Quantified Propositional Logspace Reasoning

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by  Steven Perron

Quantified Propositional Logspace Reasoning
Steven Perron
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0 February 3, 2008
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n
a Abstract
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In this paper, we develop a quantified propositional proof systems
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that corresponds to logarithmic-space reasoning. We begin by defining a
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classΣCNF(2)ofquantifiedformulasthatcanbeevaluatedinlogspace.
] Then our new proof system GL∗ is defined as G∗1 with cuts restricted to
O ΣCNF(2)formulasandnocutformulathatisnotquantifierfreecontains
L a free variable that does not appear in thefinal formula.
. To show that GL∗ is strong enough to capture log space reason-
s
ing, we translate theorems of VL into a family of tautologies that have
c
[ polynomial-size GL∗ proofs. VL is a theory of bounded arithmetic that
is known to correspond to logarithmic-space reasoning. To do the trans-
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lation, we find an appropriate axiomatization of VL, and put VL proofs
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into a new normal form.
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To show that GL∗ is not too strong, we prove the soundness of GL∗
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in such a way that it can be formalized in VL. This is done by giving a
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4 logarithmic-space algorithm that witnesses GL∗ proofs.
.
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0 1 Introduction
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Recentlytherehas beena significant amount of research looking intothe
:
v connection between computational complexity, bounded arithmetic, and
i propositionalproofcomplexity. Arecentsurveyonthistopiccanbefound
X
at [6]. The idea is that there is a hierarchy of complexity classes
r
a AC0 (TC0 ⊆NC1 ⊆L⊆NL⊆P.
The first class is the set of problems that can be solved by uniform,
polynomial-size, constant depth circuits. This class is important because
itcanbeshownthatPARITYcannotbesolvedinAC0. Infact,problems
that involvecountingcannot besolved in AC0. Thesecond class is TC0.
This set of problems is the same as AC0 except that TC0 circuits can
use counting gates. The class NC1 is the set of problems that can be
solved using polynomial-size, logarithmic-depth circuits. This class can
bethoughtofasthesetofproblemsthatcanbesolvedveryquicklywhen
workisdoneinparallel. Evaluatingboolean formulasiscompleteforthis
class. TheclassListhesetofproblemsthatcanbesolvedinlogarithmic
spaceonaTuringmachine. Theclass NListheset ofproblemsthatcan
1
be solved in logarithmic space on a non-deterministic Turing machine.
The reachability problem for directed graphs is complete for this class.
The sequence finishes with P, the set of problems that can be solved in
polynomial time on a deterministic Turing machine. Except for the first
inclusion, is it unknown if any of these inclusions are proper.
Each of these complexity classes has a corresponding theory of arith-
metic: V0,VTC0,VNC1,VL,VNL,andTV0,respectively. Eachofthese
theories can prove that the functions in their corresponding complexity
class are total. As a consequence, any information we can obtain about
the theory tells us something about thecomplexity class and vice versa.
There is also a connection with propositional proof complexity. Some
ofthetheoriesmentionedabovehaveacorrespondingpropositionalproof
system. As before, information about the proof systems tells us about
thecorresponding theory and complexity class. In thispaper, weexplore
the proof systems. The goal is to try to understand how the strength of
a proof system is affected bydifferent restrictions.
Ourfocuswillbeonquantifiedpropositionalproofsystems,but,toex-
plain ourmethod,wewill usequantifier-freepropositional proof systems.
Start with aFregeproof system,sometimes called Hilbert StyleSystems.
These systems are described in standard logic text books. A Frege proof
is a series of propositional formulas where each formula is an axiom or
canbeinferredfrompreviousformulasusingoneoftherulesofinference.
There are two common ways of restricting this proof system. The first is
to restrict all of theformulas in theproof. Forexample, one definition of
bounded-depthFregeis torestrict everyformula intheprooftoformulas
with a constant depth. This worked, but, if a proof system is defined
this way, then there are formulas that cannot be proved simply because
theyarenotallowedtoappearintheproof. Forexample,bounded-depth
Frege with formulas of depth d cannot prove any formula of depth d+1.
The other method is to restrict the formulas on which certain rules can
be applied. This solves the problem of the first method and led to other
definitions of bounded-depthFrege.
In this paper, we will look at restricting the cut rule in the tree-like
sequent calculus for quantified propositional formulas. This systems is
known as G∗. The cut rule derives Γ−→∆ from A,Γ−→∆ and Γ−→
∆,A. In G∗, A can be any quantified propositional formulas. The proof
systemG∗ isdefinedbyrestrictingAtoquantifier-freeformulas. Ifweare
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givenaG∗proofofaΣq formula(∃~zB(~z),whereBisquantifier-free),then
0 1
wecanfindawitnessforexistentialquantifiersinthisformulainuniform
NC1; moreover, this problem is complete for this class. The complexity
class NC1 is the set of problems that can be solved by polynomial-size,
logarithmic-depth circuits with fan-in 2. The interesting observation is
thatevaluatingquantifier-freeformulasisalsocompleteforNC1. Itisalso
possible to connect G∗ to NC1 indirectly through bounded arithmetic.
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There is a theory of arithmetic VNC1 that is known to correspond to
NC1 reasoning. GivenaVNC1 proofof aboundedformula itispossible
to translate the proof into a family of polynomial-size G∗ proofs. This
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tells us that the reasoning power of G∗ is at least as strong as that of
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VNC1 [5]. In the other direction, VNC1 can prove that G∗ is sound
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when proving Σq formulas. This means that, when proving Σq formulas,
1 1
2
thereasoning powerofG∗ isnot strongerthanthat ofVNC1. Sowesay
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that G∗ corresponds to NC1 reasoning.
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As well, if we restrict cut formulas to constant-depth, quantifier-free
formulas, we get a proof system that corresponds to AC0 reasoning.
The complexity class AC0 is the set of problems that can be solved by
polynomial-size, constant-depth circuits with unbounded fan-in. Again,
evaluating constant-depthformulas is completefor AC0. Weshouldnote
we are talking about the proofs of quantifier-freeformulas.
Thisgivesustwoproofsystemswhosereasoningpoweristhesameas
the complexity of evaluating their cut formulas. This raises the question
ofwhetherornotthisholdsingeneral. Thequickanswerisno. Acounter-
example is G∗. Evaluating Σq formulas is complete for NP, but the Σq
1 1 1
witnessingproblemforG∗iscompleteforP [8]. Anothercounter-example
1
is GPV∗, where cut formulas are quantifier-free or formulas of the form
∃x[x ↔ A], where A is a quantifier-free formula that does not mention
x. Evaluating a cut formula for GPV∗ is complete for NC1, but the
witnessing problem is complete for P [14].
In this paper, we define a new proof system GL∗ that corresponds to
L reasoning. The complexity class L is the set of problems that can be
solved on aTuringMachinewith aread-only inputtapeandawork tape
where the space used on the work tape is proportional to the logarithm
of the size of the input. Our proof system GL∗ is defined by restricting
cuts to ΣCNF(2) formulas, a set of formulas for which the evaluation
problem is completefor L. However,that isnot enough. Wealso restrict
thefreevariablesthatappearincutformulaswithquantifierstovariables
that appear free in the final sequent. We then prove this proof system
corresponds to L reasoning by connecting it with a theory of arithmetic
that is known to correspond to L reasoning. This definition is meant to
demonstratethatthestrengthofaproofsystemisnotrelatedtothediffi-
cultyofevaluatingasinglecutformulaintheproof,buttothecomplexity
of witnessing theeigenvariables in theproof.
In Section 2, we give definitions of theimportant concepts. Inpartic-
ular, we define two-sorted computational complexity and bounded arith-
metic. As well, we define the standard proof systems and explain the
connection between proof systems and theories of bounded arithmetic in
more detail. In Section 3, we define GL∗. This includes the definition of
theΣCNF(2)formulas. InSection4,wechangethetheoryVLandprove
anormal-formthatisnecessaryforourresults. Thisisthemosttechnical
section in the paper. In Section 5, we prove the translation theorem. In
Section 6, we prove that GL∗ is sound in the theory. This includes an
algorithm to evaluate ΣCNF(2) formulas in L.
This paperis an expanded version of theauthor’s earlier paper[13].
2 Basic Definitions And Notation
2.1 Two-Sorted Computational Complexity
Inthispaper,weusetwo-sortedcomputationalcomplexity. Thetwosorts
arenumbersandbinarystrings(akafinitesets). Thenumbersareintended
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to range over the natural numbers and will be denoted by lower-case
letters. For example, i, j, x, y, and z will often be used for number
variables; r, s, and t will be used for number terms; and f, g and h will
be used for functions that return numbers. The strings are intended to
be finitestrings over {0,1} with leading 0 removed. Sincethestrings are
finite, they can be thought of as sets where the ith bit is 1 if i is in the
set. The strings will be denoted by upper-case letters. The letters X,Y,
and Z will often be used for string variables.
We focus on the complexity class L. Let R(~x,X~) be a relation. If we
aregoingtosolvethisrelationonaTuringMachineM,thentheinputto
M will be ~x in unary and X~ as a series of binary strings. So the size of
theinput is~x+|X~|. Wesay R is in L if R can be decided bya two-tape
Turing Machine such that one tape is a read-only input tape, and less
than O(log(~x+|X~|)) squares are visited on the othertape.
Forfunctions, we saya numberfunction f(~x,X~)is in FLif thereisa
polynomial p such that f(~x,X~)<p(~x,|X~|), and therelation f(~x,X~)=y
isinL. AstringfunctionF(~x,X~)isinFLifthesizeofF(~x,X~)isbounded
by a polynomial and if the relation
R(i,~x,X~)↔ theith bit of F(~x,X~) is 1
is in L. This is equivalent to defining FL using a three-tape Turing
Machine with a write-only output tape.
2.2 Two-Sorted Bounded Arithmetic
Besides two-sorted computational complexity, we also use the two-sorted
bounded arithmetic. The sorts are the same. This notation was base on
thework of Zambella in [15],but we follow thepresentation of Cook and
Nguyen from [4, 6].
The base language is
L2 ={0,1,+,×,<,=,= ,∈,||}.
A 2
The constants 0 and 1 are number constants. The functions + and ×
take two numbers as input and return a number–the intended meanings
are the obvious ones. The language also includes two binary predicates
that take two numbers: < and =. The predicate = is meant to be
2
equality between strings, instead of numbers. In practice, the 2 will not
be written because which equality is meant is obvious from the context.
The membership predicate ∈ takes a number i and a string X. It is
meant to be true if the ith bit of X is 1 (or i is in the set X). This will
also be written as X(i). The final function |X| takes a string as input
and returns a number. It is intended to be the number of bits needed to
writeX whenleadingzerosareremoved(ortheleast upperboundofthe
set X). The set of axioms 2BASIC is the set of defining axioms for L2.
A
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B1. x+16=0 B7. x≤x+y
B2. x+1=y+1⊃x=y B8. (x≤y∧y≤x)⊃x=y
B3. x+0=x B9. 0≤x
B4. x+(y+1)=(x+y)+1 B10. x≤y∨y≤x
B5. x×0=0 B11. x≤y↔x<y+1
B6. x×(y+1)=(x×y)+x B12. x6=0⊃∃y≤x(y+1=x)
L1. X(y)⊃y<|X| L2. y+1=|X|⊃X(y)
SE. X =Y ↔[|X|=|Y|∧∀i<|X|(X(i)↔Y(i))]
Weuse∃X <bφasshorthandfor∃X[(|X|<b) ∧ φ]. Theshorthand
∀X < b φ means ∀X[(|X| < b) ⊃ φ]. The set ΣB = ΠB is the set
0 0
of formulas whose only quantifiers are bounded number quantifiers. For
i > 0, the set ΣB is the set of formulas of the form ∃X~ <~tφ where φ is
i
a ΠB formula. For i>0, the set ΠB is the set of formulas of the form
i−1 i
∀X~ <~tφ where φ is a ΣB formula.
i−1
Now we can definetwo important axiom schemes:
Φ-COMP: ∃X ≤b∀i<b[X(i)↔φ(i)],
Φ-IND:[φ(0)∧∀x<b[φ(x)⊃φ(x+1)]]⊃φ(b)
whereΦisasetofformula andφ(i)∈Φ, and,for ΣB-COMP,φdoesnot
i
contain X, but may contain otherfree variables.
We can now definethe base theory.
Definition 2.1. The theory V0 is axiomatized by the 2BASIC axioms
plus ΣB-COMP.
0
It is possible to show that V0 proves ΣB-IND (Corollary [6]). This
0
theory is typically viewed at the theory that corresponds to AC0 reason-
ing.
From time to time, we will use functions symbols that are not in L2.
A
The first is X(i,j) ≡ X(hi,ji), where hi,ji = (i+j)(i+j +1)+2j is
the pairing function. It can be thought of as a two dimensional array of
bits. The second is the row function. The notation we use is X[i]. This
functionsreturnstheithrowofthetwodimensionalarrayX. Inthesame
way,we can also describe threedimensional arrays. Wealso want topair
string. So if X = hY ,Y i, then X[0] = Y and X[1] = Y . Note that,
1 2 1 2
if we add these functions with their ΣB defining axioms to any theory T
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extending V0, we get a conservative extension. They can also be used
in the induction axioms [4]. This means that, if there is a T proof of a
formula that uses thesefunctions, there is a T proof of the same formula
that does not use these functions.
TogetatheorythatcorrespondstoLreasoning,weaddanaxiomthat
says there is an output to a function that is complete for L with respect
to AC0 reductions. This is a specific example of the method used in [4]
to construct a theory for a given complexity class. The theory we define
isΣB-recfrom[16],butwewillcallitVL. Thecompletefunctionweuse
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is: Given a graph with edge relation φ(i,j) and nodes {0,...,a}, where
every vertex in the graph has out-degree at least 1, find a path of length
b. This is expressed using theΣB-rec axiom:
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∀x≤a∃y≤aφ(x,y)⊃∃Z,∀w≤bφ(f(a,w,Z),f(a,w+1,Z)) (ΣB-rec)
0
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where f(a,w,Z) = min (Z(w,x)∨x = a) and φ is a ΣB formula. The
0
x
ideaisthatthefunctionf(a,w,Z)extractsthewthnodeinthepaththat
Z encodes.
Definition 2.2. The theory VL is the theory axiomatized by V0 plus
ΣB-rec.
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TheΣB-recaxiomhasthedisadvantagethatthepathcanstartatany
0
node. However, as Zambella pointed out in [16], to is possible to prove
that thereis a path of length b starting at a particular node a.
Lemma 2.3. Let E bethe edge relation foradirected graph onthe nodes
{0,...,n−1}. Then for all a < n and b, VL proves, if ∀i < n∃j <
n E(i,j), then there is a path of length b starting at node a.
Proof. Defineφ(hw,ii,hw′,ji) as
φ(hw,ii,hw′,ji)≡(w′ =w+1 mod b+1)∧(w′ 6=0⊃E(i,j))∧(w′ =0⊃j =a).
Takeapathoflength2binthegraphofφ. Atsomepointinthefirsthalf
ofthatpath,thepathpassesthroughthenodeh0,ai. Startingfromthere
we can extract a path of length b in E that starts at node a.
2.3 A Universal Theory For L Reasoning
Another way to get a theory for L is to define a universal theory with a
languagethatcontainsafunctionsymbolforeveryfunctioninFL. Then,
we get a theory for L by taking the defining axioms for these functions.
This is the idea behind other universal theories like PV and V0. In our
case, we characterize the FL functions using Lind’s characterization [10]
adjusted for thetwo-sort setting.
In the next definition, we define the set of function symbols in LFL
and givetheir intended meaning.
Definition 2.4. The language LFL is thesmallest language satisfying
1. L2A∪{pd,min}isasubsetofLFLandhavedefiningaxioms2BASIC,
and the axioms
pd(0)=0 (2.1)
pd(x+1)=x (2.2)
min(x,y)=z↔(z=x∧x≤y)∨(z=y∧y≤x) (2.3)
2. For every open formula α(i,~x,X~) over LFL and term t(~x,X~) over
L2A, thereis a string function Fα,t in LFL with bit definingaxiom
Fα,t(~x,X~)(i)↔i<t(~x,X~)∧α(i,~x,X~) (2.4)
3. For every open formula α(z,~x,X~) over LFL and term t(~x,X~) over
L2A, thereis a numberfunction fα,t in LFL with definingaxioms
fα,t(~x,X~)≤t(~x,X~) (2.5)
z <t(~x,X~)∧α(z,~x,X~)⊃α(fα,t(~x,X~),~x,X~) (2.6)
z<fα,t(~x,X~)⊃¬α(z,~x,X~) (2.7)
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4. For all numberfunctions g(~x,X~) and h(p,y,~x,X~) in LFL and term
t(y,~x,X~) over L2, there is a number function f (y,~x,X~) with
A g,h,t
definingaxioms
f (0,~x,X~)=min(g(~x,X~),t(~x,X~)) (2.8)
g,h,t
f (y+1,~x,X~)=min(h(f(y,~x,X~,y,~x,X~)),t(~x,X~)) (2.9)
g,h,t
Thelastschemeiscalledp-boundednumberrecursion. Thep-bounded
number recursion is equivalent to the log-bounded string recursion given
in [10]. The otherschemes come from the definition of LFAC0 in [4].
It is not difficult to see every function in LFL is in FL. The only
point we should note is that the intermediate values in the recursion are
boundedbyapolynomialinthesizeoftheinput. Thismeans,ifwestore
intermediatevaluesinbinary,thespaceusedisboundedbythelogofthe
size of the input. So the recursion can be simulated in L. To show that
everyFLfunctionhasacorrespondingfunctionsymbolinLFL,notethat
the p-bounded number recursion can be used to traverse a graph where
every node has out-degreeat most one.
Definition 2.5. VL is the theory over the language LFL with B1-B11,
SE, plus 2.1; 2.2; 2.3; axiom 2.4 for each string function Fα,t in LFL;
axioms2.5,2.6,and2.7foreachnumberfunctionfα,t inLFL;andaxioms
2.8 and 2.9 for each numberfunction fg,h,t in LFL.
An open(L) formula is a formula over the language L that does not
haveany quantifiers.
Theimportantpartofthistheoryisthatitreallyisauniversalversion
of VL.
Theorem 2.6. VL is a conservative extension of VL.
Proof. First toprovethatVLisan extensionof VL. Allthatisrequired
istoprovetheΣB-COMPandΣB-recaxioms. ToproveΣB-COMP,note
0 0 0
thateveryΣB formulaφisequivalenttoanopenformulaφ′. Forexample,
0
VL⊢∃z<bψ(z,~x,X~)↔ψ(f (~x,X~),~x,X~)
ψ,b
when ψ is an open formula. Then thefunction Fφ′,t is thewitness for
∃Z ≤t∀i<t[Z(i)↔φ(i)].
To prove the ΣB-rec axiom, we can define a function f(i,a,E) that re-
0
turnstheith nodein thepath theaxiom saysexists. Thefunction f can
be defined using p-bounded number recursion. From there, a function
witnessing the ΣB-recaxiom can be defined.
0
To provethat theextension is conservative, we show how to takeany
modelM ofVLandfindanexpansionthatisamodelofVL. Theideais
toexpandthemodelonefunctionatatime. Wecanorderthefunctionsin
LFL suchthateachfunctionisdefinedintermsofthepreviousfunctions.
Let Li be the language L2A plus the first i functions in LFL. Let Mi be
themodelobtainedbyexpandingM tothefunctionsinLi. Wewillshow
thatthemodelM∞ = Mi isamodelVL. Asimilarproofcanbefound
S
in Chapter 9 of [6] and we will not repeat it here.
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2.4 Quantified Propositional Calculus
We are also interested in quantified propositional proof systems. The
proof systems we use were originally defined in [9], and then they were
redefined in [5, 11], which is the presentation we follow.
The set of connectives are {∧,∨,¬,∃,∀,⊤,⊥}, where ⊤ and ⊥ are
constants for true and false, respectively. Formulas are built using these
connectives in the usual way. We will often refer to formulas by the
numberof quantifieralternations.
Definition 2.7. Theset offormulasΣq =Πq istheset ofquantifier-free
0 0
propositional formulas. For i > 0, the set of Σq (Πq) formulas is the
i i
smallest set of formulas that contains Πq (Σq ) and is closed under
i−1 i−1
∧, ∨, existential (universal) quantification, and if A∈Πq (A∈Σq) then
i i
¬A∈Σq (¬A∈Πq).
i i
The first proof system, from which all others will be defined, is the
proofsystemG. ThisproofsystemisasequentcalculusbasedonGentzen’s
system LK. Thesystem Gisessentially theDAG-like,propositional ver-
sion of LK. We will not give all of the rules, but will mention a few of
special interest.
The cut ruleis
A,Γ−→∆ Γ−→∆,A
cut
Γ−→∆
In this rule, we call A the cut formula. There are also four rules that
introduce quantifiers:
A(x),Γ−→∆ Γ−→∆,A(B)
∃-left ∃-right
∃zA(z),Γ−→∆ Γ−→∆,∃zA(z)
Γ−→∆,A(x) A(B),Γ−→∆
∀-left ∀-right
Γ−→∆,∀zA(z) ∀zA(z),Γ−→∆
These rules have conditions on them. In ∃-left and ∀-right, the variable
x must not appear in the bottom sequent. In these rules, x is called the
eigenvariable. Intheothertworules,theformulaBmustbeaΣq formula,
0
and novariable that appears free in B can be bound in A(x).
The initial sequents of G are sequents of the form −→ ⊤, ⊥ −→, or
x −→ x, where x is any propositional variable. A G proof is a series
of sequents such that each sequent is either an initial sequent or can be
derived from previous sequents using one of the rules of inference. The
proof system Gi is G with cut formulas restricted to Σqi formulas.
WedefineG∗ asthetreelikeversion ofG. So,aG∗ proof isaGproof
where each sequent in used as an upper sequent in an inference at most
once. A G∗ proof is a G∗ proof in which cut formulas are prenex Σq. In
i i
[11], it was shown that, for treelike proofs, it did not matter if the cut
formulas in G∗ were prenex or not. So when we construct G∗ proofs, the
i i
cut formulas will not always beprenex, butthat does not matter.
Tomakeproofs simpler,weassumethatalltreelikeproofsareinfree-
variable normal form.
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Definition 2.8. AparametervariableforaG∗ proofπ isavariablethat
i
appearsfreeinthefinalsequentofπ. Aproofπ isinfree-variable normal
formif(1)everynon-parametervariableisusedasaneigenvariableexactly
once in π,and (2) parameter variables are not used as eigenvariables.
Note that, if a proof is treelike, we can always put it in free-variable
normal form by simply renaming variables. In fact, VPV proves that
every treelike proof can beput in free-variable normal form.
Ausefulpropertyoftheseproofsystemsisthesubformulaproperty. It
canbeshowninVLthateveryformulainaG∗ proofisanancestor(and
i
thereforeasubformula)ofacutformulaoraformulainthefinalsequent.
This is useful because it tells us that any non-Σq formula in a G∗ proof
i i
must bean ancestor of a finalformula.
2.5 Truth Definitions
In order to reason about the proof systems in the theories, we must be
able to reason about quantified propositional formulas. We follow the
presentation in [8, 9, 5].
Formallyformulaswillbecodedasstrings,butwewillnotdistinguish
betweenaformulaanditsencoding. SoifF isaformula,wewilluseF as
thestringencoding theformula aswell. Themethod ofcoding aformula
can be found in [5].
In this paper, we are only interested in Σq formulas and prenex Σq
0 1
formulas. For Σq0 formulas, we are able to give an ΣB0(LFL) functions
that evaluates the formula. This formula will be referred to using A |=
0
F, where A is an assignment and F is a formula. We leave the precise
definition tothe readers.
Given a prenex Σq formula F, the truth definition is a formula that
1
says there is an assignment to the quantified variables that satisfies the
Σq part of theformula. This formula will be referred to as A|= F.
0 1
Valid formulas (or tautologies) are definedas
TAUTi(F)≡∀A,(“A is an assignment to thevariables of F”⊃A|=i F)
Thistruthdefinitioncanbeextendedtodefinethetruthofasequent.
So, if Γ−→∆ is a sequent of Σq∪Πq formulas, then
i i
(A|=i Γ−→∆)≡“there exists a formula in Γ that A does not satisfy”
∨ “there exists a formula in ∆ that A satisfies”
Anotherimportantformulawewill useisthereflection principlefora
proof system. We define the Σq reflection principle for a proof system P
i
as
Σqi-RFN(P)≡∀F∀π,(“π is a P proof of F”∧F ∈Σqi)⊃TAUTi(F)
Thisformulaessentiallysaysthat,ifthereexistsaP proofofaΣq formula
i
F, then F is valid. Another way of putting it is to say that P is sound
when provingΣq formulas.
i
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2.6 Propositional Translations
There is a close connection between the theory Vi and the proof system
G∗. YoucanthinkofG∗asthenon-uniformversionofVi. Thisideamight
i i
notmakemuchsenseatfirstuntilyourealizeyoucantranslateaViproof
into a polynomial-size family of G∗ proofs. The translation that we use
i
is described in [4, 5]. It is a modification of the Paris-Wilkie translation
[12]. Given a ΣB formula φ(~x,X~) over the language L2, we want to
i A
translateitintoafamilyofpropositionalformulas||φ(~x,X~)||[m~;~n],where
the size of the formulas is bounded by a polynomial in m~ and ~n. The
formula||φ(~x,X~)||[m~;~n]ismeanttobeaformulathatisatautologywhen
φ(~x,X~) is true in the standard model whenever xi = mi and |Xi| = ni.
Thenifφ(~x,X~)istrueinthestandardmodelforall~xandX~,thenevery
||φ(~x,X~)||[m~;~n] is a tautology.
Thevariablesm~ and~nwilloftenbeomittedsincetheyareunderstood.
Thefreevariables inthepropositional formula will bepXji forj <ni−1.
The variable pXji is meant to represent thevalue of the jth bit of Xi; we
knowthatthenithbitis1,andfor j >ni,weknowthejthbitis0. The
definition of thetranslation proceeds by structural induction on φ.
Suppose φ is an atomic formula. Then it has one of the following
forms: s = t, s < t, Xi(t), or one of the trivial formulas ⊥ and ⊤, for
termssandt. Notethatthetermssandtcanbeevaluatedimmediately.
This is because the exact value of every number variable and the size of
each string variable is known. Let val(t) be valueof theterm t.
Inthefirstcase,wedefine||s=t||astheformula⊤,ifval(s)=val(t),
and ⊥, otherwise. A similar construction is done for s < t. If φ is one
of the trivial formulas, then ||φ|| is the same trivial formula. So now, if
φ≡Xi(t), let j =val(t). Then thetranslation is defined as follows:
8pXji if j <ni−1
||φ||≡><1 if j =n1−1
0 if j >n −1
1
>:
Nowfortheinductivepartofthedefinition. Supposeφ≡α∧β. Then
||φ||≡||α||∧||β||.
When theconnectiveis∨ or¬,thedefinition issimilar. Iftheoutermost
connectiveisanumberquantifierboundbyatermt,letj =val(t). Then
the translation is definedas
j
||∃y ≤t,α(y)||≡ ||α(y)||[i]
_
i=0
j
||∀y ≤t,α(y)||≡ ||α(y)||[i]
^
i=0
j
||∃Y ≤t,α(Y)||≡∃pY,...,∃pY , ||α(Y)||[i]
0 m−2 _
i=0
j
||∀Y ≤t,α(Y)||≡∀pY,...,∀pY , ||α(Y)||[i]
0 m−2 ^
i=0
10