Undecidability of FL e in the presence of structural rules Gavin - PowerPoint PPT Presentation

Residuated frames Definition [Galatos & Jipsen 2013] A residuated frame is a structure W = ( W, W ′ , N, ◦ , � , � , 1) , s.t. ◮ ( W, ◦ , 1) is a monoid and W ′ is a set. ◮ N ⊆ W × W ′ , called the Galois relation, and ◮ � : W × W ′ → W ′ and � : W ′ × W → W ′ such that ◮ N is a nuclear , i.e. for all u, v ∈ W and w ∈ W ′ , ( u ◦ v ) N w iff u N ( w � v ) iff v N ( u � w ) . Define ⊲ : P ( W ) → P ( W ′ ) and ⊳ : P ( W ′ ) → P ( W ) via X ⊲ = { y ∈ W ′ : ∀ x ∈ X, xNy } for each X ⊆ W and Y ⊳ = { x ∈ W : ∀ y ∈ Y, xNy } for each Y ⊆ W ′ . Then ( ⊲ , ⊳ ) is a Galois connection. γ N → X ⊲⊳ is a closure operator on P ( W ) . �− − So γ N defined via X Fact: N is nuclear iff γ N is a nucleus. Gavin St.John Application 6. Residuated frames and (un)decidability 6 / 34

Residuated frames cont. Theorem [Galatos & Jipsen 2013] W + := ( γ N [ P ( W )] , ∪ γ N , ∩ , ◦ γ N , \ , /, γ N ( { 1 } )) , X ∪ γ N Y = γ N ( X ∪ Y ) and X ◦ γ N Y = γ N ( X ◦ Y ) , is a residuated latice. Gavin St.John Application 6. Residuated frames and (un)decidability 7 / 34

Residuated frames cont. Theorem [Galatos & Jipsen 2013] W + := ( γ N [ P ( W )] , ∪ γ N , ∩ , ◦ γ N , \ , /, γ N ( { 1 } )) , X ∪ γ N Y = γ N ( X ∪ Y ) and X ◦ γ N Y = γ N ( X ◦ Y ) , is a residuated latice. Comment Certain structural properties (inference rules) for the nuclear relation N are preserved by the ordering relation ⊆ on W + . Gavin St.John Application 6. Residuated frames and (un)decidability 7 / 34

Residuated frames cont. Theorem [Galatos & Jipsen 2013] W + := ( γ N [ P ( W )] , ∪ γ N , ∩ , ◦ γ N , \ , /, γ N ( { 1 } )) , X ∪ γ N Y = γ N ( X ∪ Y ) and X ◦ γ N Y = γ N ( X ◦ Y ) , is a residuated latice. Comment Certain structural properties (inference rules) for the nuclear relation N are preserved by the ordering relation ⊆ on W + . ◦ We can encode “desirable properties” we want a RL to satisfy in N . Gavin St.John Application 6. Residuated frames and (un)decidability 7 / 34

Residuated frames cont. Theorem [Galatos & Jipsen 2013] W + := ( γ N [ P ( W )] , ∪ γ N , ∩ , ◦ γ N , \ , /, γ N ( { 1 } )) , X ∪ γ N Y = γ N ( X ∪ Y ) and X ◦ γ N Y = γ N ( X ◦ Y ) , is a residuated latice. Comment Certain structural properties (inference rules) for the nuclear relation N are preserved by the ordering relation ⊆ on W + . ◦ We can encode “desirable properties” we want a RL to satisfy in N . ◦ In particular, (simple) rules in the signature �∨ , · , 1 � are preserved via ( − ) + , Gavin St.John Application 6. Residuated frames and (un)decidability 7 / 34

Rules in the signature �∨ , · , 1 � and Linearization Any equation s = t in the signature �∨ , · , 1 � is equivalent to some conjunction of simple rules . m x d j (1) · · · x d j ( n ) � ( d ) x 1 · · · x n ≤ , n 1 j =1 where d = { d 1 , ..., d m } ⊂ N n . Gavin St.John Application 6. Residuated frames and (un)decidability 8 / 34

Rules in the signature �∨ , · , 1 � and Linearization Any equation s = t in the signature �∨ , · , 1 � is equivalent to some conjunction of simple rules . m x d j (1) · · · x d j ( n ) � ( d ) x 1 · · · x n ≤ , n 1 j =1 where d = { d 1 , ..., d m } ⊂ N n . Such conjoins can be determined by the properties of CRL : ◮ x ≤ y ⇐ ⇒ x ∨ y = y ◮ x ∨ y ≤ z ⇐ ⇒ x ≤ z and y ≤ z ◮ linearization Gavin St.John Application 6. Residuated frames and (un)decidability 8 / 34

Rules in the signature �∨ , · , 1 � and Linearization Any equation s = t in the signature �∨ , · , 1 � is equivalent to some conjunction of simple rules . m x d j (1) · · · x d j ( n ) � ( d ) x 1 · · · x n ≤ , n 1 j =1 where d = { d 1 , ..., d m } ⊂ N n . Such conjoins can be determined by the properties of CRL : ◮ x ≤ y ⇐ ⇒ x ∨ y = y ◮ x ∨ y ≤ z ⇐ ⇒ x ≤ z and y ≤ z ◮ linearization E.g., the rule ( ∀ u )( ∀ v ) u 2 v ≤ u 3 ∨ uv is equivalent to, via the substitution σ : u σ → x ∨ y and v σ �− �− → z, ( ∀ x )( ∀ y )( ∀ z ) xyz ≤ x 3 ∨ x 2 y ∨ xy 2 ∨ y 3 ∨ xz ∨ yz Gavin St.John Application 6. Residuated frames and (un)decidability 8 / 34

Simple rules and Residuated Frames Let W = ( W, W ′ , N ) be a residuated frame and ( d ) be the simple rule given by m � x d j (1) · · · x d j ( n ) x 1 · · · x n ≤ . n 1 j =1 Gavin St.John Application 6. Residuated frames and (un)decidability 9 / 34

Simple rules and Residuated Frames Let W = ( W, W ′ , N ) be a residuated frame and ( d ) be the simple rule given by m � x d j (1) · · · x d j ( n ) x 1 · · · x n ≤ . n 1 j =1 = [ d ] iff for all u 1 , ..., u n ∈ W and v ∈ W ′ , the following We say W | inference rule is satisfied � n i =1 u d 1 ( i ) � n i =1 u d m ( i ) · · · N v N v i i [ d ] . � n i =1 u i N v Gavin St.John Application 6. Residuated frames and (un)decidability 9 / 34

Simple rules and Residuated Frames Let W = ( W, W ′ , N ) be a residuated frame and ( d ) be the simple rule given by m � x d j (1) · · · x d j ( n ) x 1 · · · x n ≤ . n 1 j =1 = [ d ] iff for all u 1 , ..., u n ∈ W and v ∈ W ′ , the following We say W | inference rule is satisfied � n i =1 u d 1 ( i ) � n i =1 u d m ( i ) · · · N v N v i i [ d ] . � n i =1 u i N v Proposition [Galatos & Jipsen 2013] All simple rules are preserved by ( − ) + . In particular, = [ d ] iff W + | W | = ( d ) . Gavin St.John Application 6. Residuated frames and (un)decidability 9 / 34

The Word Problem A presentation for L is a pair � X, E � where ◮ X is a set of generators , and ◮ E is a set of equations over T ( X ) . Gavin St.John Application 6. Residuated frames and (un)decidability 10 / 34

The Word Problem A presentation for L is a pair � X, E � where ◮ X is a set of generators , and ◮ E is a set of equations over T ( X ) . If both X and E are finite, we call the presentation � X, E � finite. Gavin St.John Application 6. Residuated frames and (un)decidability 10 / 34

The Word Problem A presentation for L is a pair � X, E � where ◮ X is a set of generators , and ◮ E is a set of equations over T ( X ) . If both X and E are finite, we call the presentation � X, E � finite. ◮ We denote the conjunction of equations in E by & E. Gavin St.John Application 6. Residuated frames and (un)decidability 10 / 34

The Word Problem A presentation for L is a pair � X, E � where ◮ X is a set of generators , and ◮ E is a set of equations over T ( X ) . If both X and E are finite, we call the presentation � X, E � finite. ◮ We denote the conjunction of equations in E by & E. We say V has an undecidable word problem if there exists a finite presentation � X, E � such that there is no algorithm deciding ⇒ s = t ) holds in V having s, t ∈ T ( X ) as whether the q.e. ( & E = inputs. Gavin St.John Application 6. Residuated frames and (un)decidability 10 / 34

The Word Problem A presentation for L is a pair � X, E � where ◮ X is a set of generators , and ◮ E is a set of equations over T ( X ) . If both X and E are finite, we call the presentation � X, E � finite. ◮ We denote the conjunction of equations in E by & E. We say V has an undecidable word problem if there exists a finite presentation � X, E � such that there is no algorithm deciding ⇒ s = t ) holds in V having s, t ∈ T ( X ) as whether the q.e. ( & E = inputs. Or equivalently, there is a finitely presented algebra A ∈ V generated by X such that the following set is undecidable: { ( s, t ) ∈ T ( X ) 2 : A | = s = t } . Gavin St.John Application 6. Residuated frames and (un)decidability 10 / 34

The Word Problem A presentation for L is a pair � X, E � where ◮ X is a set of generators , and ◮ E is a set of equations over T ( X ) . If both X and E are finite, we call the presentation � X, E � finite. ◮ We denote the conjunction of equations in E by & E. We say V has an undecidable word problem if there exists a finite presentation � X, E � such that there is no algorithm deciding ⇒ s = t ) holds in V having s, t ∈ T ( X ) as whether the q.e. ( & E = inputs. Or equivalently, there is a finitely presented algebra A ∈ V generated by X such that the following set is undecidable: { ( s, t ) ∈ T ( X ) 2 : A | = s = t } . ◮ undecidable word problem ⇒ undecidable q.e. theory. Gavin St.John Application 6. Residuated frames and (un)decidability 10 / 34

Counter Machines A k -CM M = ( R k , Q, P ) is a finite state automaton that has Gavin St.John Application 6. Residuated frames and (un)decidability 11 / 34

Counter Machines A k -CM M = ( R k , Q, P ) is a finite state automaton that has ◮ a set R k := { r 1 , ..., r k } of k registers (bins) that can each store a non-negative integer (tokens), Gavin St.John Application 6. Residuated frames and (un)decidability 11 / 34

Counter Machines A k -CM M = ( R k , Q, P ) is a finite state automaton that has ◮ a set R k := { r 1 , ..., r k } of k registers (bins) that can each store a non-negative integer (tokens), ◮ a finite set Q of states with designated final state q f , Gavin St.John Application 6. Residuated frames and (un)decidability 11 / 34

Counter Machines A k -CM M = ( R k , Q, P ) is a finite state automaton that has ◮ a set R k := { r 1 , ..., r k } of k registers (bins) that can each store a non-negative integer (tokens), ◮ a finite set Q of states with designated final state q f , ◮ and a finite set P of instructions p of the form: q + r q ′ ◦ Increment register r : q − r q ′ ◦ Decrement register r : q 0 r q ′ , ◦ Zero-test register r : where q, q ′ ∈ Q and r ∈ R k . E.g, Gavin St.John Application 6. Residuated frames and (un)decidability 11 / 34

Counter Machines A k -CM M = ( R k , Q, P ) is a finite state automaton that has ◮ a set R k := { r 1 , ..., r k } of k registers (bins) that can each store a non-negative integer (tokens), ◮ a finite set Q of states with designated final state q f , ◮ and a finite set P of instructions p of the form: q + r q ′ ◦ Increment register r : q − r q ′ ◦ Decrement register r : q 0 r q ′ , ◦ Zero-test register r : where q, q ′ ∈ Q and r ∈ R k . E.g, input configuration inst. output configuration q + r i q ′ � q ′ ; n 1 , ..., n i + 1 , ..., n k � � q ; n 1 , ..., n i , ..., n k � �− − − − − → q − r i q ′ � q ′ ; n 1 , ..., n i , ..., n k � � q ; n 1 , ..., n i + 1 , ..., n k � �− − − − − → q 0 r i q ′ � q ′ ; n 1 , ..., 0 , ..., n k � � q ; n 1 , ..., 0 , ..., n k � �− − − − → Gavin St.John Application 6. Residuated frames and (un)decidability 11 / 34

And-branching k -Counter Machines ( k -ACM) A k -ACM M = ( R k , Q, P ) , as introduced by Lincoln, Mitchell, Scedrov, Shankar (1992), is a type of non-deterministic parallel-computing counter machine that has ◮ a set R k := { r 1 , ..., r k } of k registers (bins) that can each store a non-negative integer (tokens), ◮ a finite set Q of states with designated final state q f , Gavin St.John Application 6. Residuated frames and (un)decidability 12 / 34

And-branching k -Counter Machines ( k -ACM) A k -ACM M = ( R k , Q, P ) , as introduced by Lincoln, Mitchell, Scedrov, Shankar (1992), is a type of non-deterministic parallel-computing counter machine that has ◮ a set R k := { r 1 , ..., r k } of k registers (bins) that can each store a non-negative integer (tokens), ◮ a finite set Q of states with designated final state q f , ◮ and a finite set P of instructions p of the form: ≤ p q ′ r ◦ Increment: q ≤ p q ′ ◦ qr Decrement: q ′ ∨ q ′′ , ≤ p ◦ Fork: q where q, q ′ , q ′′ ∈ Q and r ∈ R k . Gavin St.John Application 6. Residuated frames and (un)decidability 12 / 34

ACM’s continued ◮ A configuration C is a word which consists of a single state and a number of register tokens 2 · · · r n k C = qr n 1 1 r n 2 k . Gavin St.John Application 6. Residuated frames and (un)decidability 13 / 34

ACM’s continued ◮ A configuration C is a word which consists of a single state and a number of register tokens 2 · · · r n k C = qr n 1 1 r n 2 k . ◮ Forking instructions allow parallel computation. The “status” u of a machine at a given moment in a computation is called an instantaneous description (ID), u = C 1 ∨ C 2 ∨ · · · ∨ C n , where C 1 , ..., C n are configurations. Gavin St.John Application 6. Residuated frames and (un)decidability 13 / 34

ACM’s continued ◮ A configuration C is a word which consists of a single state and a number of register tokens 2 · · · r n k C = qr n 1 1 r n 2 k . ◮ Forking instructions allow parallel computation. The “status” u of a machine at a given moment in a computation is called an instantaneous description (ID), u = C 1 ∨ C 2 ∨ · · · ∨ C n , where C 1 , ..., C n are configurations. ◮ An instruction p is a function (relation) on ID’s that can replace a single configuration C by an ID v , i.e. C ∨ u ≤ p v ∨ u Gavin St.John Application 6. Residuated frames and (un)decidability 13 / 34

Computations We view computations as order relations on the free commutative semiring A M = ( A M , ∨ , · , ⊥ , 1) generated by Q ∪ R k , where M = ( R k , Q, P ) is a k -ACM and Gavin St.John Application 6. Residuated frames and (un)decidability 14 / 34

Computations We view computations as order relations on the free commutative semiring A M = ( A M , ∨ , · , ⊥ , 1) generated by Q ∪ R k , where M = ( R k , Q, P ) is a k -ACM and ◮ ( A M , ∨ , ⊥ ) is a commutative monoid with identity ⊥ = � ∅ , ◮ ( A M , · , 1) is a commutative monoid with identity 1 , and ◮ multiplication ( · ) distributes over “join” ( ∨ ). Gavin St.John Application 6. Residuated frames and (un)decidability 14 / 34

Computations We view computations as order relations on the free commutative semiring A M = ( A M , ∨ , · , ⊥ , 1) generated by Q ∪ R k , where M = ( R k , Q, P ) is a k -ACM and ◮ ( A M , ∨ , ⊥ ) is a commutative monoid with identity ⊥ = � ∅ , ◮ ( A M , · , 1) is a commutative monoid with identity 1 , and ◮ multiplication ( · ) distributes over “join” ( ∨ ). Each instruction p ∈ P defines a relation ≤ p closed under u ≤ p v u ≤ p v ux ≤ p vx [ · ] u ∨ w ≤ p v ∨ w [ ∨ ] , and for u, v, w, x ∈ A M . Gavin St.John Application 6. Residuated frames and (un)decidability 14 / 34

Computations We view computations as order relations on the free commutative semiring A M = ( A M , ∨ , · , ⊥ , 1) generated by Q ∪ R k , where M = ( R k , Q, P ) is a k -ACM and ◮ ( A M , ∨ , ⊥ ) is a commutative monoid with identity ⊥ = � ∅ , ◮ ( A M , · , 1) is a commutative monoid with identity 1 , and ◮ multiplication ( · ) distributes over “join” ( ∨ ). Each instruction p ∈ P defines a relation ≤ p closed under u ≤ p v u ≤ p v ux ≤ p vx [ · ] u ∨ w ≤ p v ∨ w [ ∨ ] , and for u, v, w, x ∈ A M . We define the computation relation ≤ M to be the smallest ( · , ∨ ) -compatible preorder containing � ≤ p . p ∈ P Gavin St.John Application 6. Residuated frames and (un)decidability 14 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. We say a machine M accepts an ID u (writen u ∈ Acc( M ) ) if u ≤ M v , for some v ∈ Fin( M ) . Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. We say a machine M accepts an ID u (writen u ∈ Acc( M ) ) if u ≤ M v , for some v ∈ Fin( M ) . ◮ C 1 ∨ · · · ∨ C n ∈ Acc( M ) ⇐ ⇒ C 1 , ..., C n ∈ Acc( M ) . Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. We say a machine M accepts an ID u (writen u ∈ Acc( M ) ) if u ≤ M v , for some v ∈ Fin( M ) . ◮ C 1 ∨ · · · ∨ C n ∈ Acc( M ) ⇐ ⇒ C 1 , ..., C n ∈ Acc( M ) . ◮ u ∈ Acc( M ) = ⇒ ∃ p 1 , ..., p n ∈ P and ∃ u 0 , ..., u n ∈ ID( M ) , u = u 0 ≤ p 1 u 1 ≤ p 2 · · · ≤ p n u n ∈ Fin( M ) . Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. We say a machine M accepts an ID u (writen u ∈ Acc( M ) ) if u ≤ M v , for some v ∈ Fin( M ) . ◮ C 1 ∨ · · · ∨ C n ∈ Acc( M ) ⇐ ⇒ C 1 , ..., C n ∈ Acc( M ) . ◮ u ∈ Acc( M ) = ⇒ ∃ p 1 , ..., p n ∈ P and ∃ u 0 , ..., u n ∈ ID( M ) , u = u 0 ≤ p 1 u 1 ≤ p 2 · · · ≤ p n u n ∈ Fin( M ) . Example Machine Let M = M even := ( { r } , { q 0 , q 1 , q f } , { p 1 , p 2 , p 3 } ) , with instructions q 0 r ≤ p 1 q 1 ; q 1 r ≤ p 2 q 0 ; q 0 ≤ p 3 q f ∨ q f . Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. We say a machine M accepts an ID u (writen u ∈ Acc( M ) ) if u ≤ M v , for some v ∈ Fin( M ) . ◮ C 1 ∨ · · · ∨ C n ∈ Acc( M ) ⇐ ⇒ C 1 , ..., C n ∈ Acc( M ) . ◮ u ∈ Acc( M ) = ⇒ ∃ p 1 , ..., p n ∈ P and ∃ u 0 , ..., u n ∈ ID( M ) , u = u 0 ≤ p 1 u 1 ≤ p 2 · · · ≤ p n u n ∈ Fin( M ) . Example Machine Let M = M even := ( { r } , { q 0 , q 1 , q f } , { p 1 , p 2 , p 3 } ) , with instructions q 0 r ≤ p 1 q 1 ; q 1 r ≤ p 2 q 0 ; q 0 ≤ p 3 q f ∨ q f . ◮ Note that q 0 r n ∈ Acc( M ) iff n is even. Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. We say a machine M accepts an ID u (writen u ∈ Acc( M ) ) if u ≤ M v , for some v ∈ Fin( M ) . ◮ C 1 ∨ · · · ∨ C n ∈ Acc( M ) ⇐ ⇒ C 1 , ..., C n ∈ Acc( M ) . ◮ u ∈ Acc( M ) = ⇒ ∃ p 1 , ..., p n ∈ P and ∃ u 0 , ..., u n ∈ ID( M ) , u = u 0 ≤ p 1 u 1 ≤ p 2 · · · ≤ p n u n ∈ Fin( M ) . Example Machine Let M = M even := ( { r } , { q 0 , q 1 , q f } , { p 1 , p 2 , p 3 } ) , with instructions q 0 r ≤ p 1 q 1 ; q 1 r ≤ p 2 q 0 ; q 0 ≤ p 3 q f ∨ q f . ◮ Note that q 0 r n ∈ Acc( M ) iff n is even. q 0 r 4 ≤ p 1 q 1 r 3 ≤ p 2 q 0 r 2 ≤ p 1 q 1 r ≤ p 2 q 0 ≤ p 3 q f ∨ q f ∈ Acc( M ) Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Define Fin( M ) = { � n i =1 q f : n ∈ Z + } to be the set of Final ID’s. We say a machine M accepts an ID u (writen u ∈ Acc( M ) ) if u ≤ M v , for some v ∈ Fin( M ) . ◮ C 1 ∨ · · · ∨ C n ∈ Acc( M ) ⇐ ⇒ C 1 , ..., C n ∈ Acc( M ) . ◮ u ∈ Acc( M ) = ⇒ ∃ p 1 , ..., p n ∈ P and ∃ u 0 , ..., u n ∈ ID( M ) , u = u 0 ≤ p 1 u 1 ≤ p 2 · · · ≤ p n u n ∈ Fin( M ) . Example Machine Let M = M even := ( { r } , { q 0 , q 1 , q f } , { p 1 , p 2 , p 3 } ) , with instructions q 0 r ≤ p 1 q 1 ; q 1 r ≤ p 2 q 0 ; q 0 ≤ p 3 q f ∨ q f . ◮ Note that q 0 r n ∈ Acc( M ) iff n is even. q 0 r 4 ≤ p 1 q 1 r 3 ≤ p 2 q 0 r 2 ≤ p 1 q 1 r ≤ p 2 q 0 ≤ p 3 q f ∨ q f ∈ Acc( M ) q 0 r 3 ≤ p 1 q 1 r 2 ≤ p 2 q 0 r ≤ p 3 q f r ∨ q f r �∈ Acc( M ) Gavin St.John Application 6. Residuated frames and (un)decidability 15 / 34

Undecidable Problem Theorem [LMSS 1992] There exists a 2 -ACM M such that membership of the set { u ∈ ID( M ) : u ∈ Acc( M ) } is undecidable. Gavin St.John Application 6. Residuated frames and (un)decidability 16 / 34

Undecidable Problem Theorem [LMSS 1992] There exists a 2 -ACM M such that membership of the set { u ∈ ID( M ) : u ∈ Acc( M ) } is undecidable. Let M = ( R k , Q, P ) be a k -ACM and u ∈ ID( M ) , ◮ We can define a quasi-equation acc M ( u ) in the signature �∨ , · , 1 � via & P = ⇒ u ≤ q f . Gavin St.John Application 6. Residuated frames and (un)decidability 16 / 34

ACM’s and Residuated Frames Let M = ( R k , Q, P ) be a k -ACM and W := ( Q ∪ R k ) ∗ be the free commutative monoid generated by Q ∪ R k . Gavin St.John Application 6. Residuated frames and (un)decidability 17 / 34

ACM’s and Residuated Frames Let M = ( R k , Q, P ) be a k -ACM and W := ( Q ∪ R k ) ∗ be the free commutative monoid generated by Q ∪ R k . The frame W M Inspired by Horčík (2015), we let W ′ := W and define the relation N M ⊆ W × W ′ via x N M z iff xz ∈ Acc( M ) , for all x, z ∈ W . Gavin St.John Application 6. Residuated frames and (un)decidability 17 / 34

ACM’s and Residuated Frames Let M = ( R k , Q, P ) be a k -ACM and W := ( Q ∪ R k ) ∗ be the free commutative monoid generated by Q ∪ R k . The frame W M Inspired by Horčík (2015), we let W ′ := W and define the relation N M ⊆ W × W ′ via x N M z iff xz ∈ Acc( M ) , for all x, z ∈ W . Observe that, for any x, y, z ∈ W , xy N M z ⇐ ⇒ xyz ∈ Acc( M ) ⇐ ⇒ x N M yz. Since W is commutive it follows that N M is nuclear. Gavin St.John Application 6. Residuated frames and (un)decidability 17 / 34

ACM’s and Residuated Frames Let M = ( R k , Q, P ) be a k -ACM and W := ( Q ∪ R k ) ∗ be the free commutative monoid generated by Q ∪ R k . The frame W M Inspired by Horčík (2015), we let W ′ := W and define the relation N M ⊆ W × W ′ via x N M z iff xz ∈ Acc( M ) , for all x, z ∈ W . Observe that, for any x, y, z ∈ W , xy N M z ⇐ ⇒ xyz ∈ Acc( M ) ⇐ ⇒ x N M yz. Since W is commutive it follows that N M is nuclear. Lemma W M := ( W, W ′ , N M ) is a residuated frame, W + M ∈ CRL , and there exists a valuation ν : Tm → W + M such that W + M , ν | = & P . Gavin St.John Application 6. Residuated frames and (un)decidability 17 / 34

ACM’s and Residuated Frames cont. Let M be a k -ACM and V ⊆ ( C ) RL a variety. Theorem If W + M ∈ V then for all u ∈ ID( M ) , u ∈ Acc( M ) if and only if V | = acc M ( u ) . Gavin St.John Application 6. Residuated frames and (un)decidability 18 / 34

ACM’s and Residuated Frames cont. Let M be a k -ACM and V ⊆ ( C ) RL a variety. Theorem If W + M ∈ V then for all u ∈ ID( M ) , u ∈ Acc( M ) if and only if V | = acc M ( u ) . Corollary If W + M ∈ V then the computational complexity for the word problem of V is at least as complex as the membership of Acc( M ) . Gavin St.John Application 6. Residuated frames and (un)decidability 18 / 34

ACM’s and Residuated Frames cont. Let M be a k -ACM and V ⊆ ( C ) RL a variety. Theorem If W + M ∈ V then for all u ∈ ID( M ) , u ∈ Acc( M ) if and only if V | = acc M ( u ) . Corollary If W + M ∈ V then the computational complexity for the word problem of V is at least as complex as the membership of Acc( M ) . Corollary Suppose membership of Acc( M ) is undecidable. If W + M ∈ V then V has an undecidable word problem. Gavin St.John Application 6. Residuated frames and (un)decidability 18 / 34

ACM’s and Residuated Frames cont. Let M be a k -ACM and V ⊆ ( C ) RL a variety. Theorem If W + M ∈ V then for all u ∈ ID( M ) , u ∈ Acc( M ) if and only if V | = acc M ( u ) . Corollary If W + M ∈ V then the computational complexity for the word problem of V is at least as complex as the membership of Acc( M ) . Corollary Suppose membership of Acc( M ) is undecidable. If W + M ∈ V then V has an undecidable word problem. In particular, ( C ) RL has an undecidable word problem since W + M ∈ CRL , where ˜ M is the ˜ machine from LMSS (1992). Gavin St.John Application 6. Residuated frames and (un)decidability 18 / 34

Simple rules in k -ACM’s and the relation ≤ d M Let M = ( R k , Q, P ) be a k -ACM. Given a simple rule, e.g. ( d ) : x ≤ x 2 ∨ x 4 , we add “ambient” instructions of the form � � m t ≤ d t 2 ∨ t 4 i =1 t i ≤ ¯ i =1 t d j ( i ) � n � � n d , i j =1 for each t ∈ ( Q ∪ R k ) ∗ ( t 1 , ..., t n ∈ ( Q ∪ R k ) ∗ ). Gavin St.John Application 6. Residuated frames and (un)decidability 19 / 34

Simple rules in k -ACM’s and the relation ≤ d M Let M = ( R k , Q, P ) be a k -ACM. Given a simple rule, e.g. ( d ) : x ≤ x 2 ∨ x 4 , we add “ambient” instructions of the form � � m t ≤ d t 2 ∨ t 4 i =1 t i ≤ ¯ i =1 t d j ( i ) � n � � n d , i j =1 for each t ∈ ( Q ∪ R k ) ∗ ( t 1 , ..., t n ∈ ( Q ∪ R k ) ∗ ). ◮ As with the instructions in P , we close ≤ d under the inference rules [ · ] and [ ∨ ] . Gavin St.John Application 6. Residuated frames and (un)decidability 19 / 34

Simple rules in k -ACM’s and the relation ≤ d M Let M = ( R k , Q, P ) be a k -ACM. Given a simple rule, e.g. ( d ) : x ≤ x 2 ∨ x 4 , we add “ambient” instructions of the form � � m t ≤ d t 2 ∨ t 4 i =1 t i ≤ ¯ i =1 t d j ( i ) � n � � n d , i j =1 for each t ∈ ( Q ∪ R k ) ∗ ( t 1 , ..., t n ∈ ( Q ∪ R k ) ∗ ). ◮ As with the instructions in P , we close ≤ d under the inference rules [ · ] and [ ∨ ] . ◮ Similarly, we define the relation ≤ d M to be the smallest ( · , ∨ ) -compatible preorder generated by ≤ d ∪ ≤ M . Gavin St.John Application 6. Residuated frames and (un)decidability 19 / 34

Simple rules in k -ACM’s and the relation ≤ d M Let M = ( R k , Q, P ) be a k -ACM. Given a simple rule, e.g. ( d ) : x ≤ x 2 ∨ x 4 , we add “ambient” instructions of the form � � m t ≤ d t 2 ∨ t 4 i =1 t i ≤ ¯ i =1 t d j ( i ) � n � � n d , i j =1 for each t ∈ ( Q ∪ R k ) ∗ ( t 1 , ..., t n ∈ ( Q ∪ R k ) ∗ ). ◮ As with the instructions in P , we close ≤ d under the inference rules [ · ] and [ ∨ ] . ◮ Similarly, we define the relation ≤ d M to be the smallest ( · , ∨ ) -compatible preorder generated by ≤ d ∪ ≤ M . ◮ We denote this new machine by d M . Gavin St.John Application 6. Residuated frames and (un)decidability 19 / 34

Simple rules in k -ACM’s and the relation ≤ d M Let M = ( R k , Q, P ) be a k -ACM. Given a simple rule, e.g. ( d ) : x ≤ x 2 ∨ x 4 , we add “ambient” instructions of the form � � m t ≤ d t 2 ∨ t 4 i =1 t i ≤ ¯ i =1 t d j ( i ) � n � � n d , i j =1 for each t ∈ ( Q ∪ R k ) ∗ ( t 1 , ..., t n ∈ ( Q ∪ R k ) ∗ ). ◮ As with the instructions in P , we close ≤ d under the inference rules [ · ] and [ ∨ ] . ◮ Similarly, we define the relation ≤ d M to be the smallest ( · , ∨ ) -compatible preorder generated by ≤ d ∪ ≤ M . ◮ We denote this new machine by d M . Lemma Let M = ( R k , Q, P ) be a k -ACM and ( d ) a simple rule. Then = [ d ] , and therefore W + W d M | d M ∈ CRL + ( d ) . Gavin St.John Application 6. Residuated frames and (un)decidability 19 / 34

Admissibility of simple rules for a machine Definition Let M be a k -ACM and ( d ) be a d -rule. We say ( d ) is admissible in M if Acc( M ) = Acc( d M ) , Gavin St.John Application 6. Residuated frames and (un)decidability 20 / 34

Admissibility of simple rules for a machine Definition Let M be a k -ACM and ( d ) be a d -rule. We say ( d ) is admissible in M if Acc( M ) = Acc( d M ) , i.e., W + M ∈ CRL + ( d ) . Gavin St.John Application 6. Residuated frames and (un)decidability 20 / 34

Admissibility of simple rules for a machine Definition Let M be a k -ACM and ( d ) be a d -rule. We say ( d ) is admissible in M if Acc( M ) = Acc( d M ) , i.e., W + M ∈ CRL + ( d ) . However, we will rephrase admissibility as the intermediate notions register and state admissibility . Gavin St.John Application 6. Residuated frames and (un)decidability 20 / 34

Admissibility cont. We define ≤ ¯ d to be the “ambient” instruction, for each x ∈ R ∗ k ( x 1 , ..., x n ∈ R ∗ k ), � � m d x 2 ∨ x 4 x ≤ ¯ � n i =1 x i ≤ ¯ � � n i =1 x d j ( i ) d , i j =1 and define ≤ ¯ d M as usual. Gavin St.John Application 6. Residuated frames and (un)decidability 21 / 34

Admissibility cont. We define ≤ ¯ d to be the “ambient” instruction, for each x ∈ R ∗ k ( x 1 , ..., x n ∈ R ∗ k ), � � m d x 2 ∨ x 4 x ≤ ¯ � n i =1 x i ≤ ¯ � � n i =1 x d j ( i ) d , i j =1 and define ≤ ¯ d M as usual. In this way, we see Acc( M ) ⊆ Acc(¯ d M ) ⊆ Acc( d M ) . Gavin St.John Application 6. Residuated frames and (un)decidability 21 / 34

Admissibility cont. We define ≤ ¯ d to be the “ambient” instruction, for each x ∈ R ∗ k ( x 1 , ..., x n ∈ R ∗ k ), � � m d x 2 ∨ x 4 x ≤ ¯ � n i =1 x i ≤ ¯ � � n i =1 x d j ( i ) d , i j =1 and define ≤ ¯ d M as usual. In this way, we see Acc( M ) ⊆ Acc(¯ d M ) ⊆ Acc( d M ) . We say ( d ) is register (state) admissible in M if Acc( M ) = Acc(¯ d M ) (Acc(¯ d M ) = Acc( d M ) ). Gavin St.John Application 6. Residuated frames and (un)decidability 21 / 34

Admissibility cont. We define ≤ ¯ d to be the “ambient” instruction, for each x ∈ R ∗ k ( x 1 , ..., x n ∈ R ∗ k ), � � m d x 2 ∨ x 4 x ≤ ¯ � n i =1 x i ≤ ¯ � � n i =1 x d j ( i ) d , i j =1 and define ≤ ¯ d M as usual. In this way, we see Acc( M ) ⊆ Acc(¯ d M ) ⊆ Acc( d M ) . We say ( d ) is register (state) admissible in M if Acc( M ) = Acc(¯ d M ) (Acc(¯ d M ) = Acc( d M ) ). Therefore, ( d ) is admissible in M iff it is both state and register admissible in M . Gavin St.John Application 6. Residuated frames and (un)decidability 21 / 34

Admissibility cont. We define ≤ ¯ d to be the “ambient” instruction, for each x ∈ R ∗ k ( x 1 , ..., x n ∈ R ∗ k ), � � m d x 2 ∨ x 4 x ≤ ¯ � n i =1 x i ≤ ¯ � � n i =1 x d j ( i ) d , i j =1 and define ≤ ¯ d M as usual. In this way, we see Acc( M ) ⊆ Acc(¯ d M ) ⊆ Acc( d M ) . We say ( d ) is register (state) admissible in M if Acc( M ) = Acc(¯ d M ) (Acc(¯ d M ) = Acc( d M ) ). Therefore, ( d ) is admissible in M iff it is both state and register admissible in M . Theorem Let M be a k -ACM and ( d ) a d -rule. Then ( d ) is state -admissible in M iff there is no substitution σ : Var → Var ∗ such that σ [ d ] ≡ x k ≤ x or σ [ d ] ≡ x k ≤ 1 . Gavin St.John Application 6. Residuated frames and (un)decidability 21 / 34

◮ For rules that don’t entail k -mingle ( x k ≤ x ), it suffices to show only register-admissibility for a machine. Gavin St.John Application 6. Residuated frames and (un)decidability 22 / 34

◮ For rules that don’t entail k -mingle ( x k ≤ x ), it suffices to show only register-admissibility for a machine. ◮ However, for some ACM’s M , it’s possible that C ∈ Acc(¯ d M ) but C �∈ Acc( M ) . Gavin St.John Application 6. Residuated frames and (un)decidability 22 / 34

◮ For rules that don’t entail k -mingle ( x k ≤ x ), it suffices to show only register-admissibility for a machine. ◮ However, for some ACM’s M , it’s possible that C ∈ Acc(¯ d M ) but C �∈ Acc( M ) . Example Consider M = M even and ( d ) given by x ≤ x 2 ∨ x 4 . Gavin St.John Application 6. Residuated frames and (un)decidability 22 / 34

◮ For rules that don’t entail k -mingle ( x k ≤ x ), it suffices to show only register-admissibility for a machine. ◮ However, for some ACM’s M , it’s possible that C ∈ Acc(¯ d M ) but C �∈ Acc( M ) . Example Consider M = M even and ( d ) given by x ≤ x 2 ∨ x 4 . ◮ q 0 r 3 �∈ Acc( M ) since 3 is odd. Gavin St.John Application 6. Residuated frames and (un)decidability 22 / 34

◮ For rules that don’t entail k -mingle ( x k ≤ x ), it suffices to show only register-admissibility for a machine. ◮ However, for some ACM’s M , it’s possible that C ∈ Acc(¯ d M ) but C �∈ Acc( M ) . Example Consider M = M even and ( d ) given by x ≤ x 2 ∨ x 4 . ◮ q 0 r 3 �∈ Acc( M ) since 3 is odd. ◮ However, q 0 r 3 ∈ Acc( d M ) , witnessed by q 0 r 3 = q 0 r 2 r ≤ d q 0 r 2 r 2 ∨ q 0 r 2 r 4 = q 0 r 4 ∨ q 0 r 6 ∈ Acc( M ) since q 0 r 4 ∈ Acc( M ) and q 0 r 6 ∈ Acc( M ) . Gavin St.John Application 6. Residuated frames and (un)decidability 22 / 34

Goal Given an ACM M and a d -rule ( d ) , is it possible to construct a new ACM M ′ such that ⇒ θ ( C ) ∈ Acc( M ′ ) (1) C ∈ Acc( M ) ⇐ (where θ : ID( M ) → ID( M ′ ) is some computable function), and (2) ( d ) is register-admissible in M ′ ? And if so, under what conditions? Gavin St.John Application 6. Residuated frames and (un)decidability 23 / 34

Then M K machine Let M = ( R 2 , Q, P ) be a 2-ACM and let K > 1 be given. We define the 3-ACM M K = ( R 3 , Q K , P K ) such that Gavin St.John Application 6. Residuated frames and (un)decidability 24 / 34

Then M K machine Let M = ( R 2 , Q, P ) be a 2-ACM and let K > 1 be given. We define the 3-ACM M K = ( R 3 , Q K , P K ) such that ◮ Q ⊂ Q K with q F the final state of M K and instruction ( q f r 1 r 2 ≤ F q F ∨ q F ) ∈ P K , Gavin St.John Application 6. Residuated frames and (un)decidability 24 / 34

Then M K machine Let M = ( R 2 , Q, P ) be a 2-ACM and let K > 1 be given. We define the 3-ACM M K = ( R 3 , Q K , P K ) such that ◮ Q ⊂ Q K with q F the final state of M K and instruction ( q f r 1 r 2 ≤ F q F ∨ q F ) ∈ P K , ◮ each forking instruction in P is contained in P K , Gavin St.John Application 6. Residuated frames and (un)decidability 24 / 34

Then M K machine Let M = ( R 2 , Q, P ) be a 2-ACM and let K > 1 be given. We define the 3-ACM M K = ( R 3 , Q K , P K ) such that ◮ Q ⊂ Q K with q F the final state of M K and instruction ( q f r 1 r 2 ≤ F q F ∨ q F ) ∈ P K , ◮ each forking instruction in P is contained in P K , ◮ each increment and decrement instruction of P is replaced by multiply and divide by K programs , i.e. qr ∀ ⊑ p q ′ r K ·∀ ≤ p q ′ r q ∈ P = ⇒ ⊆ P K ⊆ P K . qr ∀ ⊑ p q ′ r K \∀ ≤ p q ′ qr ∈ P = ⇒ Gavin St.John Application 6. Residuated frames and (un)decidability 24 / 34

Then M K machine Let M = ( R 2 , Q, P ) be a 2-ACM and let K > 1 be given. We define the 3-ACM M K = ( R 3 , Q K , P K ) such that ◮ Q ⊂ Q K with q F the final state of M K and instruction ( q f r 1 r 2 ≤ F q F ∨ q F ) ∈ P K , ◮ each forking instruction in P is contained in P K , ◮ each increment and decrement instruction of P is replaced by multiply and divide by K programs , i.e. qr ∀ ⊑ p q ′ r K ·∀ ≤ p q ′ r q ∈ P = ⇒ ⊆ P K ⊆ P K . qr ∀ ⊑ p q ′ r K \∀ ≤ p q ′ qr ∈ P = ⇒ Fact For each q ∈ Q , qr n 1 1 r n 2 ⇒ qr K n 1 r K n 2 ∈ Acc( M ) ⇐ ∈ Acc( M K ) . 2 1 2 Gavin St.John Application 6. Residuated frames and (un)decidability 24 / 34

Detecting applications of ≤ d Observation Consider a configuration where the contents of some register r is n = s + t , whereafer ≤ d is applied to t -many tokens, i.e., qr n = qr s r t ≤ d qr s ( r 2 t ∨ r 4 t ) = qr s +2 t ∨ qr s +4 t Gavin St.John Application 6. Residuated frames and (un)decidability 25 / 34

Detecting applications of ≤ d Observation Consider a configuration where the contents of some register r is n = s + t , whereafer ≤ d is applied to t -many tokens, i.e., qr n = qr s r t ≤ d qr s ( r 2 t ∨ r 4 t ) = qr s +2 t ∨ qr s +4 t Fact For ( d ) : x ≤ x 2 ∨ x 4 , if K > 3 , it is impossible for s + 2 t and s + 4 t to both be powers of K . Gavin St.John Application 6. Residuated frames and (un)decidability 25 / 34

Detecting applications of ≤ d Observation Consider a configuration where the contents of some register r is n = s + t , whereafer ≤ d is applied to t -many tokens, i.e., qr n = qr s r t ≤ d qr s ( r 2 t ∨ r 4 t ) = qr s +2 t ∨ qr s +4 t Fact For ( d ) : x ≤ x 2 ∨ x 4 , if K > 3 , it is impossible for s + 2 t and s + 4 t to both be powers of K . ◮ Consequently, qr n ∈ Acc(¯ d M K ) iff qr n ∈ Acc( M K ) , i.e Acc(¯ d M K ) = Acc( M K ) , so ( d ) is register-admissible in M K . Gavin St.John Application 6. Residuated frames and (un)decidability 25 / 34

Detecting applications of ≤ d Observation Consider a configuration where the contents of some register r is n = s + t , whereafer ≤ d is applied to t -many tokens, i.e., qr n = qr s r t ≤ d qr s ( r 2 t ∨ r 4 t ) = qr s +2 t ∨ qr s +4 t Fact For ( d ) : x ≤ x 2 ∨ x 4 , if K > 3 , it is impossible for s + 2 t and s + 4 t to both be powers of K . ◮ Consequently, qr n ∈ Acc(¯ d M K ) iff qr n ∈ Acc( M K ) , i.e Acc(¯ d M K ) = Acc( M K ) , so ( d ) is register-admissible in M K . ◮ ( d ) does not entail k -mingle, therefore ( d ) is M K admissible. Gavin St.John Application 6. Residuated frames and (un)decidability 25 / 34

Undecidable quasi-equational theory for 1-variable d -rules Let D 1 be the set of 1-variable d -rules defined via ( d ) ∈ D 1 iff ( d ) : x n ≤ � m ∈ X x m such that n ∈ X or | X \ { 0 }| ≥ 2 for some finite X ⊆ N . Theorem Let ( d ) ∈ D 1 . Then there exists a K > 1 such that ( d ) is admissible in M K for any 2-ACM M . Gavin St.John Application 6. Residuated frames and (un)decidability 26 / 34

Undecidable quasi-equational theory for 1-variable d -rules Let D 1 be the set of 1-variable d -rules defined via ( d ) ∈ D 1 iff ( d ) : x n ≤ � m ∈ X x m such that n ∈ X or | X \ { 0 }| ≥ 2 for some finite X ⊆ N . Theorem Let ( d ) ∈ D 1 . Then there exists a K > 1 such that ( d ) is admissible in M K for any 2-ACM M . Theorem Let Γ ⊂ D 1 be finite. Then then CRL + Γ has an undecidable quasi-equational theory. Gavin St.John Application 6. Residuated frames and (un)decidability 26 / 34

Undecidable quasi-equational theory for 1-variable d -rules Let D 1 be the set of 1-variable d -rules defined via ( d ) ∈ D 1 iff ( d ) : x n ≤ � m ∈ X x m such that n ∈ X or | X \ { 0 }| ≥ 2 for some finite X ⊆ N . Theorem Let ( d ) ∈ D 1 . Then there exists a K > 1 such that ( d ) is admissible in M K for any 2-ACM M . Theorem Let Γ ⊂ D 1 be finite. Then then CRL + Γ has an undecidable quasi-equational theory. ◮ CRL + ( x n ≤ x m ) has the FEP, and hence is decidable for any n � = m . ◮ However, the decidability of CRL + ( x n ≤ x m ∨ 1) remains open, for any n � = m > 0 . Gavin St.John Application 6. Residuated frames and (un)decidability 26 / 34

The general case. Let ( d ) be an n -variable d -rule. We define the set D via ( d ) ∈ D if there exists K > 1 such that: For all s, s ′ ∈ N n , if there exists α, α ′ ∈ N such that d • s + α and d • s ′ + α ′ are powers of K for each d ∈ d , then there exists ¯ d ∈ d d • s ′ = l n • s ′ , such that ¯ d • s = l n • s and ¯ where l n ( i ) = 1 for each i = 1 , ..., n . Gavin St.John Application 6. Residuated frames and (un)decidability 27 / 34

The general case. Let ( d ) be an n -variable d -rule. We define the set D via ( d ) ∈ D if there exists K > 1 such that: For all s, s ′ ∈ N n , if there exists α, α ′ ∈ N such that d • s + α and d • s ′ + α ′ are powers of K for each d ∈ d , then there exists ¯ d ∈ d d • s ′ = l n • s ′ , such that ¯ d • s = l n • s and ¯ where l n ( i ) = 1 for each i = 1 , ..., n . Theorem For every ( d ) ∈ D there exists a K > 1 such that ( d ) is admissible in M K , for any 2 -ACM M . Consequently, ( C ) RL + ( d ) has an undecidable quasi-equational theory. Gavin St.John Application 6. Residuated frames and (un)decidability 27 / 34