In order to show that the refutation system LCLDSpresented here does indeed cor- respond to a standard Hilbert axiom presentation for Linear Logic it is necessary to show that theorems derived within the two systems are the same (Properties 4 and 5 of Theorem 1). Similarly for RCLDSand Relevant Logic. The complete set of axioms used in the implication fragments of LL and RL is shown in Table 2.
Axioms (I2), (I3) and (I4) correspond, respectively, to contraction, distributivity and permutation. A useful axiom, (I5), is derivable also from (I3) and (I4) and is included for convenience. All axioms are appropriate for RL, whereas (I2) is omitted for LL. Respectively, Theorems 2 and 3 state that theorems in LL and RL derived from these axioms together with the rule of Modus Ponens (MP) are also theorems of AlgMG, and that theorems of LCLDS and RCLDS are also theorems in the Hilbert System(s).
Table 2.The Hilbert axioms for ICLDS
α→α (I1)
(α→(α→β))→(α→β) (I2) (α→β)→((γ→α)→(γ→β)) (I3)
(α→(β→γ))→(β→(α→γ)) (I4) (α→β)→((β→γ)→(α→γ)) (I5)
Correspondence Part I. Property (4) of AlgMGis shown in Theorem 2. An outline proof is given. For RL the appropriate Hilbert axioms are (Ax1) - (Ax5);
(Ax2) is omitted for LL.
Theorem 2. Let P be a Hilbert theorem of LL then the union of {ơ[P]∗(1)} and the appropriate set of instances of the semantic axioms (equivalences)for
ơ[P]∗(1),PS, has no models in HL. (For RL,PS has no models.)
Proof. (Outline only.) The proof is essentially the same for both logics. LetPS
be the set of defining equivalences for P and its subformulas, ∀x[[P]∗(x) ↔ R(x)] be the defining equivalence for [P]∗ and ∀x[[P]∗(x) ↔ TP(x)] be the resulting equivalence after replacing every occurrence in R(x) of an atom that has a defining equivalence inPS by the right-hand side of that equivalence. It is shown next thatTP(1) is always true and hence that there are no models ofPS
andơ[P]∗(1).
This property ofTP(1) is shown by induction on the number of (MP) steps in the Hilbert proof ofP. In caseP is an axiom and uses no applications of (MP) in its proof then the property can be seen to hold by construction. For instance, in the case of the contraction axiom (I2),T(I2)(1) is the sentence
∀y(∀zv([α]∗(z)∧[α]∗(v)→[β]∗(zyv))→ ∀u([α]∗(u)→[β]∗(uy))) In the case of LL, the equivalences include also the restrictedpredicate (short- ened to r in the illustration below). For the permutation axiom (I4), T(I4)(1), after some simplification6 , is the sentence
∀z
∀y([α]∗(y)→ ∀v(r(zyv)→([β]∗(v)→[γ]∗(zyv))))→
∀u([β]∗(u)→ ∀w(r(zuw)→([α]∗(u)→[γ]∗(zuw))))
Let the property hold for all theorems that have Hilbert proofs using < n applications of (MP), and consider a theoremP such that its proof usesn(MP) steps, with the last step being a derivation fromPandP→P. By hypothesis, TP(1) is true, andTP→P(1) is true.
Hence, since∀x[TP→P(x)↔ ∀u[TP(u)→TP(ux)]], thenTP(1) is also true.
The contrapositive of Theorem 2 allows the conclusion thatP is not a theorem to be drawn from the existence of a model for {ơ[P]∗(1)} ∪PS as found by a terminating AlgMG.
6 In particular,restricted(xy) implies alsorestricted(x) andrestricted(y).
Correspondence Part II. To show that every formula classified as a theorem by AlgMGin RCLDS or LCLDS is also derivable using the appropriate Hilbert axioms and the rule of Modus Ponens, Theorem 3 is used.
Theorem 3. Let Gα be the set of instances ofA+L for showingα(not including
ơ[α]∗(1)), then if there exists an AlgMG refutation in LCLDS ofGα∪ơ[α]∗(1)then there is a Hilbert proof in LL of α, which is therefore a theorem of LL. That is,
ifGα,ơ[α]∗(1)|=FOL then "HIα7. Similarly for RCLDS and RL.
Proof. Suppose Gα,ơ[α]∗(1) |=FOL, hence any model of Gα is also a model of [α]∗(1); it is required to show"HIα. Lemma 2 below states there is a model M of A+L (A+R), and hence ofGα, with the property that [α]∗(1) =true iff"HI α. Therefore, sinceM is a model ofA+L (A+R) it is a model of [α]∗(1) and hence"HIα is true, as required. The desired model is based on the canonical interpretation introduced in [1].
Definition 15. Thecanonical interpretationfor LCLDS is an interpretation from Mon(LP,LL)onto the power set ofLP defined as follows:
– ||cα||={z:"HIα→z}, for each parametercα;
– ||λ◦λ|| ={z : "HI α∧β →z}={z : "HI α→(β →z)} , where α∈ ||λ||
andβ∈ ||λ||;
– ||1||={z:"HIz} and
– || ||={(||x||,||y||) :||x|| ⊆ ||y||};
– ||[α]∗||={||x||:α∈ ||x||};
Similarly for RCLDS. For the case of LCLDS an interpretation of therestricted predicate is also needed. This depends on the particular theorem that is to be proven, as it makes use of the relevant indices of the parameters occurring in the translated clauses. The interpretation is given by:
||restricted||={||x||:
∀z(z∈ ||x|| →z is provable using ≤mαi occurrences ofαi)}
In other words, restricted(x) = true iff x includes ≤ mαi occurrences of pa- rameterαi. (In case a new parameter is used for each instance of Axiom (Ax3c) then the definition does not depend on the particular theorem to be proven as mαi = 1 for everycαi.)
The canonical interpretation is used to give a Herbrand model forA+L (A+R), by setting [α]∗(x) =trueiffα∈ ||x||. This means, in particular, that if [α]∗(1) = true then α ∈ ||1|| and hence "HI α. The following Lemma states that the canonical interpretation of Definition 15 is a model ofA+I (A+R).
Lemma 2. The properties of the labelling algebraAL (AR)given in Definition 2 and the semantic axioms ofA+L (A+R)are satisfied by the canonical interpretation for LCLDS (RCLDS).
7 The notationHIγindicates thatγis provable using the appropriate Hilbert axioms.
Proof. Each of the properties of the labelling algebra is satisfied by the canonical interpretation. For RCLDS the case for contraction is given here. The other cases are as given in [4]. For LCLDSthe case for Axiom (Ax3a) is given. The other cases are as given in [4] but modified to include the restrictedpredicate.
contraction Suppose that δ ∈ ||λ|| ◦ ||λ||. Then there is a Hilbert proof of α→(α→δ), where α∈ ||λ||. By axiom (I2)"HIα→δandδ∈ ||λ||.
(Ax3a) Let the maximum number of parameter occurrences allowed be fixed by the global relevant indices for the particular theorem to be proved. Suppose restricted(x), restricted(y) andrestricted(xy) and thatα∈ ||x||and α→ β∈ ||y||. Then there are Hilbert proofs ofδ→αandγ→α→βforδ∈ ||x||
and γ ∈ ||y|| such that no more than the allowed number of subformula occurrences, as given by the relevant indices for the problem, are used in the combined proofs ofδand γ. To showδ→(γ→β), and henceβ ∈ ||x◦y||, use axioms (I4) and (I5).
6 Conclusions
In this paper the method of Compiled Labelled Deductive Systems, based on the principles in [9], is applied to the two resource logics, LL and RL. The method of CLDS provides logics with a uniform presentation of their derivability relations and semantic entailments and its semantics is given in terms of a translation approach into first-order logic. The main features of a CLDS system and model theoretic semantics are described here. The notion of a configuration in a CLDS system generalises the standard notion of a theory and the notion of semantic entailment is generalised to relations between structured theories. The method is used to give presentations of LCLDS and RCLDS, which are seen to be generali- sations, respectively, of Linear and Relevance Logic through the correspondence results in Sect. 5, which shows that there is a one-way translation of standard theories into configurations, while preserving the theorems of LL and RL.
The translation results in a compiled theory of a configuration. A refutation system based on a Model Generation procedure is defined for this theory, which, together with a particular unification algorithm and an appropriate restriction on the size of terms, yields a decidability test for formulas of propositional Lin- ear Logic or Relevance Logic. The main contribution of this paper is to show how the translation approach into first order logic for Labelled Deductive Sys- tems can still yield decidable theories. This meets one of the main criticisms levelled at LDS, and at CLDS in particular, that for decidable logics the CLDS representation is not decidable.
The method used in this paper can be extended to include all operators of Linear Logic, including the additive and exponential operators. For instance, the axiom for the additive disjunction operator∨in LL is
∀x([α∨β]∗(x)↔ ∀y(([α→γ]∗(y)∧[β →γ]∗(y))→[γ]∗(x◦y))) From an applicative point of view, the CLDS approach provides a logic with reasoning which is closer to the needs of computing and A.I. These are in fact
application areas with an increasing demand for logical systems able to represent and to reason about structuresof information (see [9]). For example in [3] it is shown how a CLDS can provide a flexible framework for abduction.
For the automated theorem proving point of view, the translation method described in Section 2.2 facilitates the use of first-order therem provers for de- riving theorems of the underlying logic. In fact, the first order axioms of a CLDS extended algebraA+S can be translated into clausal form, and so any clausal the- orem proving method might be appropriate for using the axioms to automate the process of proving theorems. The clauses resulting from the translation of a particular configuration represent a partial coding of the data. A resolution refutation that simulates the application of natural deduction rules could be developed, but because of the simple structure of the clauses resulting from a subtructural CLDS theory the extended Model Generation method used here is appropriate.
References
1. M. D’Agostino and D. Gabbay. A generalisation of analytic deduction via labelled deductive systems. Part I: Basic substructural Logics. Journal of Automated Rea- soning, 13:243-281, 1994.
2. K. Broda, M. Finger and A. Russo. Labelled Natural Deduction for Substructural Logics. Logic Journal of the IGPL, Vol. 7, No. 3, May 1999.
3. K. Broda and D. Gabbay. An Abductive CLDS. In Labelled Deduction, Kluwer, Ed. D. Basin et al, 1999.
4. K.Broda and D. Gabbay. A CLDS for Propositional Intuitionistic Logic.
TABLEAUX-99, USA, LNAI 1617, Ed. N. Murray, 1999.
5. K. Broda and A. Russo. A Unified Compilation Style Labelled Deductive System for Modal and Substructural Logic using Natural Deduction. Technical Report 10/97. Department of Computing, Imperial College 1997.
6. K. Broda, A. Russo and D. Gabbay. A Unified Compilation Style Natural Deduc- tion System for Modal, Substructural and Fuzzy logics, in Dicovering World with Fuzzy logic: Perspectives and Approaches to Formalization of Human-consistent Logical Systems. Eds V. Novak and I.Perfileva, Springer-Verlag 2000
7. A. Bundy. The Computer Modelling of Mathematical Reasoning. Academic Press, 1983.
8. C. L. Chang and R. Lee. Symbolic Logic and Mechanical Theorem Proving. Aca- demic Press 1973.
9. D. Gabbay. Labelled Deductive Systems, Volume I - Foundations. OUP, 1996.
10. J. H. Gallier. Logic for Computer Science. Harper and Row, 1986.
11. R. Hasegawa, H. Fujita and M. Koshimura. MGTP: A Model Generation Theo- rem Prover - Its Advanced Features and Applications. In TABLEAUX-97, France, LNAI 1229, Ed. D. Galmiche, 1997.
12. W. Mc.Cune. Otter 3.0 Reference Manual and Guide. Argonne National Laboraq- tory, Argonne, Illinois, 1994.
13. J.A. Robinson. Logic, Form and Function. Edinburgh Press, 1979.
14. A. Russo. Modal Logics as Labelled Deductive Systems. PhD. Thesis, Department of Computing, Imperial College, 1996.
15. R. A. Schmidt. Resolution is a decision procedure for many propositional modal logics. Advances in Modal Logic, Vol.1, CSLI, 1998.
16. P. B. Thistlethwaite, M. A. McRobbie and R. K. Meyer. Automated Theorem- Proving in Non-Classical Logics, Wiley, 1988.
A Critique of Proof Planning
Alan Bundy Division of Informatics, University of Edinburgh
Abstract. Proof planning is an approach to the automation of theorem proving in which search is conducted, not at the object-level, but among a set of proof methods. This approach dramatically reduces the amount of search but at the cost of completeness. We critically examine proof planning, identifying both its strengths and weaknesses. We use this analysis to explore ways of enhancing proof planning to overcome its current weaknesses.
Preamble
This paper consists of two parts:
1. a brief ‘bluffer’s guide’ to proof planning1; and 2. a critique of proof planning organised as a 4x3 array.
Those alreadyfamiliar with proof planning maywant to skip straight to the critique which starts at §2, p164.
1 Background
Proof planning is a technique for guiding the search for a proof in automated theorem proving, [Bundy, 1988, Bundy, 1991, Kerber, 1998, Benzm¨ulleret al, 1997]. The main idea is to identify common patterns of rea- soning in families of similar proofs, to represent them in a computational fashion and to use them to guide the search for a proof of conjectures from the same family. For instance, proofs by mathematical induction share the common pat- tern depicted in figure 1. This common pattern has been represented in the proof plannersClam andλClam and used to guide a wide variety of inductive proofs [Bundy et al, 1990b, Bundyet al, 1991, Richardsonet al, 1998].
The research reported in this paper was supported by EPSRC grant GR/M/45030. I would like to thank Andrew Ireland, Helen Lowe, Raul Monroy and two anonymous referees for helpful comments on this paper. I would also like to thank other members of the Mathematical Reasoning Group and the audiences at CIAO and Scottish Theorem Provers for helpful feedback on talks from which this paper arose.
1 Pointers to more detail can be found at
http://dream.dai.ed.ac.uk/projects/proof planning.html
A.C. Kakas, F. Sadri (Eds.): Computat. Logic (Kowalski Festschrift), LNAI 2408, pp. 160–177, 2002.
c Springer-Verlag Berlin Heidelberg 2002
induction
;
;
;
@
@
@ R
base case
step case
ripple
fertilize
?
Inductive proofs start with the application of aninduction rule, which reduces the conjecture to somebaseandstep cases. One of each is shown above. In the step caseripplingreduces the difference between theinduction conclusionand theinduction hypothesis(see§1.2, p162 for more detail).Fertilizationapplies the induction hypothesis to simplify the rippled induction conclusion.
Fig. 1.ind strat: A Strategy for Inductive Proof