Responsable : Jean Goubault-Larrecq (LSV, ENS Cachan).

Lecturers: Philippe Schnoebelen then Jean Goubault-Larrecq for the course, and Guillaume Scerri for the exercise sessions (TD).

For the first half (until partial exam on November 9th) the lectures take place in room 1-I-82 at ENS Paris-Saclay each Wednesday, from 09h30 to 11h30. They are followed by exercise sessions each Friday from 14h00 to 16h00.

Planning: début du cours le 21 sept 2022. Début de la 2ème partie du cours: 16 nov.

Evaluation: Two written exams (see exams from previous years at bottom of page). The partial exam took place on Nov. 9th, 2022. Here is a version with some answers.

Final exam : Jan. 11th, 2023.

La théorie de la complexité va bien au-delà de celle de la NP-complétude. Le but de ce cours est d'aller regarder un certain nombre d'autres constructions fondamentales de la théorie de la complexité: complexité en espace, notions de machines alternantes, ou randomisées. On y verra quelques théorèmes fascinants: l'équivalence du temps alternant et de l'espace déterministe par exemple, ou le théorème IP=PSPACE de Shamir.

The course is based on the following lecture notes polycopié (1st part) and polycopié (2nd part). (Note: il semble que le premier poly ait des problèmes de polices sous Acrobat Reader 9; aucun problème connu sous xpdf ou evince.)

After each class, this section will be updated with a summary of the material actually covered. Numbers, e.g. ”(2.1)”, refer to statement in the lecture notes.

What we covered in class:

• Introduction. Deterministic and nondeterministic Turing machines
• Def (2.1) TIME(f), SPACE(f), NTIME, NSPACE, .. P, NP, L, NL, ..
• Compression Thm (2.2): SPACE(O(f(n))) ⊆ SPACE(f(n)) and NSPACE(O(f(n))) ⊆ NSPACE(f(n))
• For proper function f, NTIME(f(n)) ⊆ SPACE(f(n)) and NSPACE(f(n)) ⊆ TIME(2^{(f(n)+log n)})
• GAP (aka REACH), a problem on directed graphs
• How do we see it as a language GAP⊆Σ^*?
• (2.3) GAP is in NL, i.e. NSPACE(log(n))

The exercise sheet used on Sept. 23rd is available here.

What we covered in class:

• (2.6) GAP is in SPACE(log²(n))
• A k-worktape (N)TM in space f(n) running on input x visits at most ≈ |Q| × |Σ|^f(n) x n × f(n)^k different configurations, i.e. O(n) x 2^O(f(n))
• (2.4) NL ⊆ P (aka PTIME) and NL ⊆ SPACE(log²n)
• (2.9) Thm (Savitch 1970) NSPACE(f(n)) ⊆ SPACE(f²(n)) when f is proper & ≥log
• (2.10) Hence NPSPACE=NSPACE(n^O(1))=SPACE(n^O(1))=PSPACE
• (2.12) Def. logspace reductions, complete problems & basic properties
• GAP is NL-complete

The exercise sheet used on Sept. 30th is available here.

What we covered in class:

• What is co𝒞 and coL for 𝒞 a class of problems (or languages) and L a problem.
• (2.16) coGAP is in NL, hence (2.17) coNL = NL (Immerman-Szelepcsényi)
• SAT is NP-complete (Cook)
• QBF is PSPACE-complete
• HORNSAT and CIRCUIT-VALUE are P-complete

The exercise sheet used on Oct. 7th is available here.

What we covered in class:

• (3.1;3.2) Alternating Turing machines. The classes ATIME(f) & ASPACE(f) are closed under complementation.
• (2.21) Thm (Stockmeyer-Meyer 1973) QBF is PSPACE-complete and APTIME-complete, hence PSPACE=APTIME.
• (3.8) Padding lemma, entailing (3.9) SPACE(poly(f(n)) = ATIME(poly(f(n)) for any proper f s.t. f(n) ≥ n for large enough n.
• This yields e.g. AEXPTIME=EXPSPACE.
• (3.18) CIRCUIT_VALUE is PTIME-complete & ALOGSPACE-complete, entailing PTIME=ALOGSPACE, i.e. P=AL,
• (3.19) more generally TIME(2^O(f(n)) = ASPACE(f(n)) with f as above, e.g., EXPTIME=APSPACE.

The exercise sheet used on Oct. 14th is available here.

What we covered in class:

• TM machines that access an oracle O (can be any language ⊆A^*). In a run of M^O, there is a special query tape where the TM might write a query q. When M enters the querying state, q is erased and the TM jumps to state s_yes or s_no depending on whether q∊O. When measuring space usage, the query tape in counted.
• Def: (given a language O) ℳ^O for a class ℳ of machines, e.g. P^O, NP^O, etc., similarly P^𝒞, NP^𝒞, etc. for a class 𝒞 of oracles.
• Some problems in P^SAT: UNSAT, SAT-UNSAT, UNIQ-SAT (Boolean formulae with only one satisying valuation), PARITY-SAT ≝ {(𝜑1,...,𝜑n) such that the first satisfiable 𝜑i has odd index i}, MIN-SAT ≝ {satisfiable Boolean formulae 𝜑(x1,..,xn) such that the lexicography first satisfying valuation v_min has v_min(xn)=true}, etc.
• Def (the polynomial hierarchy): PH ≝ all levels Σ_k with Σ₀≝Δ₀=Θ₀=P(=PTIME) and Σ_k+1≝NP^Σ_k, Δ_k+1≝P^Σ_k, Θ_k+1≝P^{Σ_k[O log n]}, and Π_k≝co(Σ_k).
• Facts: Δ₁=Θ₁=P and Σ₁=NP. Also Π_k=(coNP)^Σ_k where coNP seen as a class of machines comprises all polynomial-time TMs with only universal states.
• Thm: (Σ_k-1 ∪ Π_k-1) ⊆ Θ_k ⊆ Δ_k ⊆ (Σ_k ∩ Π_k).
• Thm: Σ_k ⊆ PSPACE^PSPACE = PSPACE for all k, hence PH ⊆ PSPACE.
• Thm: If PH=PSPACE then PH has a complete problem. If PH has a complete problem then PH coincides with some Σ_n level (we say “PH collapses”).
• Some complete problems for specific levels in PH: the ∃^*∀^* fragment of QBF is Σ₂-complete, in fact the ∃^*∀^*∃^*∀^*.. (∃^*/∀^*)_k fragment is Σ_k-complete; PARITY-SAT is Θ₂-complete; MIN-SAT and UNIQ-TSP (graphs with a unique Hamiltonian circuit of minimal length) are Δ₂-complete.
• Thm: Θ_k+1 = P_‖^Σ_k, where polynomially many queries can be submitted to the oracle, but they have to be computed in advance so that they can all be answered “in parallel”, in just one round.
• Def (the Boolean hierarchy): BH ≝ all levels BH_k with BH₁≝NP, BH_2k≝BH_2k-1⋀coNP, and BH_2k+1≝BH_2k⋁NP. DP is another name for BH₂, i.e. NP⋀coNP. Recall that 𝒞⋀𝒞' is defined as {L∩L'|L∊𝒞 and L'∊𝒞'}, so that one has 𝒞∩𝒞' ⊆ 𝒞,𝒞' ⊆ 𝒞⋀𝒞' (NB: the last inclusion assumes that 𝒞 and 𝒞' contain the universal languages A^* for any alphabet). There is a similar definition for 𝒞⋁𝒞.
• SAT-UNSAT and UNIQ-SAT are DP-complete, where DP=NP⋀coNP. There exist complete problems for each level of BH. If BH itself has a complete problem, then it collapses to some level BH_k.
• Facts: BH = P^NP[O(1)] hence BH ⊆ Θ₂. If BH=Θ₂ then BH collapses.

The exercise sheet used on Oct. 21st is available here.

What we covered in class:

• Two counting problems #Cycle (≝ count the number of simple cycles in a given graph) and #SAT (≝ count the number of valuations satisfying a given boolean formula).
• #SAT is “difficult”, indeed if #SAT ∊ FP then P=NP. Perhaps #Cycle is easier, perhaps in FP (≝ the functions computable in polynomial-time).
• Fact: if #Cycle ∊ FP then in that case too P=NP. Proof is by reduction from HamiltonianGraph.
• Def. f ∊ #P iff f(x) is the number of accepting paths of a polytime NDTM on x. We can assume that all configurations of the TM have exactly 2 following configurations (or 0) and that all runs have length exactly p(n). Thus #P contains all the functions that counts the “solutions” to NP problems. E.g. #SAT and #Cycle belong to #P.
• Obviously if #P = FP then NP = P. We don't know whether the reverse implication holds. However if PSPACE = P then #P = FP.
• NB. #P is a class of functions but there are classes of decision problems based on counting accepting runs: PP, ⊕P.
• Def. f is #P-complete iff f in #P and g ≤ f for all g in #P. What reductions do we consider? Several versions of parsimonious reductions can be used. Or a more generous g ≤ f iff g in FP^f.
• Thm. #SAT is #P-complete under parsimonious reductions.
• Computing the permanent of a matrix is related to counting the cycle-covers of a directed graph. The definition looks like computing the determinant. However we showed #SAT ≤ PERMANENT ≤ PERMANENT(0/1) so PERMANENT is #P-complete [Valiant 1979]. See wikipedia for a variant proof.

The exercise sheet used on Oct. 26th is available here.

Holidays ! no class this day

The slides of the second part of the course:

• Randomized Turing machines, RP, coRP, ZPP: covers Sections 1.1 and 1.2 of the polycopié (2nd part); the shorter version, and the video (November 4th, 2020);
• BPP (part 1): covers Section 1.3 and the Sipser-Gács-Lautemann theorem (Proposition 1.24) of the polycopié (2nd part); the shorter version, and the video;
• BPP (part 2): covers Section 1.4 (except for the Sipser-Gács-Lautemann theorem) of the polycopié (2nd part); the shorter version, and the video;
• Arthur vs. Merlin games (part 1): covers Section 3 up to and including Section 3.1 of the polycopié (2nd part); the shorter version, and the video;
• Arthur vs. Merlin games (part 2): covers Section 3.2 of the polycopié (2nd part); the shorter version, and the video;
• Sipser's coding lemmas, and consequences: covers Section 3.3 and Section 3.4 of the polycopié (2nd part); the shorter version; the video, covering this, plus the beginning of the next series (including ABPP⊆IP⊆PSPACE);
• Shamir's theorem: covers Section 3.5 of the polycopié (2nd part); the shorter version; and the video, covering the hard direction PSPACE⊆ABPP.
• NP=PCP and hardness of approximation: covers a good deal of Section 2 of the polycopié (2nd part); the shorter version.

The exercise sheets for the second part of the course:

• november 16th
• november 23rd

The class will be taught in English. Le polycopié est en français et les élèves peuvent poser des questions, intervenir, etc. en français.

Les questions des sujets d'examen seront en anglais. Les étudiants seront autorisés à rédiger leur copie d'examen en français ou en anglais.

Students are expected to be familiar with the concepts and formal definitions for Turing Machines, the class P (problems decidable in deterministic polynomial time), the class NP (non-deterministic polynomial-time), Cook's Theorem (SAT is NP-complete). These notions will only be very quickly recalled during the first classes.

Complexité et Calculabilité, Cryptologie.

Livres:

• Christos H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
• Sanjeev Arora and Boaz Barak. Computational Complexity. Cambridge University Press, 2009.
  (une pré-version est gracieusement disponible ici http://www.cs.princeton.edu/theory/complexity/)

• Sylvain Périfel. Complexité algorithmique. Ellipses, 2014. Disponible en ligne.

L'examen final 2021-22, en temps limité. Voici un corrigé.

L'examen final 2022-23, en temps limité. Voici un corrigé.