Proof Complexity Olaf Beyersdorff School of Computing University - - PowerPoint PPT Presentation
Proof Complexity Olaf Beyersdorff School of Computing University of Leeds, UK 1 Outline of this tutorial Tour of proof systems Resolution Frege and beyond Cutting Planes . . . Relations to other areas Separation of
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School of Computing University of Leeds, UK
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◮ Resolution ◮ Frege and beyond ◮ Cutting Planes ◮ . . .
◮ Separation of complexity classes ◮ Analysis of SAT algorithms ◮ Proof search – Automatizability ◮ First-Order Logic – Bounded Arithmetic ◮ Further topics
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◮ correctness: rng(f) ⊆ L ◮ completeness: L ⊆ rng(f) ◮ efficiency: proofs should be easy to check,
◮ Most research in proof complexity has studied
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◮ Most proofs are exponentially long in the size of the
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◮ Most proofs are exponentially long in the size of the
◮ We look for proof systems with shorter proofs.
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◮ Introduced by Blake 1937, Davis & Putnam 1960, and
◮ Resolution proofs operate with clauses. ◮ Refutation system ◮ only one rule
◮ many subsystems studied: tree-like, regular . . .
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◮ Pigeonhole principle: n + 1 pigeons cannot sit in n holes ◮ CNF formulation PHPn+1 n
xi,j for all pigeons i ∈ [n + 1] ¬xi1,j ∨ ¬xi2,j for all distinct i1, i2 ∈ [n + 1] and j ∈ [n]
◮ PHPn+1
n
requires Resolution refutations of size 2Ω(n). [Haken 85]
◮ Combinatorial principles: ordering principle, . . . ◮ Graph-theoretic principles: Tseitin formulas, pebbling . . . ◮ Random 3-CNF’s are hard for Resolution.
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◮ ϕi is a substitution instance of an axiom, or ◮ ϕi was derived by modus ponens from ϕj, ϕk with j, k < i.
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◮ ϕi is a substitution instance of an axiom, or ◮ ϕi was derived by modus ponens from ϕj, ϕk with j, k < i.
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◮ f simulates g, if for any g-proof w there is an f-proof w′ of
◮ If w′ is computable from w in polynomial time, then f
◮ f and g are (p-)equivalent if they (p-)simulate each other.
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◮ f simulates g, if for any g-proof w there is an f-proof w′ of
◮ If w′ is computable from w in polynomial time, then f
◮ f and g are (p-)equivalent if they (p-)simulate each other.
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◮ Historically one of the first and best analyzed proof
◮ basic objects: sequents ϕ1, . . . , ϕm ⊢ ψ1, . . . , ψk . ◮ Sequents of the form
◮ An LK-proof of a propositional formula ϕ is a derivation of
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◮ The standard proof system for SAT is polynomially
◮ The truth-table system is not a polynomially bounded proof
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◮ |θn| = nO(1); ◮ θn requires super-polynomial size proofs in P.
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◮ |θn| = nO(1); ◮ θn requires super-polynomial size proofs in P.
◮ Better: . . . exponential size proofs.
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◮ |θn| = nO(1); ◮ θn requires super-polynomial size proofs in P.
◮ Better: . . . exponential size proofs.
◮ Find a sequence of polynomially constructible formulas
◮ This is usually the case: take θn as the propositional
◮ Find a large set of formulas (e.g. random 3-CNF) which
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◮ Haken (1985): exponential lower bound to the proof size in
◮ Ajtai (1988): Super-polynomial lower bound for
◮ Lower bounds for algebraic and geometric proof systems:
◮ Cutting Planes [Pudlák 97] ◮ Polynomial Calculus [Razborov 98, . . .] ◮ Nullstellensatz [Buss et al. 97] [Grigoriev 98] ◮ OBDD proof systems [Krajíˇ
cek 08] [Segerlind 08]
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◮ feasible interpolation [Krajíˇ
◮ size-width relation [Ben-Sasson & Wigderson 01] ◮ game-theoretic techniques [Pudlák, Buss, Impagliazzo,. . .] ◮ proof complexity generators [Krajíˇ
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◮ A number of combinatorial principles have been
◮ E.g., matrix multiplication: AB = I =
◮ A good candidate from logic: reflection principles ◮ Problem: hard to analyze ◮ A promising approach: formulas from pseudo-random
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◮ Cutting Planes uses the idea of linear programming. ◮ As in Resolution, CP is a refutation system that works with
◮ Clauses are translated into linear inequalities.
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◮ Clauses are translated into linear inequalities
◮ Propositional variables p are identically represented by
◮ ¬p is translated to 1 − p. ◮ A clause
◮ To get an inequality of the form (1), constants are moved to
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1p1 + . . . + a′ npn ≥ b′
1)p1 + . . . + (an + a′ n)pn ≥ b + b′
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◮ A CP refutation of a set of clauses Γ is a CP derivation of
◮ Easy to see: CP p-simulates Resolution. ◮ The converse is false. ◮ Frege systems p-simulate CP [Goerdt 91].
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Optimal Proof System? Truth Table Tree-like Resolution Nullstellen Satz Resolution Polynomial Calculus PCR Bounded- depth Frege Cutting Planes Frege Extended Frege ZFC
not polynomially bounded
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◮ is at the intersection of logic and complexity. ◮ uses concepts and intuition from algebra, geometry, . . .
◮ Separation of complexity classes ◮ Analysis of SAT algorithms ◮ Proof search – Automatizability ◮ First-Order Logic – Bounded Arithmetic ◮ Proving lower bounds is hard!
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◮ Input: a propositional formula F ◮ Question: Is F satisfiable?
◮ Cook 71: SAT is NP complete. ◮ No efficient algorithm unless P=NP
◮ Exponential-time hypothesis: SAT has worst-case running
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◮ Intensively investigated and improved since the 90’s
◮ Routinely solve industrial instances with ≥ 100.000’s of
◮ Many problems can be efficiently encoded in SAT. ◮ Applied in many areas: model checking, data bases,
◮ Why are SAT algorithms so successful? ◮ What are their limitations?
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◮ choose a variable x ◮ set x to 0/1 ◮ simplify the formula ◮ recursively iterate until satisfying assignment is found or
◮ based on DPLL ◮ improved by unit propagation, clause learning, restarts . . . ◮ implementation tuning
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◮ operates with clauses ◮ only one rule
◮ proofs are trees
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◮ Binary tree ◮ Inner nodes are labeled with variables from F. ◮ Leafs are labeled with clauses from F. ◮ Each path in the tree corresponds to a partial assignment
◮ In the tree, each path α must lead to a clause which is
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◮ A boolean decision tree solves the search problem for F:
◮ given an assignment α, ◮ find a clause from F falsified by α.
◮ Each tree-like Resolution refutation of F yields a boolean
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◮ On unsatisfiable formulas, the DPLL algorithm produces a
◮ To analyse the complexity of classical DPLL we need to
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◮ Let F be unsatisfiable. ◮ Delayer claims she knows a satisfying assignment. ◮ Prover wants to find a contradiction.
◮ Prover asks a variable x. ◮ Delayer “answers” 0/1 and gets some points for his choice. ◮ Prover wins if the partial assignment falsifies a clause from
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◮ Delayer assigns two weights p0 and p1 which satisfy:
◮ Prover chooses value b. ◮ Delayer gets log 1
◮ If Prover chooses b with pb = 0, Delayer gets ∞ points.
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◮ PHPm n uses variables xi,j with i ∈ [m] and j ∈ [n], ◮ xi,j indicates that pigeon i goes into hole j. ◮ PHPm n consists of the clauses
n has tree-like Resolution refutation of size 2θ(n log n).
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◮ Let α be a partial assignment to the variables
◮ Let
◮ Intuitively, Ei(α) corresponds to the number of holes which
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2
n 2 − Ei(α) n 2 + 1 − Ei(α)
n 2 + 1 − Ei(α) .
n 2 excluded free holes for pigeon i.
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n 2 − Ei(α) n 2 + 1 − Ei(α)
n 2 + 1 − Ei(α) . ◮ First observation: Delayer always earns more when Prover
◮ This is intuitively correct: the amount of freedom for
◮ In fact, our choice of scores can be completely explained
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◮ When Prover sends a pigeon to a hole, Delayer should
◮ When we play the game, in each round Delayer should get
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◮ When Prover sends a pigeon to a hole, Delayer should
◮ When we play the game, in each round Delayer should get
◮ Prover fixes i to a hole in the very beginning by answering
◮ Then Delayer should get the log n points immediately.
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◮ Prover has already excluded n 2 − 1 holes for pigeon i. ◮ Then it does not matter whether Prover sets xi,j to 0 or 1
◮ Consequently, Delayer gets just 1 point regardless of
n 2 − Ei(α) n 2 + 1 − Ei(α)
n 2 + 1 − Ei(α) .
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◮ If Delayer uses this strategy, then the small clauses
n will not be violated in the game. ◮ Therefore, a contradiction will always be reached on one of
j∈[n] xi,j. ◮ Let us assume now that the game ends by violating
◮ As soon as the number Ei(α) of excluded free holes for
2, Delayer will not leave the
◮ Instead, Delayer will try to place pigeon i into some hole. ◮ If Delayer still answers 0 to xi,j even after Ei(α) > n 2, it must
◮ Therefore, at the end of the game at least n 2 variables have
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◮ At the end of the game at least n 2 variables have been set
◮ We assume that these are the variables xi,ji for i = 1, . . . , n 2.
◮ We calculate the points separately for each pigeon
2. ◮ Distinguish two cases: whether xi,ji was set to 1 by Delayer
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n 2 log( n 2 +1) as a lower bound to the size of each
n .
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◮ For Resolution: Dag-like systems are more powerful
◮ For Frege systems: dag-like and tree-like versions are
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◮ Let A1, . . . , Am be a proof in (dag-like) Frege. ◮ Let
◮ We get linear-size tree-like Frege proofs of
◮ m − 1 applications of Modus Ponens give Am. ◮ The proof is tree-like.
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◮ Goerdt 92: first separation: example with poly-size dag-like
◮ Bonet, Galesi, Esteban, Johannsen 98: first exponential
◮ Ben-Sasson, Impagliazzo, Wigderson 03: simplified and
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◮ pebbling games are played on DAGs ◮ source nodes: in-degree 0 ◮ target nodes: out-degree 0 ◮ game: place pebbles on nodes according to rules ◮ aim: place a pebble at some target node
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◮ Let S be a strategy to pebble the dag G. ◮ P(G, S) = max # of pebbles placed simultaneously on G
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◮ The proof is constructive. ◮ Example: pyramidal graphs
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◮ xv for all v ∈ V ◮ Meaning: xv = 1 if v has been pebbled
u∈N−(v) xu) → xv
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◮ Peb0(G) is unsatisfiable. ◮ But: They have polynomial-size tree-like Resolution
◮ Idea: Start from the bottom and explore the graph in a
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◮ Use pebbles of two different colors: black and white ◮ Consider a node pebbled if it has a black or white pebble
◮ Source nodes can always be pebbled black or white. ◮ For an internal node v, if all its parents are pebbled black
◮ No target node is pebbled black or white.
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◮ xv,c for all v ∈ V and c ∈ {B, W} ◮ Meaning: xv,B = 1 if v has been pebbled black
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◮ Peb(G) is unsatisfiable. ◮ Proof strategy as for Peb0 does not work anymore. ◮ But: They have polynomial-size dag-like Resolution
◮ Aim: Show a lower bound for tree-like Resolution
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log n
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◮ Each run of a SAT algorithm on an unsatisfiable formula
◮ Therefore: each SAT algorithm defines a proof system for
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◮ enhances DPLL by learning new clauses from conflicts ◮ major algorithmic improvement upon DPLL
◮ Pool Resolution [Van Gelder 04] ◮ Resolution trees with lemmas
◮ . . .
◮ All these proof systems are simulated by Resolution. ◮ But: general Resolution is simulated by some of these
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◮ corresponds to running time of algorithms ◮ most studied measure
◮ maximal size of blackboard to carry out proof ◮ corresponds to memory requirements for SAT algorithms
◮ Lower bounds on space
◮ Size-space trade-offs
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◮ So far we concentrated on estimating the length of a proof
◮ But how complicated is it to actually find a proof?
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◮ So far we concentrated on estimating the length of a proof
◮ But how complicated is it to actually find a proof?
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◮ first-order arithmetic theories ◮ weak subsystems of Peano arithmetic ◮ axiomatized by
◮ a number of basic axioms describing the interplay of
+, ·, ≤, 0, 1, . . . and
◮ some controlled amount of induction
◮ I∆0 (induction for all bounded formulas) ◮ PV (formalizes poly-time computations)
◮ S1 2 ⊆ T 1 2 ⊆ S2 2 ⊆ T 2 2 ⊆ . . . ⊆ S2 = T2
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◮ A bounded universal quantifier is of the form
◮ Πb 1-formulas only contain bounded universal quantifiers. ◮ Πb 1-formulas describe coNP-sets.
1-formula ϕ(x) can be translated into a sequence of
◮ ϕn has polynomial size in n; ◮ for each a ∈ N, N |
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◮ For ϕ ∈ Πb 1, if T ⊢ (∀x)ϕ, then there are poly-size P-proofs
◮ T proves the correctness of P, i.e. T ⊢ RFN(P).
2 corresponds to extended Frege EF.
◮ construct short P-proofs (upper bounds); ◮ show lower bounds to the proof size for P [Ajtai 94]; ◮ show simulations between proof systems.
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1 formulas
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1 formulas
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◮ Refined gap theorem for Resolution
◮ Many classically hard formulas become easy. ◮ Bounded-width CNFs have short proofs (fpt size).
◮ New lower bound techniques needed.
◮ strong lower bound for pigeonhole principle
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◮ Non-classical logics such as modal logics, tree logics, or
◮ Yields better understanding of propositional proofs – we
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◮ Non-classical logics are often more expressive than
◮ Satisfiability of the modal logic K is PSPACE-complete
◮ Intuitively, lower bounds to the lengths of proofs in
◮ In contrast to classical logic, we have exponential lower
◮ new phenomena for extensions of Frege (EF vs. SF)
◮ other logics: default logic, autoepistemic logic
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◮ Main objective: understand the complexity of theorem
◮ many strong results in the last decades ◮ many ingenious techniques developed
◮ Computational Complexity: separation of complexity
◮ SAT solving: algorithmic limitations ◮ First-order logic: bounded arithmetic ◮ Proving lower bounds is hard
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◮ understand current algorithmic techniques ◮ find simple compelling proof systems ◮ analyse their strength
◮ use strong proof systems developed and analysed ◮ Cutting planes, Polynomial Calculus, . . .
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◮ understand current algorithmic techniques ◮ find simple compelling proof systems ◮ analyse their strength
◮ use strong proof systems developed and analysed ◮ Cutting planes, Polynomial Calculus, . . .