Functional completeness of weak logics with the strict negation
Ivan Chajda, Radom´ır Halaˇ s Palack´ y University Olomouc Department of Algebra and Geometry Czech Republic
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The classical propositional logic is useful for logical design and switching circuits because every boolean function can be represented as a boolean polynomial. More precisely, if f : {0, 1}n → {0, 1} is an arbitrary function (n ≥ 1) then f can be expressed as a conjunction of elementary disjunctions (or dually), see e.g. [2]. This is based on the following algebraic concepts. Definition 1. A finite algebra A = (A, F ) is called functionally complete (see e.g. [3] ) if for every integer n ≥ 1 and each mapping h : An → A there exists an n-ary polynomial p(x1, ..., xn) over A such that h(a1, ..., an) = p(a1, ..., an) holds for all a1, ..., an ∈ A. Definition 2. By the ternary discriminator (see e.g. [5]) on a non-empty set A is meant a function t : A3 → A such that t(a, a, c) = c and t(a, b, c) = a for all a, b, c ∈ A with a 6= b. An algebra A = (A, F ) is called a discriminator algebra if there is the ternary discriminator term function of A.
The following result by H. Werner [5] is the base for our considerations. Proposition. A finite algebra A is functionally complete if and only if the ternary discriminator is a polynomial of A. For example, in the light of previous concepts, the two-element Boolean algebra B = ({0, 1}; ∨, ∧, ¬, 0, 1) is a discriminator algebra since the term t(x, y, z) = (x ∧ ¬y) ∨ (x ∧ z) ∨ (¬y ∧ z) is the ternary discriminator on {0, 1}. In what follows we are interested in the problem how the classical logic can be weakened to be still functionally complete. It means we try to discern an algebra of an implication reduct of a propositional logic having the ternary discriminator as a polynomial.
Definition 3. By a weak logic with operations F will be called an algebra A = (A; {→, 0, 1} ∪ F ) such that 0, 1 ∈ A are constants, F is a set of operations on A (possibly empty ) and implication → is any binary operation on A satisfying the axioms (I1) 1 → x = x, 0 → x = 1, x → 1 = 1; (I2) (x → y = y → x = 1) iff x = y. We usually assume that among term operations of a weak logic there is a kind of conjunction, equivalence or negation (depending on our needs). Definition 4. Let A be a set with a distingushed element 1. A binary operation ¯ on A will be called a weak conjunction if it satisfies the axioms (C1) 1 ¯ x = x, x ¯ 1 = x; (C2) x ¯ y = 1 implies x = y = 1. A binary operation p(x, y) on A will be called a weak equivalence if it satisfies the axioms (E1) p(x, 1) = p(1, x) = x; (E2) p(x, y) = 1 iff x = y.
Most of propositional logical calculi known to us are equipped by implication, conjunction or equivalence that satisfy the above mentioned axioms (I1), (I2), (C1), (C2), (E1) and (E2). Thus we claim our ”logic” to be weak. Lemma 1. Let A be a weak logic with → and a weak conjunction ¯. Then p(x, y) = (x → y) ¯ (y → x) is a weak equivalence on A. In the classical two-valued propositional logic we have ¬0 = 1 and ¬1 = 0 where ¬x denotes the negation of x. It can be alternatively expressed by ¬1 = 0 and ¬x = 1 for x 6= 1. We extend this concept to the general case as follows.
Definition 5. Let A be a set with two distinguished elements 0 and 1. A unary operation ¬ will be called strict negation if ¬1 = 0 and ¬x = 1 for x 6= 1. A unary operation 4 will be called globalization (see [4 ], [1 ]) if 4(1) = 1 and 4(x) = 0 for x 6= 1. Remark 1. If ¬ is the strict negation then 4(x) = ¬¬x is the globalization. Hence, the assumption of the existence of a globalization is weaker than that of a strict negation. Thus we will often assume that A is a weak logic with globalization rather than A is with strict negation although the negation is a basic propositional connective. Theorem 1. Let A be a weak logic with a weak conjunction and globalization. Then A is a discriminator algebra whose ternary discriminator is the term t(x, y, z) = (4((x → y) ¯ (y → x)) → z) ¯ ((4((x → y) ¯ (y → x)) → 0) → x).
A weak logic A is called finite if its carrier A is finite. We can characterize functional completeness of weak logics by the existence of globalization. Theorem 2. A finite weak logic with a weak conjunction is functionally complete if and only if the globalization is its polynomial. Consider a relatively pseudocomplemented semilattice S = (S; ∧, ∗). Then x → y = x ∗ y clearly satisfies (I1) and (I2) and x ¯ y = x ∧ y satisfies (C1) and (C2). Hence, every relatively pseudocomplemented semilattice is a weak logic with a weak conjunction: Hence, we have Corollary 3. Let A = (A; ∧, ∗) be a finite relatively pseudocomplemented semilattice with the least element 0 and greatest element 1. If A is an algebra on A such that ∧ and ∗ are polynomials of A, then A is functionally complete if and only if the globalization is its polynomial.
Consider a chain C (as a ∨-semilattice). Denote by x+ the dual pseudocomplement of x ∈ C, i.e. x+ is the least element of C satisfying x ∨ x+ = 1. Then clearly ¬x = x+ is the strict negation on C. By using of Remark 1 and Corollary 3, we conclude Corollary 4. Every finite chain considered as a relatively pseudocoplemented lattice with dual pseudocomplementation is functionally complete. In what follows we show that the existence of a weak equivalence is a necessary and sufficient condition for a weak logic to be a discriminator algebra under the additional condition that the implication → satisfies the identity y → ((x → y) → y) = 1.
(B)
To the authors’ knowledge (B) is satisfied in the majority of reasonable propositional logics.
Theorem 5. Let A be a weak logic with the strict negation satisfying (B). Then A is a discriminator algebra if and only if a weak equivalence is a term function of A. Proof. If a weak equivalence p(x, y) is a term function of A then, by (T) in the proof of Theorem 1, the term function t(x, y, z) = p(¬¬(p(x, y)) → z, (¬¬(p(x, y)) → 0) → x) is the ternary discriminator. Conversely, assume that the ternary discriminator t(x, y, z) is a term function of A. Define the term function p(x, y) = t(x, (x → y) → y, ((y → x) → x) → y). We prove (E2). One can easily verify p(x, x) = 1. For the converse, suppose p(x, y) = 1. We have two cases: (a) If x = (x → y) → y then, by (B), y → x = 1 and, moreover, 1 = p(x, y) = t(x, x, (1 → x) → y) = = (1 → x) → y = x → y. Applying (I2) we conclude x = y.
(b) Suppose to the contrary that x 6= (x → y) → y. Then 1 = p(x, y) = = t(x, (x → y) → y, ((y → x) → x) → y) = x and hence by (I1) and (I2) (x → y) → y = (1 → y) → y = y → y = 1 = x, a contradiction. Thus p(x, y) = 1 if and only if x = y proving (E2). For (E1), we proceed as follows: p(1, y) = t(1, (1 → y) → y, ((y → 1) → 1) → y) = t(1, y → y, (1 → 1) → y) = t(1, 1, y) = y and p(x, 1) = t(x, (x → 1) → 1, ((1 → x) → x) → 1) = t(x, 1 → 1, (x → x) → 1) = t(x, 1, 1 → 1) = t(x, 1, 1) = x. Thus p(x, y) is really a weak equivalence on A. 2
References
[1] Chajda I., Vychodil V.: A note on residuated lattices with globalization, preprint. [2] Gr¨ atzer G.: On Boolean functions (Notes on lattice theory II ), Revue de Math. Pures et Appl. 7 (1962), 693-697. [3] Kaarli K., Pixley A. F.: Polynomial completeness in algebraic systems, Boca Raton, Chapman & Hall, CRC, 2000. [4] Takeuti G., Titani S.: Globalization of intuitionistic set theory, Ann. Pure Appl. Logic, 33 (1967), 195-211. [5] Werner H.: Discriminator algebras, Studien zur Algebra und ihre Anwendungen, Band 6, Akademie-Verlag, Berlin, 1978.
