Abstract Reduction Systems and Idea of Knuth-Bendix Completion Algorithm

Summary Educational content for abstract reduction systems concerning reduction, convertibility, normal forms, divergence and convergence, Church- Rosser property, term rewriting systems, and the idea of the Knuth-Bendix Completion Algorithm. The theory is based on [1].


Reduction and Convertibility
We consider ARS's which extend 1-sorted structures and are systems a carrier, a reduction where the carrier is a set, the reduction is a binary relation on the carrier.
Let A be a non empty set and r be a binary relation on A. Observe that A, r is non empty and there exists an ARS which is non empty and strict.
Let X be an ARS and x, y be elements of X.We say that x → y if and only if (Def. 1) x, y ∈ the reduction of X.
We introduce y ← x as a synonym of x → y.
We say that x → 01 y if and only if grzegorz bancerek (Def.2) (i) x = y, or (ii) x → y.One can verify that the predicate is reflexive.We say that x → * y if and only if (Def.3) The reduction of X reduces x to y.
Let us observe that the predicate is reflexive.From now on X denotes an ARS and a, b, c, u, v, w, x, y, z denote elements of X.
Now we state the propositions: (1) If a → b, then X is not empty.
(3) If x → * y → * z, then x → * z.The scheme Star deals with an ARS X and a unary predicate P and states that (Sch. 1) For every elements x, y of X such that x → * y and P[x] holds P [y] provided • for every elements x, y of X such that x → y and P[x] holds P [y].
The scheme Star1 deals with an ARS X and a unary predicate P and elements a, b of X and states that (Sch.2) P [b] provided • a → * b and

• P[a] and
• for every elements x, y of X such that x → y and P[x] holds P [y].
The scheme StarBack deals with an ARS X and a unary predicate P and states that (Sch.3) For every elements x, y of X such that x → * y and P [y] holds P [x] provided • for every elements x, y of X such that x → y and P [y] holds P[x].
The scheme StarBack1 deals with an ARS X and a unary predicate P and elements a, b of X and states that (Sch.4) P [a] provided • a → * b and

• P[b] and
• for every elements x, y of X such that x → y and P [y] holds P[x].
Let X be an ARS and x, y be elements of X.We say that x → + y if and only if (Def.4) There exists an element z of X such that x → z → * y.
Now we state the proposition: (4) x → + y if and only if there exists z such that x → * z → y.Proof: If x → + y, then there exists z such that x → * z → y.Define P[element of X] ≡ there exists u such that $ 1 → u → * y.For every y and z such that y → z and P[z] holds P [y].For every y and z such that y → * z and P[z] holds P [y] from StarBack.Let us consider X, x, and y.We introduce y ← 01 x as a synonym of x → 01 y and y ← * x as a synonym of x → * y and y ← + x as a synonym of x → + y.
We say that x ↔ y if and only if (Def.5) (i) x → y, or (ii) x ← y.One can check that the predicate is symmetric.Now we state the proposition: (5) x ↔ y if and only if x, y ∈ (the reduction of X) ∪ (the reduction of X) .Let us consider X, x, and y.We say that x ↔ 01 y if and only if (Def.6) (i) x = y, or (ii) x ↔ y.Observe that the predicate is reflexive and symmetric.We say that x ↔ * y if and only if (Def.7) x and y are convertible w.r.t. the reduction of X.
One can check that the predicate is reflexive and symmetric.Now we state the propositions: The scheme Star2 deals with an ARS X and a unary predicate P and states that (Sch.5) For every elements x, y of X such that x ↔ * y and P[x] holds P [y] provided • for every elements x, y of X such that x ↔ y and P[x] holds P [y].
The scheme Star2A deals with an ARS X and a unary predicate P and elements a, b of X and states that (Sch. ( ( The theorem is a consequence of ( 2) and (3).
Let us assume that x ↔ 01 y.Now we state the propositions: The theorem is a consequence of (36).

Examples of an Abstract Reduction System
The scheme ARSex deals with a non empty set A and a binary predicate R and states that (Sch.7) There exists a strict non empty ARS X such that the carrier of X = A and for every elements (ii) the reduction of ARS 02 = {0} × {0, 1, 2}, respectively.One can check that ARS 01 is non empty and ARS 02 is non empty.
From now on i, j, k denote elements of ARS 01 .Now we state the propositions: (87) Let us consider a set s. Then s is an element of ARS 01 if and only if s = 0 or s = 1.(88) i → j if and only if i = 0.The theorem is a consequence of (87).
In the sequel l, m, n denote elements of ARS 02 .Now we state the propositions: (89) Let us consider a set s. Then s is an element of ARS 02 if and only if s = 0 or s = 1 or s = 2. (90) m → n if and only if m = 0.The theorem is a consequence of (89).

Normal Forms
Let us consider X and x.We say that x is a normal form if and only if (Def.11) There exists no y such that x → y.
Now we state the proposition: (91) x is a normal form if and only if x is a normal form w.r.t. the reduction of X. Proof: If x is a normal form, then x is a normal form w.r.t. the reduction of X by [13, (87)].Let us consider X, x, and y.We say that x is a normal form of y if and only if (Def.12) (i) x is a normal form, and (ii) y → * x.Now we state the proposition: (92) x is a normal form of y if and only if x is a normal form of y w.r.t. the reduction of X.The theorem is a consequence of (91).Let us consider X and x.We say that x is normalizable if and only if (Def.13) There exists y such that y is a normal form of x.Now we state the proposition: (93) x is normalizable if and only if x has a normal form w.r.t. the reduction of X.The theorem is a consequence of (92).Let us consider X.We say that X is WN if and only if (Def.14) x is normalizable.
We say that X is SN if and only if (Def.15) Let us consider a function f from N into the carrier of X.Then there exists a natural number i such that f (i) → f (i + 1).We say that X is UN* if and only if (Def.16) If y is a normal form of x and z is a normal form of x, then y = z.
We say that X is UN if and only if (Def.17) If x is a normal form and y is a normal form and x ↔ * y, then x = y.
We say that X is NF if and only if (Def.18) If x is a normal form and x ↔ * y, then y → * x.
Now we state the propositions: (94) X is WN if and only if the reduction of X is weakly-normalizing.The theorem is a consequence of (93).
(95) If X is SN, then the reduction of X is strongly-normalizing.
(96) If X is not empty and the reduction of X is strongly-normalizing, then X is SN.From now on A denotes a set.Now we state the proposition: (97) X is SN if and only if there exists no A and there exists z such that z ∈ A and for every x such that x ∈ A there exists y such that y ∈ A and x → y.The scheme notSN deals with an ARS X and a unary predicate P and states that (Sch.8) X is not SN provided • there exists an element x of X such that P[x] and • for every element x of X such that P[x] there exists an element y of X such that P [y] and x → y.

Now we state the propositions:
(98) X is UN if and only if the reduction of X has unique normal form property.Proof: Set R = the reduction of X.If X is UN, then R has unique normal form property by (91), [6, (28) Proof: Set R = the reduction of X.If X is NF, then R has normal form property by ( 91), [6, (28), ( 31), ( 12)].
Let us consider X and x.Assume there exists y such that y is a normal form of x and for every y and z such that y is a normal form of x and z is a normal form of x holds y = z.The functor nf x yielding an element of X is defined by (Def.19) it is a normal form of x.

Now we state the propositions:
(100) Suppose there exists y such that y is a normal form of x and for every y and z such that y is a normal form of x and z is a normal form of x holds grzegorz bancerek y = z.Then nf x = nf α (x), where α is the reduction of X.The theorem is a consequence of (92).
(101) If x is a normal form and x → * y, then x = y.The theorem is a consequence of (85).
(102) If x is a normal form, then x is a normal form of x.
(103) If x is a normal form and y → x, then x is a normal form of y.
(104) If x is a normal form and y → 01 x, then x is a normal form of y.
(105) If x is a normal form and y → + x, then x is a normal form of y.
(106) If x is a normal form of y and y is a normal form of x, then x = y.
(107) If x is a normal form of y and z → y, then x is a normal form of z.
(108) If x is a normal form of y and z → * y, then x is a normal form of z.
(109) If x is a normal form of y and z → * x, then x is a normal form of z.
Let us consider X.One can check that every element of X which is a normal form is also normalizable.Now we state the propositions: (110) If x is normalizable and y → x, then y is normalizable.
(111) X is WN if and only if for every x, there exists y such that y is a normal form of x.
(112) If for every x, x is a normal form, then X is WN.The theorem is a consequence of (102).
One can verify that every ARS which is SN is also WN.Now we state the propositions: (113) If x = y and for every a and b, a → b iff a = x, then y is a normal form and x is normalizable.The theorem is a consequence of (2).
(114) There exists X such that (i) X is WN, and (ii) X is not SN.
Consider X being a strict non empty ARS such that the carrier of X = {0, 1} and for every elements x, y of X, One can verify that every ARS which is NF is also UN* and every ARS which is NF is also UN and every ARS which is UN is also UN*.Now we state the proposition: (115) If X is WN and UN* and x is a normal form and x ↔ * y, then y → * x.
For every y and z such that y ↔ z and P [y] holds P[z].For every y and z such that y ↔ * z and P [y] holds Observe that every ARS which is WN and UN* is also NF and every ARS which is WN and UN* is also UN.Now we state the propositions: (116) If y is a normal form of x and z is a normal form of x and y = z, then x → + y.The theorem is a consequence of (85) and ( 101).(117) If X is WN and UN*, then nf x is a normal form of x. (118) If X is WN and UN* and y is a normal form of x, then y = nf x.
Let us assume that X is WN and UN*.Now we state the propositions: (119) nf x is a normal form.The theorem is a consequence of (117).(120) nf nf x = nf x.The theorem is a consequence of ( 119

Divergence and Convergence
Let us consider X, x, and y.We say that x * y if and only if (Def.20) There exists z such that x ← * z → * y.
Observe that the predicate is symmetric and reflexive.We say that x * y if and only if (Def.21) There exists z such that x → * z ← * y.
One can check that the predicate is symmetric and reflexive.We say that x 01 y if and only if (Def.22) There exists z such that x ← 01 z → 01 y.
Observe that the predicate is symmetric and reflexive.We say that x 01 y if and only if (Def.23) There exists z such that x → 01 z ← 01 y.
One can check that the predicate is symmetric and reflexive.One can verify that every ARS which is DIAMOND is also CONF and every ARS which is DIAMOND is also CR and every ARS which is CR is also WCR and every ARS which is CR is also CONF and every ARS which is CONF is also CR.Now we state the proposition: (157) If X is non CONF and WN, then there exists x and there exists y and there exists z such that y is a normal form of x and z is a normal form of x and y = z.The theorem is a consequence of (108).
Newman Lemma: Every ARS which is SN and WCR is also CR and every ARS which is CR is also NF and every ARS which is WN and UN is also CR and every ARS which is SN and CR is also COMP and every ARS which is COMP is also CR WCR NF UN UN* and WN.Now we state the proposition: (158) If X is COMP, then for every x and y such that x ↔ * y holds nf x = nf y.
Observe that every ARS which is WN and UN* is also CR and every ARS which is SN and UN* is also COMP.

Term Rewriting Systems
We consider TRS structures which extend ARS's and universal algebra structures and are systems a carrier, a characteristic, a reduction where the carrier is a set, the characteristic is a finite sequence of operational functions of the carrier, the reduction is a binary relation on the carrier.
One can verify that there exists a TRS structure which is non empty, nonempty, and strict.
Let S be a non empty universal algebra structure.We say that S is group-like if and only if (Def.29) (i) Seg 3 ⊆ dom(the characteristic of S), and We say that S is (R10) if and only if We say that S is (R11) if and only if We say that S is (R12) if and only if We say that S is (R13) if and only if We say that S is (R14) if and only if We say that S is (R15) if and only if In the sequel S denotes a group-like quasi total partial invariant non empty non-empty TRS structure and a, b, c denote elements of S. Now we state the propositions: (171) If S is (R6) and (R7), then a • 1 S * a.The theorem is a consequence of (2).
Let us consider X, x, and y.We say that x ↔ + y if and only if (Def.8) There exists z such that x ↔ z ↔ * y.
(12)If x → y → z, then x → * z.The theorem is a consequence of (2) and ) If x → * y, then x ↔ * y.Proof: Define P[element of X] ≡ x ↔ * $ 1 .For every y and z such that y → z and P[y]holds P[z].P[y]from Star1.
(83) Suppose for every x and y such that x → z and x → y holds y → z.If x → z and x → * y, then y → z.Proof: Define P[element of X] ≡ $ 1 → z.For every u and v such that u → * v and P[u] holds P[v] from Star.(84) If for every x and y such that x → y holds y → x, then for every x and y such that x ↔ * y holds x → * y.Proof: Define P[element of X] ≡ x → * $ 1 .For every u and v such that u ↔ v and P[u] holds P[v].For every u and v such that u ↔ * v and P[u] holds P[v] from Star2.(85) If x → * y, then x = y or x → + y.Proof: Define P[element of X] ≡ x = $ 1 or x → + $ 1 .For every y and z such that y → z and P[y] holds P[z].P[y] from Star1.(86) If for every x, y, and z such that x → y → z holds x → z, then for every x and y such that x → + y holds x → y.Proof: Consider z such that x → z and z → * y.Define P[element of X] ≡ x → $ 1 .P[y] from Star1.
X is NF if and only if the reduction of X has normal form property.
(31))].x is a normal form w.r.t.R and y is a normal form w.r.t.R and x and y are convertible w.r.t.
Now we state the propositions: (124) x * y if and only if x and y are divergent w.r.t. the reduction of X. (125) x * y if and only if x and y are convergent w.r.t. the reduction of X. (126) x 01 y if and only if x and y are divergent at most in 1 step w.r.t. the reduction of X. (154) Let us consider a non empty ARS X.Then X is COMP if and only if the reduction of X is complete.The theorem is a consequence of (151), (95), and (96).(155) If X is DIAMOND and x ← * z → 01 y, then there exists u such that x → 01 u ← * y.Proof: Define P[element of X] ≡ there exists u such that $ 1 → 01 u ← * y.For every u and v such that u → v and P[u] holds P[v].For every u and v such that u → * v and P[u] holds P[v] from Star.(156) If X is DIAMOND and x ← 01 y → * z, then there exists u such that x → * u ← 01 z.The theorem is a consequence of (155).