Automata and Logic

After discussing some organisational matters we gave some motivation and context for the topic of the lecture. Thereafter we very quickly recalled some facts about regular languages and finite automata (slides). We then started to consider finite monoids and their connection to regular languages. In particular we have shown that a language is regular if and only if it is accepted by a finite monoid.

Quiz

• Is every language accepted by some monoid?
• Why is Σ* called the free monoid over Σ?
• Why is the mapping φ in Proposition 1.8 a homomorphism?

We defined the syntactic congruence of a language, and showed that the syntactic monoid of a regular language L is isomorphic to the quotient monoid of the syntactic congruence of L. We considered an example where we computed the syntactic monoid of a regular language. We also showed that not every finite monoid is isomorphic to the syntactic monoid of a regular language. Finally, we defined the class of languages defined by classes of monoids, and we introduced the notion of an M-variety.

Quiz

• Why is the quotiont monoid well-defined?
• Why is the set N from Corollary 1.13 well-defined?
• Is the empty set an M-variety? Is the class of all finite monoids an M-variety?

We defined equations, sequences of equations, and considered classes of finite monoids ultimately defined by sequences of equations. By a theorem of Schützenberger and Eilenberg these are exactly the M-varieties. We also showed that the classes of languages defined by M-varieties are closed under Boolean operations. We noted that instead of monoids we can also consider semigroups, essentially obtaining the same theory. (slides) Finally, we started to look into logics and languages defined by logical formulas.

Quiz

• Is every submonoid of a finite group a group?
• Let V be a class of finite monoids such that there exists a countable set K of equations such that a monoid belons to V iff it satisfies all but finitely many equations of K. Is then V an M-variety?
• Why is it sometimes reasonable to consider semigroups instead of monoids?
• Is the set Σ* definable by some formula φ? Is the empty set definable by some formula φ?

We finished our general considerations about how logics can be used to define languages by means of some examples. In particular, we discussed (but did not show) that a(aa)* is not FOL-definable, but definable by some monadic second-order formula. We then started to investigate the example of generalized-definite languages, and show that this class coincides with the Boolean closure of all languages of the form uΣ* and Σ*u' for u, u' ∈ Σ. After this we considered idempotent elements in semigroups. In particular we showed that every finite semigroup has an idempotent element. We then introduced the class 𝔻̂ of all finite semigroups S such that eSe = e is true for all idempotent e ∈ S. We showed that 𝔻̂ is ultimately defined by some sequence of equations, and is thus an S-variety. Finally, we discussed some special properties of semigroups in 𝔻̂.

Quiz

• Why are finite languages always generalized-definite? (find a direct argument here, without using Lemma 2.2) What is one possible constant?
• Why is 𝔻̂ defined to be a class of semigroups, and not of monoids? Put another way: which monoids M satisfy eMe = e for all idempotent elements e ∈ M?
• Is the class of all finite cyclic groups an M-variety?
• Let S be a finite commutative semigroup. Show that the set of all idempotent elements in S then forms a non-empty subsemigroup of S.

We showed that the class 𝔻̂ of semigroups defines exactly the class of generalized-definite languages. From this we immediately obtained the decidability of the problem to determine whether a finitely-represented regular language is generalized-definite or not. We also showed that a language L is generalized-definite if and only if there exists some closed quantifier-free formula φ over <, =, P₁, ..., Pₖ, min, max, s, p, such that L\{ε} = L(φ). We then started to look into star-free languages, considered some examples, and introduced aperiodic monoids.

Quiz

• What is the syntactic semigroup of the empty language ∅? What is the syntactic semigroup of the complete language Σ*?
• What is L(p^n(max) < s^m(min))?
• Is a(bc)* star-free?
• Which groups are aperiodic?
• "The empty set never causes trouble": which languages are accepted by the empty semigroup?

We started the lecture by showing that finite monoids are aperiodic if and only if all groups contained in the monoid are trivial. Thereafter we stated (but did not prove) Schützenberger's Theorem, which yields that the star-free languages are exactly those accepted by aperiodic monoids. Using this theorem we were able to show that a(aa)* is not star-free. Finally, we started proving that the star-free languages (without the empty word) are exactly those definable by closed first-order formulas.

Quiz

• Is (aaa)* star-free?
• Why is it necessary to demand in the relativization of a formula φ to the predicate ≤ z that the variable z does not occur in φ?
• In the proof of Corollary 3.14, can we omit the disjunction of the negated formulas ¬ψ?

We finished the proof of Proposition 3.9, 2 ⇒ 1. The only thing that was left to complete the proof was then to show Proposition 3.12 and Proposition 3.16. The proof of the former we did directly, and for the later we started to consider Ehrenfeucht-Fraïssé Games. We introduced games of length n (with and without already played moves), introduced the notion of a winning strategy for Player II, and defined the relation ~_{k,n} to say that Player II has a winning strategy for the games of length n on two words where k moves have already been played. We showed that the relation ~_{k,n} is an equivalence relation.

Quiz

• What is the exact argument why we can restrict our attention in the proof of Proposition 3.12 to the case that φ has only free variables in {y₁, …, yₖ}?
• What are the equivalence classes of ~_{0,0} on Σ*?

We proved that two structures satisfy the same first-order formulas with k free variables and quantifier-depth at most n if and only if there exists a winning strategy for Player II on the game on those words with k moves already played and n moves left to play. As an application of this we then showed that it is decidable whether a first-order formula together with the theory of linear orders has a finite model, thus ending Chapter 3. We then started Chapter 4, and introduced infinite words, some notation and some operations on infinite words.

Quiz

• In the base case of the proof of Lemma 3.25, in the second step, why can we infer the validity of (**) b) from the validity of (**) a) and (**) c)?
• In the definition of the formulas ψ_{t_{k+1}}, why can we assume that (u,s) satisfies ψ_{t_{k+1}}, but (v,t) does not (and not the other way around)?
• Does there exist a first-order formula that is true in all finite words, but not in any infinite word?
• What is ε^ω?
• What is the limit of a*ba*?

We introduced Büchi-automata as an automata model for ω-languages. After considering some example we examined the structure of languages accepted by Büchi-automata and showed some simple closure properties. We showed that the emptiness of the language accepted by Büchi-automata is decidable. After this we discussed how to decide equality of Büchi-recognizable languages. We noted that the idea from the corresponding proof of regular languages does not work, since Büchi-automata (among others) are not necessarily determinizable.

Quiz

• What is the difference between finite automata and Büchi-automata?
• Which of the following words satisfies the condition that there is always an even number of 'b' and 'c' between every two occurrences of an 'a': aabba, abcbcabcba, abcbcabcb?
• We claimed that for a regular language U there always exists an automaton accepting U that has a single initial state that is not reachable from any state. Why is this true?
• Is (a^*)^ω = a^ω?

We finished our discussion from yesterday by showing that for deterministic Büchi-automata exchanging final and non-final states does not necessarily lead to a Büchi-automaton accepting the complement of the original language. We then showed that an ω-regular language is accepted by a deterministic Büchi-automaton if and only if it is the limes of a regular language. We also noted that ω-regular languages accepted by deterministic Büchi-automata are not closed under complement. We then started to proof that ω-regular languages are closed under complement. For this we introduced a relation ~_A with the goal to show that the ω-regular language accepted by A, as well as its complement, can be written as a finite union of languages UV^ω for regular U and V, which itself are equivalence classes of ~_A. We showed that ~_A is a congruence with finite index. To show the desired decomposition, we introduced graph-colorings and the Ramsey's Theorem. We proved Ramsey's Theorem.

Quiz

• In the proof of Proposition 4.13, where did we use the fact that the Büchi-automaton is deterministic?
• What is the general idea to show that ω-regular languages are closed under complement?
• Why are the languages L_{p,q}^F regular?
• A hypergraph is a graph where an edge can contain more then two vertices. Restate Ramsey's Theorem in terms of colorings of hypergraphs on infinite sets of vertices.

We proved Proposition 4.19 using Ramsey's Theorem, finishing the proof that ω-regular languages are closed under complement. Then we observed that this proof is actually effective, giving us a procedure to construct for a given Büchi-automaton A a Büchi-automaton B that recognizes the complement of the language recognized by A. From this we obtained the decidability of the equivalence problem of ω-regular languages. We then briefly considered Muller-automata, and we sketched the proof of McNaughton's Theorem stating that ω-regular languages are exactly those ω-languages accepted by deterministic Muller-automata. With this we finished Chapter 4. We then started Chapter 5 by introducing monadic second-order logic of one successor (S1S). We considered some examples and showed that we can dispense with the symbols \underline{0} and <.

Quiz

• Why can one effectively construct finite automata for the equivalence-classes of ~_A?
• In the proof-sketch of McNaughton's Theorem, why are the languages lim L_{q₀, q} exactly the sets of all words labeling infinite paths that visit q infinitely often?
• If A = (Q, Σ, q₀, δ, \mathcal{F}) is a deterministic Muller-automaton, then why is B = (Q, Σ, q₀, δ, \mathcal{P}(Q)\setminus\mathcal{F}) a deterministic Muller-automaton accepting the complement of the language accepted by A?
• Is it true that

  x < y iff ∃X.(X(x) ∧ ¬X(y) ∧ ∀z.(¬X(z) ⇒ ¬X(s(z)))) ?

We showed that ω-regular languages are exactly those definable by S1S-formulas over infinite words. Parts of the proof we illustrated by an example. As the proof was constructive we obtain that validity for S1S-formulas over infinite words is decidable. By adapting the proof we also obtained that regular languages are exactly those definable by S1S-formulas over finite words. In particular, decidability of finite validity for S1S-formulas is decidable.

Quiz

• Define an S1S₀-formula Empty(X) saying that X has to be interpreted as the empty set.
• In the proof of Proposition 5.4, why do we obtain an automaton for the ω-language defined by ∃Y.ψ(Y,X₁,…,Xₙ) by dropping the first component in every transition of the automaton A accepting ψ(Y,X₁,…,Xₙ)?

We introduce weak monadic second-order logic WS1S as a variation of monadic second-order logic where second-order quantifiers only range over finite subsets of ℕ. We used this logic to investigate decidability of Presburger Arithmetic, which is the set of all FOL-formulas with equality over \underline{0} and + (no successor-function!) that are valid in (ℕ, 0, +). Thereafter we investigated the generalization of star-free languages to infinite words. We noted that those languages are exactly those ω-languages that are definable by closed S1S-formulas without second-order quantifiers. We also introduced a suitable generalization of the syntactic congruence ≈_L for infinite words and noted an ω-regular language is star-free if and only if the monoid Σ*/≈_L is aperiodic. We applied this result to show that an example ω-language is not first-order definable.

Quiz

• Does the proof of Theorem 5.11 still work if we encode the number in base 3 instead of base 2? How about base 1?
• Can we omit in the definition of φ_+(X,Y,Z) the condition that R does not have to contain 0?
• In Example 5.18 we did not make use of the letter c. Can this letter be omitted?

We started this lecture by correcting an error about the complexity of deciding Presburger Arithmetic. We noted that there exists c > 0 such that every algorithm deciding Presburger Arithmetic has to run in time more than 2^2^{cn} for n being the length of the input formula. We also noted some interesting connection of Presburger Arithmetic to Gödel's First Incompleteness Theorem. After that we started to investigate dynamics logics, and in particular the logic 1DPDL that allows us to talk about properties of configurations obtained by one deterministic program p. We motivated this logic by means of some examples, and then introduced the syntax and semantics of 1DPDL. Finally we started to investigate a method to decided satisfiability of 1DPDL configuration formulas.

Quiz

• Express the following fact in 1DPDL: if A₁ holds in some configuration, then by a finite number of applications of p we reach a configuration in which A₂ holds, and from which any number of even applications of p reaches a configuration in which A₁ does not hold.
• In the interpretation of example after the definition of the semantics of 1DPDL, what is the set of configurations that satisfy [p*](A₁ ∧ ¬A₁)?

We continued our argumentation from last week to decide satisfiability of configuration formulas. For this we introduced the Fischer-Ladner-Completion of a configuration formula φ̂₀. From this set of formulas we defined words α(k₀) for some k₀ ∈ (φ₀)^I. Then we introduced Hintikka-Words for φ₀ and showed that φ₀ is satisfiable if and only if there exist Hintikka-Words for φ₀.

Quiz

• Would the argumentation of today's lecture also work if one defines the alphabet Σ to be the set of all proper subsets of the Fischer-Ladner-Completion FL(φ₀)?

The goal of this lecture was to find a Büchi-automaton that accepts exactly the Hintikka-words of a configuration formula φ₀. Such a Büchi-automaton would then immediately yield a decision procedure for the satisfiability of φ₀. This is because φ₀ is satisfiable iff there exists a Hintikka-word for φ₀ iff the Büchi-automaton accepts a non-empty language, and the last fact is decidable (in linear time). To construct such a Büchi-automaton we first considered the local automaton of φ₀ that replaced the condition 6 of the definition of Hintikka-words by a weaker local condition 6'. To ensure that the local condition 6' is equivalent to the global condition 6, we augmented the local automaton to obtain the Hintikka-automaton. Because this automaton uses the Büchi-acceptance condition we were able to show that a word is accepted by the Hintikka-automaton of φ₀ iff it is a Hintikka-word for φ₀.

Quiz

• Why is the size of the Hintikka-automaton linear in the size of the configuration formula?

We started Chapter 6 on automata on finite trees. We began by defining what we mean by a finite tree, and we considered some examples. We noted in particular that every finite word can be considered as a finite tree. We then introduced the notion of a LR-tree automaton (leaf-to-root), considered a small example, and noted that non-deterministic and deterministic LR-tree automata accept the same class of tree languages. We also considered RL-tree automata (root-to-leaf), and showed that non-deterministic LR-tree automata and RL-tree automata accept the same class of languages. We closed the lecture with a small example illustrating the proof of this claim.

Quiz

• Are runs of LR-tree automata (or RL-tree automata) itself trees?
• Why is the domain of a tree prefix-closed?
• How many subtrees can a tree have?
• How many paths does a tree have?

We started this lecture by showing that deterministic RL-tree automata are strictly weaker than non-deterministic RL-tree automata. Thereafter we defined the notion of recognizable tree languages as those languages that are recognized by LR-tree automata. We showed that there exist non-recognizable tree languages. We then started our endeavor of finding a suitable notion of regular tree languages. For this we first investigated closure properties of the class of recognizable languages. We then introduced the notion of concatenation of tree languages, and showed that the class of recognizable languages is closed under concatenation. We also introduce a suitable generalization of the Kleene-Star for tree languages, and showed that recognizable languages are also closed under this operation.

Quiz

• Suppose the alphabet Σ is such that T_Σ is finite. Do there then exist non-recognizable tree-languages over Σ?

We introduced the set Reg(T_Σ, Z) and defined the notion of a regular tree-language over Σ. We then showed that this set coincides with the set of recognizable tree-languages of Σ. We also showed that regular/recognizable tree-languages are closed under alphabet renaming. Finally, we noted that the emptiness problem for regular tree-languages is decidable. We then turned our attention to automata on infinite trees. For this we introduced the notion of an ω-tree over Σ, and we introduced the infinite iteration of concatenation of finite tree languages. We also introduced Büchi- and Rabin tree-automata.

Quiz

• Can one represent every tree-concatenation be represented in terms of a concatenation at a single position and applying an n-ary symbol to some tree-languages (cf. before Def. 6.17)?
• How does the proof of Prop. 6.22 look like in the case of words (i.e., when we consider words as trees)?
• Can a tree obtained by the "ω-iteration" from Def. 7.1 be a limit of more than one sequence?

We started the lecture by showing that every Büchi-recognizable language is also Rabin-recognizable (this was simple). Then we considered languages L₁ and L₂ and showed that L₁ is Büchi-recognizable, and that L₂ is Rabin-recognizable but not Büchi-recognizable. Since L₂ was the complement of L₁ we thus showed that Büchi-recognizable languages are not closed under complement. We remarked that Rabin-recognizable languages are indeed closed under complement, but we did not go into the proof. We instead gave an intuitive argument why the proof must be complicated. After that we started to investigate decidability of the emptiness problem of Büchi tree-automata. We did not start the proof yet, but we proved an auxiliary result, namely König's Lemma.

Quiz

• In Example 7.6 the Rabin tree-automaton had as acceptance condition the set {({i,f}, {f})}. What happens if one removes the f from the first component, leaving {({i}, {f})} as acceptance condition?
• How does König's Lemma prove that the tree D_u is finite?
• Right or Wrong: every infinite and connected graph where every node has only finite degree contains an infinite simple path that visits every node in the graph?

We finished the proof on the decidability of the emptiness problem for Büchi tree-automata. For this we showed a characterization of Büchi recognizable languages that is analogous to the one of the word case (compare Prop. 4.8 and Prop. 7.11). Based on this observation we showed that we can decided emptiness of a Büchi tree-automaton by first successively removing all states that cannot occur in a successful run, and then checking whether some initial state is left. Then we started to proof that emptiness of Rabin tree-automata is decidable. For this we introduced the notion of an active state.

Quiz

• We observed that if ℓ(u) = q, then t̂_u ∈ L_q. Why is this true?
• In the proof of Prop 7.10 we encountered the expression

  {t₀} ·^(f₁,…,fₘ) ({t₁}, …, {tₘ})^{ω,(f₁,…,fₘ)}.

  How many trees are contained in this set?

We finished the proof that emptiness of Rabin tree-automata is decidable and we also gave a summary of the resulting decision procedure.

In the last chapter of this lecture we turned out attention to logics over infinite binary trees. We introduced the logics S2S and WS2S as direct generalizations of S1S and WS1S to the case of two successors. After considering some examples we sketched the proof of the equivalence of Rabin-recognizability and definability in S2S. We immediately obtained the decidability of validity in S2S. We also noted Rabin's characterization of Büchi-recognizability and of definability in WS2S. In particular we noted that WS2S is strictly less expressive than S2S over infinite trees.

Quiz

• Do we need to replace the constant 0̲ by the constant ε̲ in the definition of S2S-formulas?
• Why is the prefix-closure of a finite set of nodes also finite?

We considered the logic DPDL, which is a generalization of 1DPDL to the case of multiple deterministic programs. We showed that a similar argumentation to the case of 1DPDL also yields that validity in DPDL is decidable in exponential time. For this we noted that emptiness of Büchi-tree automata can be decided in polynomial time. We also considered the logic PDL that drops the requirement of the atomic programs to be deterministic. We showed by a nice reduction to the satisfiability in DPDL that satisfiability in PDL is decidable in exponential time as well.