Levels of Undecidability in Infinitary Rewriting: Normalization and Reachability

Lev els of Undecidabilit y in Infinitary Re writing: Normalization and Reac habilit y J¨ org Endrullis F ree Universit y Amsterdam, The Netherlands joerg@few. vu.nl Abstract. In [ 2 ] i t has been sho wn that infinitary strong normalization ( SN ∞ ) is Π 1 1 -complete. Suprisingly , it turn s out that infinitary weak nor- malization ( W N ∞ ) is a h arder problem, b eing Π 1 2 -complete, and thereby strictly higher in the analyt ical hierarch y . W e assume familiarity with infinitary term rewr iting; w e further reading we refer to [5,4]. 1 Infinitary Strong Normalization and Reac habilit y Definition 1. A T uring machine M is a quadruple h Q , Γ , q 0 , δ i c onsisting of: – finite s e t of states Q , – an initial state q 0 ∈ Q , – a finite alphab et Γ containing a designated symbol ✷ , called blank , and – a par tial tr ansition function δ : Q × Γ → Q × Γ × { L , R } . A c onfigur ation o f a T uring machine is a pa ir h q , t ap e i consisting of a state q ∈ Q and the tap e conten t tap e : Z → Γ such that the ca r rier { n ∈ Z | tap e ( n ) 6 = ✷ } is finite. The set of all configurations is denoted C onf M . W e define the r elation → M on the set of configura tions C onf M as follows: h q , tap e i → M h q ′ , tap e ′ i whenever: – δ ( q , tap e (0)) = h q ′ , f , L i , tap e ′ (1) = f a nd ∀ n 6 = 0 . tap e ′ ( n + 1) = tap e ( n ), or – δ ( q , tap e (0)) = h q ′ , f , R i , tap e ′ ( − 1) = f and ∀ n 6 = 0 . tap e ′ ( n − 1) = tap e ( n ). Without loss of gene r ality we assume that Q ∩ Γ = ∅ , that is, the set of states and the a lphab et are disjo in t. This e nables us to denote co nfigurations as h w 1 , q , w 2 i , denoted w − 1 1 q w 2 for shor t, with w 1 , w 2 ∈ Γ ∗ and q ∈ Q , which is shorthand for h q , tap e i wher e tap e ( n ) = w 2 ( n + 1) for 0 ≤ n < | w 2 | , and tap e ( − n ) = w 1 ( n ) fo r 1 ≤ n ≤ | w 1 | and tap e ( n ) = ✷ for a ll other pos itio ns n ∈ Z . The T uring ma chines we consider are deterministic. As a conseq ue nc e , final states are unique (if they e xist), which justifies the following definition. Definition 2. Let M b e a T uring ma chine and h q , t ap e i ∈ C onf M . W e denote by final M ( h q , tap e i ) the → M -normal form of h q , tap e i if it exists and undefined, otherwise. Whenever fin al M ( h q , tap e i ) exists then we say that M halts on h q , tap e i with fi n al c onfigura tion final M ( h q , tap e i ). F urthermo r e we say M halts on t ap e a s shorthand for M halts on h q 0 , tap e i . T uring machines ca n compute n -a ry functions f : N n → N o r relations S ⊆ N ∗ . W e need only unary functions f M and binary > M ⊆ N × N rela tions. Definition 3. Let M = h Q, Γ , q 0 , δ i be a T uring machine with S , 0 ∈ Γ . W e define a partial function f M : N ⇀ N for all n ∈ N by: f M ( n ) = ( m if final M ( q 0 S n 0 ) = . . . q S m 0 . . . undefined otherwise and for M to tal (i.e. M ha lts on all tap es) we define the binary r elation > M ⊆ N × N by: n > M m ⇐ ⇒ final M ( 0S n q 0 S m 0 ) = . . . q 0 . . . . Note that, the set { > M | M a T uring machine that halts on all tap es } is the s et of recurs ive binar y r elations on N . W e use the tr anslation of T ur ing machines M to TRSs R M from [3]. Definition 4. F or every T ur ing machine M = h Q, Γ , q 0 , δ i we define a TRS R M as follows. The signa ture is Σ = Q ∪ Γ ∪ { ⊲ } where the symbols q ∈ Q hav e ar it y 2, the symbo ls f ∈ Γ have arity 1 and ⊲ is a constant sy m b ol, whic h repr esents an infinite num b er of blank symbols . The r ewrite rules of R M are: q ( x, f ( y )) → q ′ ( f ′ ( x ) , y ) for every δ ( q , f ) = h q ′ , f ′ , R i q ( g ( x ) , f ( y )) → q ′ ( x, g ( f ′ ( y ))) for every δ ( q , f ) = h q ′ , f ′ , L i together with four rules for ‘extending the tap e’: q ( ⊲, f ( y )) → q ′ ( ⊲, ✷ ( f ′ ( y ))) for every δ ( q , f ) = h q ′ , f ′ , L i q ( x, ⊲ ) → q ′ ( f ′ ( x ) , ⊲ ) for every δ ( q , ✷ ) = h q ′ , f ′ , R i q ( g ( x ) , ⊲ ) → q ′ ( x, g ( f ′ ( ⊲ ))) for every δ ( q , ✷ ) = h q ′ , f ′ , L i q ( ⊲, ⊲ ) → q ′ ( ⊲, ✷ ( f ′ ( ⊲ ))) for every δ ( q , ✷ ) = h q ′ , f ′ , L i . In [2 ] the TRSs R M has be e n extended as follow to prove Π 1 1 -completeness of finiteness of dep endency pair pro blems: Definition 5 ([2] ). F or every T uring machine M = h Q, Γ , q 0 , δ i we define the TRS R • M as follows. The signature Σ = Q ∪ Γ ∪ { ⊲, • , T } whe r e • is a unar y symbol, T is a consta n t symbol, and the rewrite rules of R • M are: ℓ → • ( r ) for every ℓ → r ∈ R M and rules for rew r iting to T after successful termination: q ( x, 0 ( y )) → T whenever δ ( q , S ) is undefined • ( T ) → T . Moreov er , we define the TRS R pickn to consist of the following r ule s : pickn → c ( pickn ) pickn → ok ( 0 ( ⊲ )) c ( ok ( x )) → ok ( S ( x )) . 2 Prop ositio n 6. L et M b e an arbitr ary T uring machine. We define the TRS S to get her to c onsist of the rules of R • M ⊎ R pickn to get her with: run ( T , ok ( x ) , ok ( y )) → run ( q 0 ( x, y ) , ok ( y ) , pickn ) , (1) and define a term t := run ( T , pickn , pi ckn ) . Then it holds: SN ∞ S ⇐ ⇒ SN ∞ S ( t ) ⇐ ⇒ > M is wel l-founde d . Pr o of. See [2]. ⊓ ⊔ Theorem 7. Uniform infinitary str ong normalization, SN ∞ R , and for single terms, SN ∞ R ( s ) is Π 1 1 -c omplete. Pr o of. The Π 1 1 -hardness ha s b een shown in [2] using that well-foundedness is Π 1 1 -complete. It remains to b e shown tha t the prop erty is in Π 1 1 (in [2] this has b een done only reductions of leng th ω ). A finite or infinite term t can be enco ded as a function t : N → N (fro m pos itions to symbols from the signa ture). An infinite reduction can b e rendered as a function σ : α → (( N → N ) × N ) from an ordinal α to terms together with the rewrite p os ition (here we assume that an o rdinal is the set of all smaller or dina ls) where σ ( β ) is the β -th ter m of the sequence together with the rew r ite pos ition, and we requir e: (i) σ ( β ) r ewrites to σ ( β + 1) for all β < α , and (ii) for all limit ordinals β < α , a nd γ approaching β from b elow, we ha ve: – σ ( γ ) conv erge s to σ ( β ), and – the depth of the γ -th rewrite steps tends to infinity . If c ondition (ii) ho lds for all limit ordinals β ≤ α then the rewrite sequence σ is called strong ly conv erg ent. An or dina l α ca n be view ed as a well-founded r e la tion α ⊆ N × N . The pro p e r ty of a relation to b e well-founded can b e expressed by a Π 1 1 -formula, and the ab ov e pr op erties on rewrite sequences are ar ithmetic. By [4] the prop erty SN ∞ R ( s ) holds if and o nly if all reductions admitted b y s are strongly conv er gent. Hence SN ∞ R and SN ∞ R ( s ) can be express ed by a Π 1 1 -formula since the ab ov e conditions (i) and (ii) a re arithmetic. ⊓ ⊔ Using a minor mo difica tion of the term r ewriting sys tem from Prop ositio n 6 we obtain that weak normaliza tio n for single ter ms and r e achability are Σ 1 1 - complete, that is, the pr oblem of deciding on the input of a TRS S and ter ms s , t whether s ։ ։ S t . Theorem 8. Infinitary we ak normalization for single terms, WN ∞ R ( s ) , and r e ach- ability in infi nitary re writing ar e Σ 1 1 -c omplete. Pr o of. Let M b e an arbitrar y T uring machine. W e define the TRS S ′ together to consist of the rules of R • M ⊎ R pickn together with: run ( T , ok ( x ) , ok ( y )) → • ( run ( q 0 ( x, y ) , ok ( y ) , pickn )) , (2) 3 and define a term t := run ( T , pickn , pickn ). W e have t ։ ։ S ′ • ∞ if a nd only if t admits a rewrite seq ue nc e containing infinitely ma n y ro ot steps with r esp ect to the rewrite system S fro m Prop osition 6. As a consequence we have: t ։ ։ S ′ • ∞ ⇐ ⇒ ¬ SN ∞ S ( t ) ⇐ ⇒ > M is not well-founded . Hence reachability is Σ 1 1 -hard. W e a dd one more rule to S ′ : run ( x, y , z ) → run ( x, y , z ) , (3) Note that this r ule has no impact on r e a chabilit y . Then WN ∞ S ′ ( t ) holds if and only if t ։ ։ S ′ • ∞ , and hence WN ∞ S ′ ( t ) is Σ 1 1 -complete. Moreov er , weak nor ma lization for single ter ms and reachability are in Σ 1 1 . W e have WN ∞ S ′ ( t ) if a nd only if there exists a normal form t ′ such that t ։ ։ t ′ , and we hav e rea chabilit y s ։ ։ t if and only if there exists a reduction from s to t . The quantification ov er terms and r ewrite sequences a r e ex istent ia l s et or function quantifiers (which can b e compre s sed to one single quantifier), a nd all other prop erties are arithmetic; see the enco ding of reduction sequences see the pro of of Theore m 7. ⊓ ⊔ 2 Uniform Infinitary W eak N ormalization F or σ ∈ Γ ∞ and i ∈ N w e w r ite σ 0 i ′ = i − 1, a nd σ ′ = σ j. i h ′ = n . 4 Definition 10. A run is called ac c epting if it is complete and non-oscilla ting. The ω -language L ω ( M ) accepted by a no n-deterministic T ur ing machine M is: L ω ( M ) = { w ∈ Γ ω | ther e exists an ac c epting ru n of M on w } Notice that accepting runs vis its every symbol at least once, but only finitely often. The following is a prop osition from [1]: Prop ositio n 11 ([1] ). The set { M | L ω ( M ) = Γ ω } is Π 1 2 -c omplete. ⊓ ⊔ W e use the tra nslation of T uring machines M to string rewriting s ystems S M from [5]. Definition 12. F or every (non-deterministic) T ur ing machine M = h Q, Γ , q 0 , δ i we define a TRS S M as follows. The signa tur e Σ consists of s ymbols from Q ∪ Γ all having ar it y 1. The rewr ite rules of S M are: q ( f ( x )) → f ′ ( q ′ ( x )) for every h q ′ , f ′ , R i ∈ δ ( q , f ) g ( q ( f ( x ) , )) → q ′ ( g ( f ′ ( x ))) for every h q ′ , f ′ , L i ∈ δ ( q , f ) Definition 13. L et M b e a non-deterministic T uring machine. We define a mapping ϕ : ( Γ ∪ Q ) ∗ → T er ∞ ( Γ ∪ Q ) by ϕ ( a w ) := a ( ϕ ( t )) , and we extend ϕ to c onfigura tions h q , σ , i i of M by defining: ϕ ( h q , σ, i i ) = ϕ ( σ

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