Generalised elimination rules and harmony

Transcription

1 Generalised elimination rules and harmony Roy Dyckhoff Based on joint work with Nissim Francez Supported by EPSR grant EP/D064015/1 St ndrews, May 26, Introduction Standard natural deduction rules for Int (intuitionistic predicate logic) in the style of Gentzen and Prawitz are presumed to be familiar. The theory of cut-elimination for sequent calculus rules is very clear; whether a derivation in a sequent calculus is cut-free or not is easily defined, according to the presence or absence of instances of the ut rule. Normality of a (natural) deduction is less clear; there are many inequivalent definitions in the literature. For implicational logic it is easy; but rules such as the elimination rule for disjunction cause problems with the notion of maximal formula occurrence, and more problems when minor premisses have vacuous discharge of assumptions. One proposed solution is the use of generalised elimination rules, i.e. GE-rules. These can also be motivated in terms of Prawitz inversion principle: the conclusion obtained by an elimination does not state anything more than what must have already been obtained if the major premiss of the elimination was inferred by an introduction [17]. See the invaluable survey [11] for a full discussion, including the idea s antecedents in the work of Lorenzen. The standard elimination rule for disjunction is already a GE rule: B and the same pattern was proposed (as a GE-rule) in the early 1980s for conjunction B [, B] by various authors, notably Martin-Löf and Schroeder-Heister, inspired in part by type theory [7] (where conjunction is a special case of the Σ-type constructor) and in part by linear logic (where conjunction appears in two forms, multiplicative and additive &) [4]. To this one can add [2] a GE-rule for implication B and similarly for universal quantification, with the standard rule for existential quantification. One thus has a calculus of rules in natural deduction style for Int; such calculi have been studied by von Plato and (independently but later) by Tennant. With the definition that a deduction is normal iff the major premiss of every elimination step is an assumption, the main results are 1. a weak normalisation result (every deduction can be replaced by a normal deduction of the same conclusion from the same assumptions) [12]; 1

2 2. a strong normalisation result (for the implicational fragment) (the obvious set of rules for reducing non-normal deductions is SN, i.e. every reduction sequence terminates) [5, 20]; 3. a strong normalisation result (for the full language): reportedly a straightforward extension of the proof for implication [14]; 4. a 1-1 correspondence with intuitionistic sequent calculus derivations [13]. Despite the above results, I see some disadvantages: 1. s commented in [2], the GE-rule for implication does not appear to generalise to the type-theoretic constant Π of which is a special case. 2. Sequent calculus has (for each derivable sequent) rather too many derivations, in comparison to natural deduction, since there are many permutations of derivations each of which is, when translated to natural deduction, replaced by an identity of deductions. The GE-rules have the same problem, which interferes with rather than assists in proof search. (On the other hand, it has to be admitted that there are some nice properties [9].) 3. s shown below, the general method (hoped for by many) for generating an elimination rule from an arbitrary set (possibly empty) of introduction rules for a logical constant leads to a mess. 4. In particular, and given the difficulty for implication, I see no obvious mechanism for proving the normalisation results in a mechanical way for elimination rules generated from an arbitrary set of introduction rules. Without such results the ideas are of limited interest. Some of the ideas in this paper area already in [3]. 2 Bullet Read [18] has, following a suggestion of Schroeder-Heister, proposed as a logical constant a nullary operator with the single (but, alas, impure) introduction rule The obvious GE-rule justified by this is then [ ] which, given the usual E rule and the unnecessary duplication of premisses, can be simplified to So, by this E rule, the premiss of the I rule is deducible, hence is deducible, hence is deducible. There is however a weakness (other than that it leads to inconsistency) in the alleged justification of as a logical constant. We follow Martin-Löf [8] in accepting that we understand a proposition when we understand what it means to have a canonical proof of it, i.e. what forms a canonical proof can take. In the case of, there is a circularity: the introduction rule gives us a canonical proof only once we have a proof of from the assumption of, i.e. have a method for transforming arbitrary proofs of into proofs of. The reference here to arbitrary proofs of is the circularity. There are similar ideas about type formers, and it is instructive to consider another case, an apparent circularity: the formation rule [7] for the type N of natural numbers. That is a type that we understand when we know what its canonical elements are; these are 0 and, when we have an element n of N, the term s(n). The reference back to an element n of N looks like a circularity of the same kind; but it is very different we don t 2 [ ]

3 need to grasp all elements of N to construct a canonical element by means of the rule, just one of them, namely n. formal treatment of this issue has long been available in the type theory literature, e.g. [10, 6, 1]. We will try to give a simplified version of the ideas. With the convention that propositions are interpreted as types (of their proofs), we take type theory as a generalisation of logic, with ideas and restrictions in the former being applicable to the latter. The simplest case (N) has just been considered and the recursion explained as harmless. What about more general definitions? The definition of the type N can be expressed as saying that N is the least fixed point of the operator Φ N = λx.(1+x), i.e. N = µx.(1+x). Similarly, the type of lists of natural numbers is µl.(1+n L), and the type of binary trees with leaves in and node labels in B is µt. + (T B T ). unary operator Φ = λx.... is said to be positive iff the only occurrences of the type variable X in the body... are positive, where an occurrence of X in the expression B is positive (resp., negative) iff it is a positive (resp. negative) occurrence in B or a negative (resp. positive) occurrence in ; a variable occurs positively in itself, and occurs positively (resp. negatively) in + B and in B just where it occurs positively (resp. negatively) in or in B. definition of a type as the least fixed point of an operator is then positive iff the operator is positive. Read s, then, is defined as µx.(x ). This is not a positive definition; the negativity of the occurrence of X in the body X captures the circular idea that can only be grasped once we already have a full grasp of what the proofs of might be. In practice, a stronger requirement is imposed, that the definition be strictly positive, i.e. the only occurrences of the type variable X in the body... are strictly positive, where an occurrence of X in the expression B is strictly positive iff it is a strictly positive occurrence in B; a variable occurs strictly positively in itself, and occurs strictly positively in + B and in B just where it occurs strictly positively in or in B. definition of a type as the least fixed point of an operator is then strictly positive iff the operator is strictly positive. This preliminary paper/talk is not the place for more detail; there is however a substantial theory, including strong normalisation theorems [10, 6, 1]. 3 GE rules The idea that the grounds for asserting a proposition collectively form some kind of structure which can be used as the assumptions in the minor premiss of a GE-rule is attractive, as illustrated by the idea that, where two formulae, B are used as the grounds for asserting B, one may make the pair, B the assumptions of the minor premiss of GE. In practice, there is an explosion of possibilities, which we analyse in order as follows: a logical constant (such as,, or (exclusive or)) can be introduced by zero or more rules; each of these rules can have zero or more premisses; each such premiss may discharge zero or more assumptions (as in E or E); each such premiss may abstract over one or more variables, as in I; and a premiss may take a term as a parameter (as in I). It is not suggested that this list is exhaustive: conventions such as those of linear logic about avoiding multiple or vacuous discharge will extend it; but it is long enough to illustrate the explosion. 3.1 Several rules Where a logical constant (such as is introduced by several alternative rules, one can formulate appropriate GE-rule as having several minor premisses, one for each of the I-rules, giving a case analysis. This is very familiar from the case of and the usual E rule: B so an appropriate generalisation for n 0 alternative I-rules is to ensure that the GE-rule has n minor premisses. 3

4 3.2 Rule has several premisses Now there are two possibilities following the general idea that the conclusion of a GE-rule is arbitrary. Let us consider the intuitionistic constant (with its only I-rule having two premisses) as an example. The first possibility is as illustrated above on page 1. The second is to have two GE-rules: the second rule is similar to the first rule, which is B and it is routine to show that the ordinary GE-rule for is derivable in a system including these two rules. Tradition goes for the first possibility; examples below show however that this doesn t work and that the second is required. 3.3 Premiss discharges some assumptions Natural deduction s main feature is that assumptions can be discharged, as illustrated by the I-rule for and the E-rule for. This raises difficulties for the construction of the appropriate GE-rules; Prawitz [16] got it wrong, Schroeder-Heister [19] gave an answer in the form of a system of rules of higher level, allowing discharge not just of assumptions but of rules (which may themselves discharge..., etc) but this system seems to have found only a few adherents. n alternative was proposed in [2] and adopted more widely by [13], the GE-rule for being B Let us now consider the position when two premisses discharge an assumption: consider the logical constant with one I-rule, namely B B ccording to our methodology, we have two possibilities for the GE-rule; first, have the minor premiss of the rule with two assumptions B, being discharged and some device to ensure that there are other premisses with and B as conclusions. I see no way of doing this coherently, i.e. with somehow tied to the discharge of B and vice-versa. The alternative is to have two GE-rules, along the lines discussed above for, and these are clearly and B B B by means of which it is clear that, from the assumption of B, one can construct a proof of B using the introduction rule as the last step, suggesting the completeness of this set of rules in a sense explored in [15]. We are thus committed in general to the use of the second rather than the first possibility of GE-rules; the use of two such rules rather than one when there are two premisses in the I-rule. Let us now consider a combination of such ideas, e.g. two I-rules each of which discharges an assumption, e.g. the pair 4

5 and What is the appropriate GE-rule? It might be B but that only captures, as it were, the first of the two cases; so we have to try also B but then these two need to be combined into a single rule (at which point the style file for Buss proofs is inadequate GRRR), so it is (using a mixed notation) something like B B which is weird; with only the second and last of these premisses, already can be deduced. The meaning of is thus surely not being captured. In other words, the use of the methodology of [2] for a GE rule for implication when either the constant being defined has several introduction rules or one or more of such rules have several premisses can lead (a) to a number (> 1) of GE rules, none of which on its own suffices, and (b) to a disharmonious mess. lready there are enough problems, before we start considering the cases where the premiss abstracts over several variables or the premiss recurses on the constant being defined. The standard solution [1, 6] is to reject the idea of [2] that the rule for handling implication (and other situations where assumptions are discharged) be treated as illustrated above and instead to take implication (and its generalisation, universal quantification) as primitive, with traditional special elimination rules (e.g. M P ) but to allow generalised rules elsewhere (e.g. for and its generalisations Σ and ). This deals fairly nicely with ; likewise, it deals with as [ B] [B ] but no doubt more examples can be developed to upset even this kind of response. 4 onclusion lthough the idea that the grounds for asserting a proposition are easily collected together as a unit is attractive, the different ways in which it can be done (conjunctive, disjunctive, with assumption discharge,..., recursion) generate (if the GE rules pattern is followed) many problems for the programme of mechanically generating one (or more) elimination rules for a logical constant, other than in simple cases. Without success of such a programme, it is hard to see what generalised harmony can amount to, except as carried out in the work of e.g. [1] where strictly positive inductive type definitions lead automatically to rules for reasoning by induction and case analysis over objects of the types thus defined. 5

6 References [1] Bertot, Y. & asteran, P., Interactive Theorem Proving and Program Development, Springer [2] Dyckhoff, R., Implementing a simple proof assistant, Leeds Logic Teaching Workshop [3] Francez, N. & Dyckhoff, R., note on harmony, Submitted, [4] Girard, J.-Y., Linear logic, Theor. omputer Science 50, 1 102, [5] Joachimski, F. & Mathes, R., Short proofs of normalization for the simply-typed lambda-calculus, permutative conversions and Gödel s T. rchive for Mathematical Logic, 42, 59 87, [6] Luo, Z.-H., omputation and Reasoning, OUP [7] Martin-Löf, P., Intuitionistic Type Theory, Bibliopolis, Naples, [8] Martin-Löf, P., On the meanings of the logical constants and the justifications of the logical laws, (the Siena Lectures), Nordic Journal of Philosophical Logic 1, 11 60, [9] Mathes, R., Interpolation for natural deduction with generalized eliminations, LNS 2183, , Springer [10] Mendler, P.F., Inductive Definition in Type Theory, PhD thesis, ornell [11] Moriconi, E. & Tesconi, L., On inversion principles, History and Philosophy of Logic 29, , [12] Negri, S. & von Plato, J., Structural Proof Theory, UP [13] von Plato, J., Natural deduction with general elimination rules. rchive for Mathematical Logic, 40, , [14] von Plato, J., Personal communication, May [15] Pfenning, F. & Davies, R., judgmental reconstruction of modal logic, Mathematical Structures in omputer Science 11, , [16] Prawitz, D., Proofs and the meaning and completeness of the logical constants, in Essays on Mathematical and Philosophical Logic, Reidel: Dordrecht, [17] Prawitz, D., Ideas and results in proof theory, in Second Scandinavian Logic Symposium Proceedings (ed. Fenstad), , North-Holland, [18] Read, S.L., Harmony and autonomy in classical logic, Journal of Philosophical Logic 29, , [19] Schroeder-Heister, P., natural extension of natural deduction, Journal of Symbolic Logic 49, , [20] Tesconi, L., strong normalization theorem..., PhD thesis, Pisa