A NEW FOUNDATION OF A COMPLETE BOOLEAN EQUATIONAL LOGIC

Transcription

1 Bulletin of the Section of Logic Volume 38:1/2 (2009), pp George Tourlakis A NEW FOUNDATION OF A COMPLETE BOOLEAN EQUATIONAL LOGIC Abstract We redefine the equational-proofs formalism of [2], [3], [7], [5], [6] and present a direct proof that it is complete in the following strong and precise sense: Every tautology has an equational proof. As a side-effect we establish that equational logic can be founded on a single rule of inference (Leibniz) without adding any axioms beyond those present in the foregoing references. 1. Introduction Equational logic structures mathematical reasoning as a sequence of equivalences that are either given (axioms, known theorems) or obtained by the application of a single rewriting rule, loosely attributed to Leibniz: To obtain the next formula in a proof one replaces occurrences of some subformula of the current formula by some other formula known to be equivalent to the one it replaces. The idea is not new in algebra or trigonometry, where it is applied to prove equality of terms (expressions that represent objects such as numbers). The approach is relatively new in logic, originating with [1] and made accessible to university undergraduates by [2]. The basic question about the power of any new logic is its adequacy with respect to what it attempts to do, namely, to formalise the discovery of This research was partially supported by NSERC grant No

2 14 George Tourlakis truth through its proofs. First, do proofs only discover truths (soundness) and, second, can proofs of the accepted type discover all truths (completeness)? These issues have been addressed for versions of equational propositional ([3]) and predicate ([4], [7], [5], [6]) logic. Why this new treatment, or, what is new with it? We define the structure of an equational proof carefully and directly, rather than indirectly by treating it as a proof layout subsumed within the Hilbert-proofs paradigm. We prove completeness (Post s theorem) directly and explicitly for equational proofs, purging knowledge of other formalisations (Hilbert s, Gentzen s) from the metatheory. By contrast, all the cited references reduce the issue of completeness to that of some Hilbertstyle logic. By working a bit harder, and directly with equational proofs throughout, we learn a bit more about their proof theory. In particular, we show that using only one primary rule of inference, that of Leibniz, and using essentially the same axioms as [2], we can prove all tautologies. By contrast, [2] had four primary rules: Leibniz, transitivity, substitution and equanimity 1, while [4] had two (for propositional logic), each a hybrid of equanimity and Leibniz 2. We also give a self-contained proof of the deduction theorem if A proves B, then we can prove A B without assuming A directly for equational proofs, rather than via the Hilbert paradigm. Our proof avoids the flaw in the proof given in [4] (cf. 6.3). In the interest of brevity we skip over the proof of soundness, a rather straightforward result where we have nothing new to suggest. 2. The language The alphabet of the language of propositional logic includes exactly the following symbols (commas not included): p, 0,,,,,,, (, ) (1) 1 From A and A B one infers B. 2 Presumably the rule was given in two versions to ensure that the commutativity of is built-in a redundant precaution (cf. 5.11).

3 A New Foundation of a Complete Boolean Equational Logic 15 The first two symbols are used to build the sub-language of the propositional or Boolean variables, whose set we denote by var, via the grammar rules (in BNF 3 ) var ::= p zeros p where zeros ::= 0 0 zeros The language of propositional logic is the set of well-formed-formulae, wff or, in BNF notation, wff given by the rules wff ::= var ( wff ) ( wff wff ) ( wff wff ) ( wff wff ) Metamathematically, members of wf f are denoted by capital Latin letters with or without primes (A, B, D (n) 4, etc.), while the variables (members of var ) are denoted by the bold-face metasymbols p, q, r with or without primes. Thus, these metasymbols stand for expressions such as p0p, p00000p, etc. 5. The variables along with the symbols (pronounced bot ) and (pronounced top ) are the atomic formulae. We assume the meta-convention that Boolean connectives have decreasing strengths (priorities) in the order,,, and that they are all rightassociative. This allows us to get rid of many brackets and still be able, if needed, to reconstruct the implied formally correctly written formula. For example, p q r is short for (p (q r)) and p q r is short for ((p q) r). We have already adopted more connectives than is usual. In the end, more primitive connectives means more axioms; fewer connectives means fewer axioms but additional definitions. There is no real trade-off. 3 Backus Naur Form is a standard notation that finitely describes a formal language via grammar rules. ::= is pronounced is defined to be, while the big vertical lines are metasymbols, pronounced or, delimiting alternative definitions of the left hand side. Names in angular brackets, as in var, name a set of syntactic objects. 4 A superscript (n) denotes n primes. 5 Compare: In the Algol 60 programming language, the variables are finite strings that start with a letter of the Latin alphabet and continue with any letter or any decimal digit. This is the only acceptable way we can write down variables in a computer program that we expect to work. On the other hand, if we are only discussing programs, then we will likely resort to shorter metavariables, such as as X, Y, x, y.

4 16 George Tourlakis By the way, we do not need to reach the completeness result, thus we introduced neither axioms nor definitions for it. 3. Substitution The main reasoning tool is the rule of Leibniz according to which we may infer an equivalence C[A] C[B], if we already have the equivalence A B, where C[A] indicates our interest in the subformula A of C and, correspondingly and somewhat ambiguously, C[B] denotes the result of replacing (by straight search-and-replace ) some occurrences of A in C by B. But which some? For example, suppose I want to replace the first A in A A (denoted for short as C ) by B. An easy way to do this is to consider p A where p is new (or fresh ; or simply does not occur in A) and replace p by B. The metanotation C[p := A] means search-and-replace all occurrences of the Boolean variable p in C by A. We do nothing, and the result of this substitution is C, if p does not occur in C. We consider the compound symbol [p := A] as the most agressive of all connectives (least scope, or highest priority). Thus, X Y [p := Z] means that the substitution affects Y only; not X Y as a whole. With this notation out of the way, the inference rule Leibniz reads A B C[p := A] C[p := B] Leib We will refer in what follows to C as the C-part of the rule, or as its denominator. 4. Equational Proofs The concept of proof requires starting points (axioms), and rules (such as Leib) about how to extend proofs. So does the concept of theorems. The axioms are expressed as schemata (with fewer brackets than formality requires) using metavariables for formulae:

5 A New Foundation of a Complete Boolean Equational Logic 17 Associativity of ((A B) C) (A (B C)) (1) Symmetry of (A B) (B A) (2) vs. (3) Introduction of A A (4) Associativity of (A B) C A (B C) (5) Symmetry of A B B A (6) Idempotency of A A A (7) Distributivity of over A (B C) A B A C (8) Excluded Middle A A (9) Implication A B A B B (10) We will denote by Λ the set of logical axioms, that is, the infinite set of instances of schemata (1) (10). We have one rule of inference, the aforementioned Leibniz. In any given instance of the rule (i.e., where all its parts have been consistently instantiated) we say that the formula in the denominator is the result of an application (or just, result) of the rule applied on the formula on its numerator. Definition 4.1. An equational proof of rank n from a set of hypotheses (formulae) Γ a Γ-proof is defined as follows: (a) For n = 0, such a proof is a sequence of formulae A 1,..., A k, k 2, such that for all i = 1,... k 1, A i A i+1 is in Γ Λ. (b) For n > 0, a proof of rank n is a sequence of formulae A 1,..., A m, m 2, such that for all i = 1,... m 1, A i A i+1 is the result of Leibniz applied to some X Y that is either a member of Γ Λ, or occurs at the right end of some proof of rank n 1, which has at its left end a member of Γ Λ. Remark 4.2. It is clear that an equational proof has a tree structure, with a main branch, A 1,..., A k, of highest rank. This branch is a sequence of equivalences each of which is either known outright (axiom, or in Γ), or is obtained by applying once the rewriting rule replace equals by equals (X in A i replaced by Y, to obtain A i+1 ), using already constructed knowledge

6 18 George Tourlakis of the form X Y. A k is the root of the tree and its leaves (except possibly A 1 ) are in Γ Λ. Note that A B is the result of Leib applied to A B ( C-part is p). Definition 4.3. A formula A is a theorem from Γ, or a Γ-theorem, if it appears at the right end of some Γ-proof whose left end is in Γ Λ 6. We write Γ A. The notation A means that Γ =. We say that A is an absolute or logical theorem in this case. Reader-friendly proofs are written vertically as A 1 reason A 2 reason. reason A k where is a conjunctional 7 alias of. The subsidiary (sub)trees that complete the tree structure of the proof are referenced to by the reason entries. 5. Some Theorems and Metatheorems Metatheorem 5.1. If Γ A and Γ, then A. This so-called hypothesis strengthening is a metatheorem-schema (like most of our theorems and metatheorems), as the claim holds for all A, Γ and. Proof. The role of Γ in any given Γ-proof is to provide non-λ ( nonlogical ) axioms. This can do as well for the given proof. 6 This end is the only leaf not guaranteed a priori by 4.1 to be in Γ Λ. 7 That is, A B C means A B and B C. Of course, is not conjunctional.

7 A New Foundation of a Complete Boolean Equational Logic 19 Metatheorem 5.2. so is A 1,..., A m. If A 1,..., A k and A k+1,..., A m are Γ-proofs, then Proof. Every link A i, A i+1 of each of the two given chains (proofs) checks alright according to 4.1, by assumption. This is true whether the chains are checked separately, or after they are concatenated. Metatheorem 5.3. If A, B is a proof from Γ, then, for any formula C and any binary connective {,, }, so are C A, C B and A C, B C. Proof. If A B is in Γ Λ, suffice it to note that C A, C B and A C, B C are obtained from A B by one application of Leib, with denominator C p and p C respectively, where p does not occur in C 8. In general (cf. 4.1) let A B be obtained by Leib from X Y proved with lower rank with denominator D and substitution variable p. Without loss of generality, p does not occur in C, and thus C A C B and A C B C are also obtained by Leib (substitution in p) from X Y and denominator C D and D C respectively. Corollary 5.4. If A, B is a proof from Γ, then so are A A, A B and A A, B A. Corollary 5.5. If A 1,... A k is a proof from Γ, then so are A 1 A 1, A 1 A 2,..., A 1 A k and A 1 A 1, A 2 A 1,..., A k A 1. Using for we obtain the useful (cf. proof of 6.1): Corollary 5.6. If A 1,... A k is a proof from Γ, then so are A 1 A 1, A 1 A 2,..., A 1 A k and A 1 A 1, A 2 A 1,..., A k A 1. Theorem 5.7. A A. Proof. A A A Leib + axiom (7); C-part : p A A A See 4.1 and 4.3 to conclude. 8 We note that for the substitution to work as expected, leaving C invariant, p must be restricted as stated. We will take such newness or freshness of substitution variables for granted, without further notice.

8 20 George Tourlakis Corollary 5.8. Every axiom is an absolute theorem. Every A Γ satisfies Γ A. Proof By 5.7 A, A is an absolute proof. When the left A is an axiom (or in Γ), the right A has been proved. Metatheorem 5.9. If Γ A and Γ A B, then Γ B. Proof. The sequence A, B is a proof by the second assumption and Remark 4.2. Let the proof C 1,..., C r, A establish A, as in 4.3. In particular, C 1 Γ Λ. By 5.2, C 1,..., C r, A, B proves B from Γ. Corollary If A 1,..., A k is a Γ-proof and Γ A 1, then Γ A k. Corollary If A 1,..., A k is a proof from Γ, then Γ A 1 A k and Γ A k A 1. Proof. By 5.7, 5.10 and 5.5 (cf. 4.3). Note that if Γ A B then A, B is a Γ-proof by Remark 4.2. By 5.11 it follows that Γ B A as well. Of course, this does not prove axiom (2) since the latter is available even if A B. Corollary If Γ A B, then Γ C[p := A] [p := B]. Proof. If A B is Γ-proved with rank n, then C[p := A], [p := B] is a Γ-proof of rank n + 1 by 4.1. We conclude by Metatheorem A k,..., A 1. If A 1,..., A k is a Γ-proof, then so is its reversal, Proof. By 5.2 it suffices to prove that if the chain A, B is a proof, then so is B, A. So, let A, B be a proof from Γ. Then Γ B A by Thus, B, A is a Γ-proof by Remark 4.2. The import of 5.13, in practise, is that one can start with what one wants to prove and by a chain of equivalences reduce it to something in our dynamic knowledgebase (axiom, hypothesis, or theorem), thereby proclaiming the task completed. We state below some easy but fundamental results (some without proof) that we need toward proving completeness of our proof calculus. The reader can find complete (equational) proofs in [8].

9 A New Foundation of a Complete Boolean Equational Logic 21 Metatheorem In a chain such as A 1 A 2 A n placement of brackets, provably, does not matter: That is, if B and C are formulae obtained via two different placements then we have an absolute proof B, C. The above uses axiom schemata (1) and (2). Metatheorem In a chain such as A 1 A 2 A n placement of brackets, provably, does not matter: That is, if B and C are formulae obtained via two different placements then we have an absolute proof B, C. The above uses axiom schemata (5) and (6). Metatheorem In a chain such as A 1 A 2 A n the relative position of the A i, provably, does not matter: That is, if B and C are formulae obtained via two different permutations then we have an absolute proof B, C. The above uses axiom schemata (1) and (2). Metatheorem In a chain such as A 1 A 2 A n the relative position of the A i, provably, does not matter: That is, if B and C are formulae obtained via two different permutations then we have an absolute proof B, C. The above uses axiom schemata (5) and (6). Theorem A A B B. Proof. By 5.14 (that allows us to use brackets only for visual suggestion) and 5.16, (A B) (B A), A A B B is an absolute proof. Corollary A A and A A. Proof. A A,, is a proof (5.18 and axiom (3) with 5.13 and 5.2). Corollary Corollary (Redundant True) Γ A iff Γ A. Proof. By 5.19 A, A is a proof. Now invoke 5.10 and Theorem B.

10 22 George Tourlakis Proof. Done (cf. 5.7). B Leib + axiom (3); C-part : B p B ( ) axiom (8) B B Metatheorem If Γ A, then also Γ B A, for any B. Proof. By 5.21, and the obvious Leib, B A, B is a proof. The right end is a theorem by Theorem A A. Theorem A B A B. Theorem Theorem A A. (A B) A B and (A B) A B. Metatheorem If Γ A and Γ B, then Γ A B. Proof. By 5.21 and the note following 5.11 we have Γ A and Γ B. Thus A,, B is a proof from Γ. Corollary If Γ A B and Γ A B, then Γ B. Proof. Consider the proof that starts with a Γ-theorem (assumption plus 5.28) A B A B doing two obvious Leib at once using 5.25 A B A B doing two obvious Leib at once using axiom (6) B A B A axiom (2) B A B A axiom (8) B ( A A)

11 A New Foundation of a Complete Boolean Equational Logic 23 Leib + axiom (4) B 5.27 B 6. The Deduction Theorem The deduction theorem states: Metatheorem 6.1. If Γ {A} B, then also Γ A B. The proof hinges on the following result. Lemma 6.2. If Γ A (B C), then Γ A (D[p := B] D[p := C]). Proof. By induction on the complexity of D, that is, the number of Boolean connectives appearing in D. Basis. D has complexity 0, so it may be p in which case the claim is trivial, or one of q (other than p),,. The claim now is if Γ A (B C), then Γ A (D D). By 5.7 and 5.23 we have A (D D). Hence A (D D), since A (D D), A (D D) is a proof (5.25). By 5.1 we are done. Let now D have complexity n + 1, and take the I.H. 9 that the claim is true for all formulae with complexity n or less. We continue to assume Γ A (B C). We have one case for each binary connective: (i) D is E. We indicate a (condensed) proof A ( E[p := B] E[p := C]) Leib, twice, along with 5.26 A (E[p := B] E[p := C]) Leib ; C-part : A q A (E[p := B] E[p := C]) 9 Induction hypothesis.

12 24 George Tourlakis (ii) D is E G. In view of 5.25 (cf. also axiom (8)), we need to prove A D[p := B] A D[p := C] from Γ: A E[p := B] G[p := B] Leib + I.H.; C-part : q G[p := B]; 5.15 and 5.17 used as well A G[p := B] E[p := C] Leib + I.H.; C-part : q E[p := C]; 5.15 and 5.17 used as well A E[p := C] G[p := C] Having established the proof idea, the reader can easily handle the remaining two cases on D: E G, and E G (cf. [8]). The proof of 6.1 itself is now easy: Let A 1,..., A n, B (1) be a proof of B from Γ and A. Let us fix attention to one particular prooftree that completes (1) in the sense of Definition 4.1 (cf. also 4.2). We form a new tree, replacing each formula F that occurs in the complete tree of (1) by A F. If this new tree is a proof-tree, then we are done, for it is based on Γ only A has been eliminated as a hypothesis: Indeed, the left end of each contributing branch now is A R, where R is in Γ or in Λ or is A (cf. 4.1(b)). By 5.25 and 5.23, we have Γ A R in the first two cases, while by 5.25 and axiom (9) the same holds in the last case. But is this new tree with main branch A A 1,..., A A n, A B a proof? To answer affirmatively we need to validate the arbitrary link of this tree. Well, any link X j X j+1 on a branch of the original tree is (cf. 4.1) one of: (a) A member of Γ Λ {A}. Then Γ A (X j X j+1 ) by 5.8, 5.25 and 5.23, and therefore Γ A X j A X j+1 via the proof A (X j X j+1 ), A X j A X j+1 (cf. axiom (8)). This validates the corresponding link A X j A X j+1, from Γ alone, in the new tree, using earlier remarks and 4.1(b). 1. Result of Leib on a C D, which we got either as a member of Γ Λ {A} or as the right end of a sub-proof. Since by 6.2 we can

13 A New Foundation of a Complete Boolean Equational Logic 25 prove A (X j X j+1 ) from A (C D), we are done in either case (cf. 4.2). Remark 6.3. Clearly, if we had a reason to expect that A occurs in the main branch only we do not then neither 6.2 nor this argument would be needed and we would be done by 5.6. Presumably [4] implicitly so assume in their proof, which leaves the general case unsettled. 7. Completeness We say that Γ tautologically implies A, in symbols Γ = taut A, precisely when every valuation from the set {t, f} that satisfies Γ also satisfies A. Metatheorem 7.1. (Post s Tautology Theorem) If Γ = taut A, then Γ A. In particular, every tautology is an absolute theorem. Proof. The proof is the standard argument of the compactness of propositional logic. The aim here is to fit the proof entirely within our equationalproof tools. We prove the equivalent If Γ A, then Γ = taut A (1) and subdivide the proof into a sequence of claims. Claim One. There is an enumeration G 0, G 1, G 2,... (2) of a all formulae of Boolean logic. This is a trivial exercise in elementary set theory. We next turn to the fundamental construction toward the proof of (1). To this end we assume the hypothesis side, Γ A (3) and define a set of formulae,, which is maximal such that Γ and also A (4) is defined by stages. We set 0 = Γ, and, for n 0,

14 26 George Tourlakis if n {G n } A then n {G n } n+1 = else if n { G n } A then n { G n } else We now set = n 0 n. Claim Two. Γ. n Claim Three. For n 0, n A. Trivial induction on n. Claim Four. The last else case in the definition of n+1 never applies. Indeed, for the case to apply we must have n {G n } A and n { G n } A. By 6.1 these two lead to n G n A and n G n A. By 5.29, n A, which contradicts Three. Claim Five. A. Otherwise, let n, for some large enough n, contain all the -formulae used in the proof of A. Then n A, contrary to Three. Claim Six. For every formula B, either B is in, or B is in, but not both. Indeed, every B is some G m in the sequence (2). By the definition of n due to the inactive last else we note that B or B will be added to m to form m+1. Could both be added at different stages? If so, B A, and B A (both by 5.23 and 5.17). Thus (5.29) A, contradicting Five. Claim Seven. is deductively closed, that is, if B, then B. Indeed, if B /, then (by Six) B will be in. As in Six, A, contradicting Five. Now define v by setting, for each variable p, v(p) = t iff p. (5) Main Claim 7.2. For all formulae B, v(b) = t iff B. Proof. By induction on the complexity of B. We have the following cases: (i) B is a variable. The claim is (5). (ii) B is the formula. Since v( ) = t, we want. By Claim Seven, it suffices to have. We have this by 5.20 and 5.1. (iii) B is the formula. Since v( ) = f, we want /. If not, we would get A, via 5.23, thus A by 5.27 (cf. 5.9). (iv) B is C. Say v( C) = t. Then v(c) = f and the I.H. yields C / ; hence C is in by Claim Six.

15 A New Foundation of a Complete Boolean Equational Logic 27 Conversely, if C is in, then C / by Claim Six. By the I.H. we have v(c) = f; hence v( C) = t. (v) B is C D. Say v(c D) = t. There are two similar cases. Here is one: v(c) = t. By the I.H. C. It follows that C D by Hence C D by Claim Seven. Conversely, let C D. It must be that at least one of C or D is in. Say, instead, C and D are in. Thus (5.23) C D; hence D as by As in Six we now have A, which cannot be. Say then C. By I.H. v(c) = t; hence v(c D) = t. (vi) B is C D. Say v(c D) = t. There are two similar cases, v(c) = f or v(d) = t. We only consider the first: By I.H., C / ; thus C is in. By 5.23 (and 5.17) C D; hence C D by Thus C D is in (Claim Seven). Conversely, let C D be in. Thus, C D, by By case (v), we have C or D (possibly both) are in. If the former, then C / by Six; hence v(c) = f by I.H. If the latter, D hence v(d) = t by I.H. Either way, v(c D) = t. (vii) B is C D. Say v(c D) = t. There are two similar cases. Case where v(c) = v(d) = t. By I.H. C and D are in. Thus C D through a proof C D Leib twice, using the assumptions C, D and (5.21) Case where v(c) = v(d) = f. By I.H. neither of C and D are in. Thus both C and D are in, and, as before, C D. Using 5.26 twice and 5.24 we conclude C D and are reduced to the previous case. Via Claim Seven, both yield that C D is in. Conversely, let C D be in. We argue that it is impossible to have exactly one of C and D in. Indeed, say that C is in and D is not. Thus D is in. As above, this yields C D and by 5.26 (C D). Along with the assumption this yields A as in case Claim Six, contradicting Claim Five. Thus, either both C and D are in, where the I.H. furnishes v(c) = t = v(d), or neither is in, where the I.H. furnishes v(c) = f = v(d). We can now easily conclude the proof of Post s Theorem as follows: By 7.2, every formula B in and hence in Γ satisfies v(b) = t. On the

16 28 George Tourlakis other hand, as A it must be A /, therefore, by 7.2, v(a) = f. Thus Γ = taut A, which proves (1). Post s theorem is the completeness theorem of propositional calculus. Our development culminating to its proof shows the adequacy of the equational proof calculus that has as its only rule the replacement of equals for equals. An adaptation of this technique in the case of predicate calculus, and totally within the equational proof paradigm with a single primary rule of inference (Leibniz) will be the subject of a future article. References [1] Edsger W. Dijkstra and Carel S. Scholten, Predicate Calculus and Program Semantics, Springer-Verlag, New York, [2] David Gries and Fred. B. Schneider, A Logical Approach to Discrete Math, Springer-Verlag, New York, [3] David Gries and Fred. B. Schneider, Equational propositional logic, Information Processing Letters 53 (1995), pp [4] V. Lifschitz, On calculational proofs, Annals of Pure and Applied Logic 113 (2002), pp [5] G. Tourlakis, A basic formal equational predicate logic Part I, Bulletin of the Section of Logic, 29:1/2 (2000), pp [6] G. Tourlakis, A basic formal equational predicate logic Part II, Bulletin of the Section of Logic, 29:3 (2000), pp [7] G. Tourlakis, On the soundness and Completeness of Equational Predicate Logics, J. Logic Computat. 11/4 (2001), pp [8] G. Tourlakis, Mathematical Logic, John Wiley & Sons, Inc., New York, Department of Computer Science York University Toronto, Ontario M3J 1P3 Canada gt@cs.yorku.ca