Transfer Theorems Igor Walukiewicz Bordeaux University 1 / 59
Recursion ≡ stacks 2 / 59
Recursion ≡ stacks F ≡ λ x . if x = 0 then 1 else F ( x − 1) · x . 3 / 59
Recursion ≡ stacks F ≡ λ x . if x = 0 then 1 else F ( x − 1) · x . [Courcelle PhD] : Recursive schemes ≡ deterministic pushdown automata. 4 / 59
Recursion ≡ stacks F ≡ λ x . if x = 0 then 1 else F ( x − 1) · x . [Courcelle PhD] : Recursive schemes ≡ deterministic pushdown automata. Thm [Senizergues]: Equivalence of schemes (in terms of trees they generate) is decidable. Thm [Courcelle]: MSOL theory of trees generated by schemes is decidable. 5 / 59
What about higher-order schemes? Second-order scheme Map ≡ λ f .λ x . if x = nil then nil else f ( hd ( x )) · Map ( f , tl ( x )) Thm [Knapik, Niwi´ nski, Urzyczyn] : Higher-order pushdown automata ≡ higher-order safe schemes Thm [Parys] : Safety is a true restriction Here: On decidability of MSO theory of trees generated by higher-order schemes. 6 / 59
In this talk Consider an operation F on models Transfer property for F For every ϕ one can effectively construct � ϕ , s.t., for every M : F ( M ) � ϕ ϕ. iff M � � We say in this case that F is MSO-compatible . 7 / 59
Transfer theorems Transduction 8 / 59
MSO interpretations Graph with labelled edges: G = � V , { E a } a ∈ Σ � Graph with edge labels from Σ ↓ graph with edge labels form ∆ determined by formulas: { ϕ a ( x , y ) } a ∈ ∆ 9 / 59
MSO-interpretations are MSO compatible. For every ϕ one can effectively construct � ϕ , s.t., for every M : I ( M ) � ϕ ϕ. iff M � � ϕ ≡ ϕ [ ϕ a ( x , y ) �→ E a ( x , y )] � 10 / 59
k -copying Duplicating k -times a graph G = � V , { E a } a ∈ Σ � . 2 k 1 G ′ = � V ′ , { E ′ a } a ∈ Σ , { E i } i ∈ [ k ] � ; where V ′ = V × [ k ] ; E ′ a (( v , i ) , ( w , i )) for ( v , w ) ∈ E a and i ∈ [ k ] ; E i (( v , i ) , ( v , j )) for v , w ∈ V and j ∈ [ k ] . The operation of k -copying is MSO compatible. 11 / 59
MSO-transductions MSO-transduction is a sequence of copying and MSO interpretations Fact: MSO-transduction is MSO compatible. copy copy I I − → − → . . . − → − → M 0 M 1 M 2 M k − 1 M k ϕ 0 ← − ϕ 1 ← − ϕ 2 . . . ← − ϕ k − 1 ← − ϕ k M 0 � ϕ 0 iff M k � ϕ k Example: from one node graph we can construct any finite graph. Remark: Actually it suffices to do one copying and one interpretation. 12 / 59
Transfer theorems Transduction Unfolding (=> Buchi and Rabin Thms) 13 / 59
Unfolding: the tree of all the paths in the graph from a given node. v vv vvv v gives 14 / 59
Unfolding: the tree of all the paths in the graph from a given node. v vv vvv v gives l r l gives lr ll lll llr 15 / 59
Unfolding: the tree of all the paths in the graph from a given node. v vv vvv v gives l r l gives lr ll lll llr Unf ( G , v 0 ) = � V U , { E ∗ a } a ∈ Σ � where V U = paths in G starting from v 0 E ∗ a ( w v , w vu ) if E a ( v , u ) , and w ∈ V U . 16 / 59
Theorem [Courcelle & W.,Muchnik] : Unfolding is MSO-compatible. For every ϕ ( x ) there is (effectively) � ϕ ( v 0 ) such that for every graph G and its vertex v 0 : G � � ϕ ( v 0 ) iff Unf ( G ) � ϕ ( v 0 Remark 1: Unfolding cannot be defined by a transduction. Remark 2: MSO-compatibility of the unfolding implies Büchi and Rabin’s Theorems. 17 / 59
Tree with substitutions: function symbols a , f , g , . . . ; variables x , y , . . . ; and explicit substitutions sub x . eval ( sub x ( s , t )) = s [ t / x ] Theorem [Courcelle & Knapik] : For fixed finite set of variables: eval is MSO-compatible 18 / 59
Transfer theorems Transduction Unfolding (=> Buchi and Rabin Thms) Muchnik Iteration 19 / 59
Stupp iteration ( ) is 0 1 2 00 01 02 10 11 12 St ( G ) = � V + , { E ∗ a } a ∈ Σ , son � where for w ∈ V ∗ , u , v ∈ V : son ( w , w v ) , E ∗ a ( w u , w v ) when E a ( u , v ) . 20 / 59
Remark 1: Stupp iteration of the two node graph gives two full binary infinite trees. Remark 2: Unfolding of a graph may not be definable in the Stupp iteration of the graph. Remark 3: Stupp iteration of the full binary tree is MSO definable in the full binary tree. 21 / 59
Muchnik iteration 0 1 2 00 01 02 10 11 12 G + = � V + , { E ∗ } a ∈ Σ , E # , son � E # ( wu , wuu ) for w ∈ V ∗ and u ∈ V . Theorem [Muchnik,W.] : Muchnik iteration is MSO-compatible. 22 / 59
2 -tree: Muchnik iteration of the full binary tree. 23 / 59
Some things interpretable in k -trees Interpreting n ( n + 1) / 2 in the iteration of the sequence. Some other things interpretable in k -trees [Fratani & Senizergues]: � N , +1 , n √ n � � N , +1 , n log( n ) � � N , +1 , n k 1 , n k 1 k 2 , . . . , n k 1 .. k m � 24 / 59
Transfer theorems Transduction Unfolding (=> Buchi and Rabin Thms) Caucal hierarchy Muchnik Iteration 25 / 59
Caucal hierarchy Level-0: finite graphs Level- k : MSO-transductions of k -tree. Equivalently: Level- k : MSO transductions of unfoldings of Level- ( k − 1) graphs. Cor: All graphs in the Caucal hierarchy have decidable MSO-theory. 26 / 59
Caucal hierarchy is infinite For a function f : N → N we define graph T f : n a a a a b b b b b f ( n ) b Thm [Engelfriet, Carayol & Wöhrle] : T exp k graph is a k -level graph but not ( k − 1) -level graph. Let exp ω ( n ) = exp n ( n ) . Cor: T exp ω graph is not in the Caucal hierarchy but has decidable MSO theory. 27 / 59
Transfer theorems Transduction Unfolding (=> Buchi and Rabin Thms) Caucal hierarchy Muchnik Iteration Machine characterization HPDA 28 / 59
General idea A graph of configurations of a machine: nodes are configurations of the machine; edges represent a step of the computation. Finite automaton: its graph of configurations is just graph of the automaton Pushdown automaton: nodes qa 1 . . . a k edges qa w → q w or qa w → qba w . 29 / 59
Configuration graph of a pushdown automaton is interpretable in a tree Cor: It has decidable MSO-theory Rem: Turing Machine graphs may have undecidable MSO-theory. 30 / 59
2 -nd order stack: example A 2 -stack is a stack of stacks. [ a 1 1 . . . a 1 k 1 ][ a 2 1 . . . a 2 k 2 ] . . . [ a n 1 . . . a n k n ] New operation of copying the top-most stack: q [ w 1 ] . . . [ w i ] → q [ w 1 ][ w 1 ] . . . [ w i ] . A system where all paths are of the form q k 1 q k 2 q k 3 . Remark: The 2 -stack gives additional power. Remark: The above automaton recognizes { a k b k c k : k ∈ N } . 31 / 59
Higher order pushdowns ≡ Caucal hierarchy Configuration graph of a pushdown automaton is interpretable in a tree Configuration graph of a k -pushdown automaton is interpretable in a k -tree. Cor : All these graphs have decidable MSO-theory. Thm [Carayol & Wöhrle] : Graphs of Caucal level k are configuration graphs of k -th order pushdown automata. (when ε -transitions are contracted). 32 / 59
Transfer theorems Transduction Unfolding (=> Buchi and Rabin Thms) Caucal hierarchy Muchnik Iteration Machine characterization HPDA 33 / 59
Languages, Schemes Higher-order pushdowns + Ianov’58 “The logical schemas of algorithms” + Park PhD’68 Recursive schemes + Aho’68 indexed languages + Scott, Elgot abstraction Program Scheme + Maslov’74 ’76 higher-order indexed solution in a languages and higher order pushdown automata. free algebra Interpretation Meaning In fi nite tree + Milner’73 Plotkin’77 PCF + Courcelle’76 for trees: 1-st order schemes=CFL + Engelfriet Schmidt’77 IO/OI + Damm’82 for languages: rec schemes= higher-order pusdowns + Kanpik Niwinski Urzyczyn’02 Safe schemes = higher-order pusdown + Senizergues’97 Equivalence of 1st order schemes is decidable +Statman’04 Equivalence of PCF terms is undecidable +Loader’01: Lambda-definability is undecidable 34 / 59
Simply typed λ -calculus with fixpoints Types: 0 is a type, and α → β is a type if α, β types. Constants: c α of type α . Terms: c α , x α , λ x α . M . MN , Example: c , d : 0 , g : 0 → 0 , f : 0 → 0 → 0 f ( gc ) d : 0 λ x . f ( gx ) d : 0 → 0 λ x . f f g g d d c x λ z . z ( gc ) d : (0 → 0 → 0) → 0 λ z . z g d c 35 / 59
β -reduction: ( λ x . M ) N = β M [ N / x ] ( λ x . f ( gx ) d ) c → β f ( gc ) d ( λ z . z ( gc ) d )( λ xy . y ) → β ( λ xy . y )( gc ) d → β d Substitution is as in logic: one should avoid variable capture ( λ h .λ x . g ( hx ))( fx ) → β λ y . g ( fxy ) and not λ x . g ( fxx ) f : 0 → 0 → 0 , g , h : 0 → 0 36 / 59
A Böhm tree of a term M : We reduce M to head normal form: M → ∗ β λ� x . KN 1 . . . N i with K a variable or a constant. BT ( M ) is λ x . K . . . BT ( N 1 ) BT ( N i ) Böhm tree of ( λ y . g ( hxy )) is λ y . g h y x 37 / 59
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