Propositional Proof Compression
Part I, Lecture 1
L. Gordeev
2008
Tuebingen – Rio de Janeiro
§1. Proof Systems
Proof system in a given language ℒ is
characterized by a semi-finite collection of
inference rules
➢ Each inference rule consists of:
➢ finite (possibly empty) list of premises
1 ∈ℒ, ... , k ∈ℒ
➢ exactly one conclusion  ∈ℒ
➢ Abbr.: 1 &  & k  
➢ Axiom := rule with empty list of premises
➢
§1.1. Regular rules and systems
An inference rule 1 &  & k   is regular
iff implication 1 ∧∧ k →  is valid
➢
Abbr.: ⊨ 1 ∧∧ k → 
➢ 1 &  & k   is strongly regular iff it is
regular and its inversion is regular too, i.e.
➢
⊨ 1 ∧∧ k ↔ 
➢ Regular (strongly regular) proof systems have
regular (strongly regular) inference rules
➢
§1.2. Sound rules and systems
An inference rule 1 &  & k   is sound iff
inference (⊨ 1 && ⊨ k ) ⇒ ⊨ holds true
➢ 1 &  & k   is strongly sound iff it is sound
and its inversions are sound too, i.e.
➢
(⊨ 1 &  & ⊨ k ) ⇔ ⊨
➢ Sound (strongly sound) proof systems have
sound (strongly sound) inference rules
➢ Clearly (strongly) regular is (strongly) normal
➢
Remarks
We consider only sound rules and systems.
➢ Some familiar rules are not strongly sound
➢ (: e.g. plain weakening   ,B is regular, and
hence sound, but not strongly sound)
➢ Some useful rules are not regular. For example
the rule of substitution is sound, but not regular
➢ (: e.g. x  y for different prop. variables x, y)
➢
§1.3. Provabilty in Proof Systems
Standard recursive definition:
1. If   is axiom, then  is provable
2. If 1 &  & k   is inference rule
and every i (1≤ i ≤ k) is provable,
then  is provable
§1.4. Completeness
Proof system is complete if every valid object (:
formula, sequent, etc.) is provable.
We consider proof systems which are both sound
and complete
§1.5. Sequent calculi
Basic objects = finite configurations of formulas
Gentzen's original designation:  → 
where  ,  are strings of formulas
Working in classical logic one can drop → and
regard sequents simply as strings of formulas
(, , , etc.)
Basic Sequent Calculus SEQ
➢
Formulas (, B, C, F, ...):
closure or literals (x, ¬x) under positive
connectives (∧ , ∨)
➢ Sequents (, , , ...): strings of formulas
➢ Axioms: x,¬x,
➢
Disjunction rule: ,B,  ∨B,
Conjunction rule: , & B,  ∧B,
➢ Note: everything modulo permutation
➢ THEOREM: SEQ is strongly regular and complete
➢
§2. Proof encoding -1
Proofs are encoded as deductions
Basic form: Deduction of  is a finite sequence of
sequents 1 , ... , n such that n =  and for every
1≤ i≤ n one of the following two options holds:
1. i =  is conclusion of axiom
2. i =  is conclusion of inference rule
1 &  & k   and for every 1≤ j≤ k there
exists 1≤ p≤ i such that p = j
§2. Proof encoding -2
Deduction (or proof) of  in basic form 1 , ... , n
is naturally presented as circuit, i.e. directed
acyclic rooted graph (: dag), and/or, in particular,
rooted tree whose nodes have labels i (1≤ i≤ n)
such that the root has label , and any i is placed
higher than any j , provided that i<j.
§2. Proof encoding -3
Note: In a circuit-structured deduction different
conclusion-nodes can have identical premisenodes (possibly referring to different inference
rules).This can't happen in a tree-structured
deduction.
Thus circuit-structured deductions can be smaller/
shorter than corresponding tree-structured
deductions (of the same sequent).
§3. Proof search
This applies to any sound proof system, but for
brevity consider our basic “most efficient” sequent
calculus SEQ .
Given any sequent  search for inference rule (in
particular axiom) 1 &  & k   with  = . Quit
if there is no solution. Stop if such axiom exists. In
the case of proper rule let  := i for every 1≤ i≤ k
and continue the same procedure. It terminates,
for each step lowers complexity of .
§4. Proof compression
Given any deduction ∂ go upwards and search for
identical sequents occurring in it. For any two ruleoccurrences 1 &  & k & 1 &  & q   and
1 &  & k & '1 &  &'r  ' identify each
pair of different nodes having the same sequentlabels i and delete one of the corresponding two
subdeductions of i . Proceed so as long as possible.
This yieds a desired compressed circuit-structured
deduction ∂'. Clearly ∂' should be smaller than ∂.
§4+. Proof search with
compression
Combine proof search with proof compression, in
order to optimize and speed-up proof search in
circuit structured mode.
§5. Cut rule
CUT is the following rule , & ,  
➢ CUT is strongly regular, but not good for proof
search (: how to pass from  to  ?)
➢
➢
But: adding CUT to SEQ can exponentially reduce
weight/size of deductions (even in tree-structured
mode)
§6. Weakened substitution rule
WS is the following rule   (), for any
 : Var → For (hom. extended to  : Seq → Seq)
➢ WS is sound, but not regular
➢ WS is good for proof search
➢ And: adding WS to SEQ can exponentially reduce
weight/size of circuit-structured deductions
Suggestion: Forget CUT, use circuit-structured WS!
➢
Conclusions
1. In the presence of weakened substitution rule,
WS, circuit-structured deducibility, a.k.a. dagdeducibility turns out to be much more powerful
that standard tree-structured deducibility.
2. Like CUT, WS can exponentially reduce weight/
size of (this time dag-) deductions.
3. Unlike CUT, WS is good for proof search.
4. Open problem. Is every valid formula (sequent)
polynomial-size dag-deducible in SEQ+CUT
+WS ? If so, then NP = CoNP !