mean?
>> Ori Lahav: For transitivity yes. I don’t have transitivity here.
>>: So you need this…
>> Ori Lahav: Then you have to exactly to go on over all pairs like you…
>>: So you can do that incrementally on SAT’s solver. So you can check the model that it satisfies
transitivity [inaudible].
>>: So there’s some progress in your last formula?
>> Ori Lahav: Yeah the very last formula is the only one which, let me finish with this theorem and then
I’ll speak with the, over the last formula here. Then we don’t want to do this for each by hand. We tried
to find something a bit more general. So what we were able to prove this is specific for primal logic, is
that if A or B is the sequent, is provable in primal logic. Then the addition of this axiom scheme to
primal logic would preserve analyticity of the calculus. This gives us all of these because if you take the
corresponding A or B all of them are provable in classic, in primal logic.
What is nice is this can be done again. Now you have an extended system with these axioms. Then if
you are able to prove something like this you can take this as another axiom and you can get even closer
to classical logic. It depends on the depth that you want to have in your formula. The same thing for
disjunction if you have A arrow C and B arrow C both provable in primal logic, C should be some subformal of A or B. Then the addition of the axiom A or B arrow C to primal logic preserves its analyticity
and this gives you all of this. This requires a special treatment but this is different from Uri what you
told me about your ESPIL Project, because this only speaks about the most, main formula, not in sight.
For this we still have to do some hard hock solution because this [indiscernible] does not have this
natural structure that is spoke about that you have this material, main formula, that all other formulas
are [indiscernible] formulas, this is not true here.
>>: What I wanted to mention that this may not generalize because in the case [indiscernible] last
formula was more general consider conjunction sets. Then a reduction [indiscernible] procedure is
linear only expected time, right. Linistically it’s probably not…
>> Ori Lahav: But I don’t know what happens if disjunction is only the most, the outer connected. If you
take these sets, if you have A or B implies C this should be exactly the same as C, B or A implies C. This is
not what this rule gives you. It just gives you disjunction on the most outer level of the formula.
>>: Yeah but this particular maybe…
>> Ori Lahav: Yeah.
>>: Okay but in general, but it shows the core that there was a real problem there.
>> Ori Lahav: Yeah, now the final thing that we are able to add is to add the bottom connective which is
always false. So we can add this rule to primal logic. This is problematic from the, in the Hilbert’s
calculus but in Gentzen calculus there is no problem to do that. You can add, so this means that bottom
is always false. But since we have some weird implication in this junction we have to, we can add to
recover some of the properties. So we can add this axiom and this, and this. All of them are doing
classical logic, but you can add in to primal logic if you want to have this also there. All these extensions
are still, we still have linear time decision procedure using a HORN-SAT solver.
That’s it on most of the things I mentioned. So if the further work that we still have to do you want to
allow weaker notions of analyticity and paraconsistent logics you don’t have the usual analyticity of this
negation and of sub-formulas.
We wonder if there are other useful logics that can be reduced to polynomial SAT fragments or whether
primal logic was the only important one, I don’t know. Here’s your question about variables so I still
haven’t studied that. Of course what you just mentioned the sets that we can’t say anything about it
yet.
That’s it, thank you.
[applause]
>>: Thank you.
>> Nikolaj Bjorner: We’ve taken the [indiscernible]
>> Ori Lahav: Great thank you.