Automated Theorem Proving

Transcription

1 Atomated Theorem Proving Frank Pfenning Carnegie Mellon University raft of Agst 23, 1999 Material for the corse Atomated Theorem Proving at Carnegie Mellon University, Fall 1999 This incldes revised versions from the corse notes on Linear Logic (Spring 1998) and Comptation and edction (Spring 1997) Material for this corse is available at Please send comments to This material is in rogh draft form and is likely to contain errors Frthermore, citations are in no way adeqate or complete Please do not cite or distribte this docment This work was spported by NSF Grants CCR and CCR Copyright c 1999, Frank Pfenning

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3 Contents 1 Introdction 1 2 Natral edction 3 21 IntitionisticNatraledction 5 22 ClassicalLogic LocalizingHypotheses Exercises 20 Bibliography 22 raft of Agst 23, 1999

4 iv CONTENTS raft of Agst 23, 1999

5 Chapter 1 Introdction Logic is a science stdying the principles of reasoning and valid inference Atomated dedction is concerned with the mechanization of formal reasoning, following the laws of logic The roots of the field go back to the end of the last centry when Frege developed his Begriffsschrift 1, the first comprehensive effort to develop a formal langage sitable as a fondation for mathematics Alas, Rssell discovered a paradox which showed that Frege s system was inconsistent, that is, the trth of every proposition can be derived in it Rssell then devised his own system based on a type theory and he and Whitehead demonstrated in the monmental Principia Mathematica how it can serve as a fondation of mathematics Later, Hilbert developed a simpler alternative, the predicate calcls Gentzen s formlation of the predicate calcls in a system of natral dedction provides a major milestone for the field In natral dedction, the meaning of each logical connective is explained via inference rles, an approach later systematically refined by Martin-Löf This is the presentation we will follow in these notes Gentzen s seminal work also contains the first 2 consistency proof for a formal logical system As a technical device he introdced the seqent calcls and showed that it derives the same theorems as natral dedction The famos Haptsatz 3 establishes that all proofs in the seqent calcls can be fond according to a simple strategy It is immediately evident that there are many propositions which have no proof according to this strategy, thereby garanteeing consistency of the system Most search strategies employed by atomated dedction systems are either directly based on or can be derived from the seqent calcls We can broadly classify procedres as either working backwards from the proposed theorem toward the axioms, or forward from the axioms toward the theorem Among the backward searching procedres we find tableax, connection methods, matrix methods and some forms of resoltion Among the forward searching proce- 1 literally translated as concept notation 2 [?] 3 literally jst main theorem, often called the ct elimination theorem raft of Agst 23, 1999

6 2 Introdction dres we find classical resoltion and the inverse method The prominence of resoltion among these methods is no accident, since Robinson s seminal paper represented a major leap forward in the state of the art It is natral to expect that a combination of forward and backward search cold improve the efficiency of theorem proving system Sch a combination, however, has been elsive p to now, de to the largely incompatible basic choices in design and implementation of the two kinds of search procedres In this corse we stdy both types of procedres We investigate high-level qestions, sch as how these procedres relate to the basic seqent calcls We also consider low-level isses, sch as techniqes for efficient implementation of the basic inference engine There is one frther dimension to consider: which logic do we reason in? In philosophy, mathematics, and compter science many different logics are of interest For example, there are classical logic, intitionistic logic, modal logic, relevance logic, higher-order logic, dynamic logic, temporal logic, linear logic, belief logic, and lax logic (to mention jst a few) While each logic reqires its own considerations, many techniqes are shared This can be attribted in part to the common root of different logics in natral dedction and the seqent calcls Another reason is that low-level efficiency improvements are relatively independent of higher-level techniqes For this corse we chose intitionistic logic for a variety of reasons First, intitionistic propositions correspond to logical specifications and proofs to fnctional programs, which means intitionistic logic is of central interest in the stdy of programming langages Second, intitionistic logic is more complex than classical logic and exhibits phenomena obscred by special properties which apply only to classical logic Third, there are relatively straightforward interpretations of classical in intitionistic logic which permits s to stdy logical interpretations in connection with theorem proving procedres The corse is centered arond a project, namely the joint design and implementation of a sccession of theorem provers for intitionistic logic We start with natral dedction, followed by a seqent calcls, and a simple tablea prover Then we trn toward the inverse method and introdce sccessive refinements consisting of both high-level and low-level optimizations 4 The implementation component is important to gain a deeper nderstanding of the techniqes introdced in or abstract stdy The goal of the corse is to give stdents a thorogh nderstanding of the central techniqes in atomated theorem proving Frthermore, they shold nderstand the systematic development of these techniqes and their correctness proofs, thereby enabling them to transfer methods to different logics or applications We are less interested here in an appreciation of the pragmatics of highly efficient implementations or performance tning 4 The precise order and extent of the improvements possible in a one-semester gradate corse has yet to be determined raft of Agst 23, 1999

7 Chapter 2 Natral edction Ich wollte znächst einmal einen Formalisms afstellen, der dem wirklichen Schließen möglichst nahe kommt So ergab sich ein,,kalkül des natürlichen Schließens 1 Gerhard Gentzen Unterschngen über das logische Schließen [Gen35] In this chapter we explore ways to define logics, or, which comes to the same thing, ways to give meaning to logical connectives Or fndamental notion is that of a jdgment based on evidence For example, we might make the jdgment It is raining based on visal evidence Or we might make the jdgment A implies A is tre for any proposition A based on a derivation The se of the notion of a jdgment as conceptal prior to the notion of proposition has been advocated by Martin-Löf [ML85a, ML85b] Certain forms of jdgments freqently recr and have therefore been investigated in their own right, prior to logical considerations Two that we will se are hypothetical jdgments and parametric jgments (the latter is sometimes called general jdgment or schematic jdgment) A hypothetical jdgment has the form J 2 nder hypothesis J 1 We consider this jdgment evident if we are prepared to make the jdgment J 2 once provided with evidence for J 1 Formal evidence for a hypothetical jdgment is a hypothetical derivation where we can freely se the hypothesis J 1 in the derivation of J 2 Note that hypotheses need not be sed, and cold be sed more than once A parametric jdgment has the form J for any a wherea is a parameter which may occr in J We make this jdgment if we are prepared to make the jdgment [O/a]J for arbitrary objects O of the right category Here [O/a]J is or notation for sbstitting the object O for parameter a in the jdgment J Formal evidence for a parametric jdgment J is a parametric derivation with free occrrences of the parameter a 1 First I wanted to constrct a formalism which comes as close as possible to actal reasoning Ths arose a calcls of natral dedction raft of Agst 23, 1999

8 4 Natral edction Formal evidence for a jdgment in form of a derivation is sally written in two-dimensional notation: J if is a derivation of J For the sake of brevity we sometimes se the alternative notation :: J A hypothetical jdgment is written as J 1 J 2 where is a label which identifies the hypothesis J 1 We se the labels to garantee that hypotheses which are introdced dring the reasoning process are not sed otside their scope The separation of the notion of jdgment and proposition and the corresponding separation of the notion of evidence and proof sheds new light on varios styles that have been sed to define logical systems An axiomatization in the style of Hilbert [Hil22], for example, arises when one defines a jdgment A is tre withot the se of hypothetical jdgments Sch a definition is highly economical in its se of jdgments, which has to be compensated by a liberal se of implication in the axioms When we make proof strctre explicit in sch an axiomatization, we arrive at combinatory logic [Cr30] A categorical logic [LS86] arises when the basic jdgment is not trth, bt entailment A entails B 2 Once again, presentations are highly economical and do not need to seek recorse in complex jdgment forms (at least for the propositional fragment) Bt derivations often reqire many hypotheses, which means that we need to lean rather heavily on conjnction here Proofs are realized by morphisms which are an integral part of the machinery of category theory While these are interesting and in many ways sefl approaches to logic specification, neither of them comes particlarly close to captring the practice of mathematical reasoning This was Gentzen s point of departre for the design of a system of natral dedction [Gen35] From or point of view, this system is based on the simple jdgment A is tre, bt relies critically on hypothetical and parametric jdgments In addition to being extremely elegant, it has the great advantage that one can define all logical connectives withot reference to any other connective This principle of modlarity extends to the meta-theoretic stdy of natral dedction and simplifies considering fragments and extension of logics Since we will consider many fragments and extension, this orthogonality of the logical connectives is a critical consideration There is another advantage to natral dedction, namely that its proofs are isomorphic to the terms in a λ- calcls via the so-called Crry-Howard isomorphism [How69], which establishes many connections to fnctional programming 2 [This has been dispted by practitioners of the field and shold be re-evalated] raft of Agst 23, 1999

9 21 Intitionistic Natral edction 5 Finally, we arrive at the seqent calcls (also introdced by Gentzen in his seminal paper [Gen35]) when we split the single jdgment of trth into two: A is an assmption and A is tre While we still employ the machinery of parametric and hypothetical jdgments, we now need an explicit rle to state that A is an assmption is sfficient evidence for A is a tre The reverse, namely that if A is tre then A may be sed as an assmption isthect rle which he proved to be redndant in his Haptsatz For Gentzen the seqent calcls was primarily a technical device to prove consistency of his system of natral dedction, bt it exposes many details of the fine strctre of proofs in sch a clear manner that many logic presentations employ seqent calcli The laws governing the strctre of proofs, however, are more complicated than the Crry-Howard isomorphism for natral dedction might sggest and are still the sbject of stdy [Her95, Pfe95] We choose natral dedction as or definitional formalism as the prest and most widely applicable Later we jstify the seqent calcls as a calcls of proof search for natral dedction and explicitly relate the two forms of presentation We begin by introdcing natral dedction for intitionistic logic, exhibiting its basic principles 21 Intitionistic Natral edction The system of natral dedction we describe below is basically Gentzen s system NJ [Gen35] or the system which may be fond in Prawitz [Pra65] The calcls of natral dedction was devised by Gentzen in the 1930 s ot of a dissatisfaction with axiomatic systems in the Hilbert tradition, which did not seem to captre mathematical reasoning practices very directly Instead of a nmber of axioms and a small set of inference rles, valid dedctions are described throgh inference rles only, which at the same time explain the meaning of the logical qantifiers and connectives in terms of their proof rles A langage of (first-order) terms is bilt p from variables x, y, etc, fnction symbols f, g, etc, each with a niqe arity, and parameters a, b, etc in the sal way Terms t ::= x a f(t 1,,t n ) A constant c is simply a fnction symbol with arity 0 and we write c instead of c() Exactly which fnction symbols are available is left nspecified in the general development of predicate logic and only made concrete for specific theories, sch as the theory of natral nmbers However, variables and parameters are always available We will se t and s to range over terms The langage of propositions is bilt p from predicate symbols P, Q, etc and terms in the sal way Propositions A ::= P (t 1,,t n ) A 1 A 2 A 1 A 2 A 1 A 2 A x A x A raft of Agst 23, 1999

10 6 Natral edction A propositional constant P is simply a predicate symbol with no argments and we write P instead of P () We will se A, B, andc to range over propositions Exactly which predicate symbols are available is left nspecified in the general development of predicate logic and only made concrete for specific theories The notions of free and bond variables in terms and propositions are defined in the sal way: the variable x is bond in propositions of the form x A and x A We se parentheses to disambigate and assme that and bind more tightly than It is convenient to assme that propositions have no free individal variables; we se parameters instead where necessary Or notation for sbstittion is [t/x]a for the reslt of sbstitting the term t for the variable x in A Becase of the restriction on occrrences of free variables, we can assme that t is free of individal variables, and ths captring cannot occr The main jdgment of natral dedction is C is tre written as, from hypotheses 1,, n We will model this as a hypothetical jdgment This means that certain strctral properties of derivations are tacitly assmed, independently of any logical inferences In essence, these assmptions explain what hypothetical jdgments are Hypothesis If we have a hypothesis than we can conclde Weakening Hypotheses need not be sed plication Hypotheses can be sed more than once Exchange The order in which hypotheses are introdced is irrelevant In natral dedction each logical connective and qantifier is characterized by its introdction rle(s) which specifies how to infer that a conjnction, disjnction, etc is tre The elimination rle for the logical constant tells what other trths we can dedce from the trth of a conjnction, disjnction, etc Introdction and elimination rles mst match in a certain way in order to garantee that the rles are meaningfl and the overall system can be seen as captring mathematical reasoning The first is a local sondness property: if we introdce a connective and then immediately eliminate it, we shold be able to erase this detor and find a more direct derivation of the conclsion withot sing the connective If this property fails, the elimination rles are too strong: they allow s to conclde more than we shold be able to know The second is a local completeness property: we can eliminate a connective in a way which retains sfficient information to reconstitte it by an introdction rle If this property fails, the elimination rles are too weak: the do not allow s to conclde everything we shold be able to know We provide evidence for local sondness and completeness of the rles by means of local redction and expansion jdgments, which relate proofs of the same proposition One of the important principles of natral dedction is that each connective shold be defined only in terms of inference rles withot reference to other raft of Agst 23, 1999

11 21 Intitionistic Natral edction 7 logical connectives or qantifiers We refer to this as orthogonality of the connectives It means that we can nderstand a logical system as a whole by nderstanding each connective separately It also allows s to consider fragments and extensions directly and it means that the investigation of properties of a logical system can be condcted in a modlar way We now show the introdction and elimination rles, local redctions and expansion for each of the logical connectives in trn The rles are smmarized on page 21 Conjnction A B shold be tre if both A and B are tre Ths we have the following introdction rle I B If we consider this as a complete definition, we shold be able to recover both A and B if we know A B We are ths led to two elimination rles B EL B ER To check or intition we consider a dedction which ends in an introdction followed by an elimination: E B EL Clearly, it is nnecessary to first introdce the conjnction and then eliminate it: a more direct proof of the same conclsion from the same (or fewer) assmptions wold be simply Formlated as a transformation or redction between derivations we have and symmetrically E B EL E B ER I I I = R = R E raft of Agst 23, 1999

12 8 Natral edction The new jdgment = R E relates derivations with the same conclsion We say locally redces to E Since local redctions are possible for both elimination rles for conjnction, or rles are locally sond To show that the rles are locally complete we show how to reintrodce a conjnction from its components in the form of a local expansion B = E B E L B I E R B Implication To derive B we assme and then derive Written as a hypothetical jdgment: I B We mst be carefl that the hypothesis is available only in the derivation above the premiss We therefore label the inference with the name of the hypothesis, which mst not be sed already as the name for a hypothesis in the derivation of the premiss We say that the hypothesis labelled is discharged at the inference labelled I A derivation of B describes a constrction by which we can transform a derivation of into a derivation of : we sbstitte the derivation of wherever we sed the assmption in the hypothetical derivation of The elimination rle expresses this: if we have a derivation of B and also a derivation of, thenwecan obtain a derivation of B E The local redction rle carries ot the sbstittion of derivations explained above I B E E = R E raft of Agst 23, 1999

13 21 Intitionistic Natral edction 9 The final derivation depends on all the hypotheses of E and except, for which we have sbstitted E An alternative notation for this sbstittion of derivations for hypotheses is [E/] :: The local redction described above may significantly increase the overall size of the derivation, since the dedction E is sbstitted for each occrrence of the assmption labeled in and may ths be replicated many times The local expansion simply rebilds the implication B = E B E I B isjnction A B shold be tre if either A is tre or B is tre Therefore we have two introdction rles IL IR B B If we have a hypothesis B, we do not know how it might be inferred That is, a proposed elimination rle B? wold be incorrect, since a dedction of the form E I R B? cannot be redced As a conseqence, the system wold be inconsistent: ifwe have at least one theorem (B, in the example) we can prove every formla (A, in the example) How do we se the assmption A B in informal reasoning? We often proceed with a proof by cases: we prove a conclsion C nder the assmption A and also show C nder the assmption B We then conclde C, since either A or B by assmption Ths the elimination rle employs two hypothetical jdgments B 1 2 E 1,2 raft of Agst 23, 1999

14 10 Natral edction Now one can see that the introdction and elimination rles match p in two redctions First, the case that the disjnction was inferred by I L I L B E 1 1 E 2 2 E 1,2 = R E 1 1 The other redction is symmetric I R B E 1 1 E 2 2 E 1,2 = R E 2 2 As in the redction for implication, the reslting derivation may be longer than the original one The local expansion is more complicated than for the previos connectives, since we first have to distingish cases and then reintrodce the disjnction in each branch B = E B 1 IL B B 2 IR B E 1,2 Negation In order to derive A we assme A and try to derive a contradiction Ths it seems that negation reqires falsehood, and, indeed, in most literatre on constrctive logic, A is seen as an abbreviation of A In order to give a self-contained explanation of negation by an introdction rle, we employ a jdgment that is parametric in a propositional parameter p: Ifwe can derive any p from the hypothesis A we conclde A p A I p, A E raft of Agst 23, 1999

15 21 Intitionistic Natral edction 11 The elimination rle follows from this view: if we know A and then we can conclde any formla C is tre In the form of a local redction: p A I p, E E = R E [C/p] The sbstittion [C/p] isvalid, since isparametricin p The localexpansion is similar to the case for implication A = E A p A E I p, Trth There is only an introdction rle for : I Since we pt no information into the proof of, we know nothing new if we have an assmption and therefore we have no elimination rle and no local redction It may also be helpfl to think of as a 0-ary conjnction: the introdction rle has 0 premisses instead of 2 and we correspondingly have 0 elimination rles instead of 2 The local expansion allows the replacement of any derivation of by I = E I Falsehood Since we shold not be able to derive falsehood, there is no introdction rle for Therefore, if we can derive falsehood, we can derive everything E Note that there is no local redction rle for E It may be helpfl to think of as a 0-ary disjnction: we have 0 instead of 2 introdction rles and we correspondingly have to consider 0 cases instead of 2 in the elimination rle Even thogh we postlated that falsehood shold not be derivable, falsehood cold clearly be a conseqence of contradictory assmption For example, raft of Agst 23, 1999

16 12 Natral edction A A is derivable While there is no local redction rle, there still is a local expansion in analogy to the case for disjnction = E E Universal Qantification Under which circmstances shold x A be tre? This clearly depends on the domain of qantification For example, if we know that x ranges over the natral nmbers, then we can conclde x A if we can prove [0/x]A, [1/x]A, etc Sch a rle is not effective, since it has infinitely many premisses Ths one sally retreats to rles sch as indction However, in a general treatment of predicate logic we wold like to prove statements which are tre for all domains of qantification Ths we can only say that x A shold be provable if [a/x]a is provable for a new parameter a abot which we can make no assmption Conversely, if we know x A, we know that [t/x]a for any term t [a/x]a I a x A x A E [t/x]a The label a on the introdction rle is a reminder the parameter a mst be new, that is, it may not occr in any ncancelled assmption in the proof of [a/x]a or in x A itself In other words, the derivation of the premiss mst parametric in a The local redction carries ot the sbstittion for the parameter [a/x]a I x A E [t/x]a = R [t/a] [t/x]a Here, [t/a] is or notation for the reslt of sbstitting t for the parameter a throghot the dedction For this sbstittion to preserve the conclsion, we mst know that a does not already occr in A Similarly, we wold change the hypotheses if a occrred free in any of the ndischarged hypotheses of This might render a larger proof incorrect As an example, consider the formla x y P (x) P (y) which shold clearly not be tre for all predicates P The raft of Agst 23, 1999

17 21 Intitionistic Natral edction 13 following is not a dedction of this formla P (a) I a? x P (x) E P (b) I P (a) P (b) I b y P (a) P (y) I a x y P (x) P (y) The flaw is at the inference marked with?, where a is free in the hypothesis labelled Applying a local proof redction to the (incorrect) I inference followed by E leads to the the assmption [b/a]p (a) which is eqal to P (b) The reslting derivation P (b) I P (a) P (b) I b y P (a) P (y) I a x y P (x) P (y) is once again incorrect since the hypothesis labelled shold read P (a), not P (b) The local expansion for niversal qantification is mch simpler x A = E x A E [a/x]a I a x A Existential Qantification term t sch that [t/x]a is tre We conclde that x A is tre when there is a [t/x]a I x A When we have an assmption x A we do not know for which t it is the case that [t/x]a holds We can only assme that [a/x]a holds for some parameter a abot which we know nothing else Ths the elimination rle resembles the raft of Agst 23, 1999

18 14 Natral edction one for disjnction x A [a/x]a E a, The restriction is similar to the one for I: the parameter a mst be new, that is, it mst not occr in x A, C, or any assmption employed in the derivation of the second premiss In the redction rle we have to perform two sbstittions: we have to sbstitte t for the parameter a and we also have to sbstitte for the hypothesis labelled [t/x]a x A [a/x]a I E E a, = R [t/x]a [t/a]e The proviso on occrrences of a garantees that the conclsion and hypotheses of [t/a]e have the correct form The local expansion for existential qantification is also similar to the case for disjnction x A [a/x]a I = E x A x A E a, x A Here is a simple example of a natral dedction We attempt to show the process by which sch a dedction may have been generated, as well as the final dedction The three vertical dots indicate a gap in the derivation we are trying to constrct, with hypotheses and their conseqences shown above and the desired conclsion below the gap (A B) B ; (A B) I (A B) B raft of Agst 23, 1999

19 21 Intitionistic Natral edction 15 ; (A B) EL I (A B) B ; (A B) EL (A B) B (A B) ER B I ; (A B) (A B) ER EL B E I (A B) B ; (A B) ER B I (A B) B (A B) EL E The symbols A and B in this derivation stand for arbitrary propositions; we can ths established a jdgment parametric in A and B In other words, every instance of this derivation (sbstitting arbitrary propositions for A and B) is a valid derivation Below is a smmary of the rles of intitionistic natral dedction raft of Agst 23, 1999

20 16 Natral edction Introdction Rles Elimination Rles B I B EL B ER IL B IR B B 1 2 E 1,2 I B B E p A I p, I no introdction [a/x]a I a x A A E no elimination E x A E [t/x]a [t/x]a I x A x A [a/x]a E a, raft of Agst 23, 1999

21 22 Classical Logic Classical Logic The inference rles so far only model intitionistic logic, and some classically tre propositions sch as A A (for an arbitrary A) are not derivable, as we will see in Section?? There are three commonly sed ways one can constrct a system of classical natral dedction by adding one additional rle of inference C is called Proof by Contradiction or Rle of Indirect Proof, C is the oble Negation Rle, and XM is referred to as Exclded Middle A A C A C A XM A A The rle for classical logic (whichever one chooses to adopt) breaks the pattern of introdction and elimination rles One can still formlate some redctions for classical inferences, bt natral dedction is at heart an intitionistic calcls The symmetries of classical logic are mch better exhibited in seqent formlations of the logic In Exercise 23 we explore the three ways of extending the intitionistic proof system and show that they are eqivalent Another way to obtain a natral dedction system for classical logic is to allow mltiple conclsions (see, for example, Parigot [Par92]) 23 Localizing Hypotheses In the formlation of natral from Section 21 correct se of hypotheses and parameters is a global property of a derivation We can localize it by annotating each jdgment in a derivation by the available parameters and hypotheses We give here a formlation of natral dedction for intitionistic logic with localized hypotheses, bt not parameters For this we need a notation for hypotheses which we call a context Contexts Γ ::= Γ,:A Here, represents the empty context, and Γ,:A adds hypothesis labelled to Γ We assme that each label occrs at most once in a context in order to avoid ambigities The main jdgment can then be written as Γ, where, 1 :A 1,, n :A n stands for 1 1 n n raft of Agst 23, 1999

22 18 Natral edction in the notation of Section 21 We se a few important abbreviations in order to make this notation less cmbersome First of all, we may omit the leading and write, for example, 1 :A 1, 2 :A 2 instead of, 1 :A 1, 2 :A 2 Secondly, we denote concatenation of contexts by overloading the comma operator as follows Γ, = Γ Γ, (Γ,:A) = (Γ, Γ ),:A With these additional definitions, the localized version of or rles are as follows Introdction Rles Elimination Rles Γ Γ I Γ B Γ B EL Γ Γ B ER Γ Γ IL Γ B Γ IR Γ B Γ B Γ, 1 :A Γ, 2 :B E 1, 2 Γ Γ,:A I Γ B Γ B Γ Γ E Γ,:A p I p, Γ A I Γ no introdction Γ [a/x]a I a Γ x A Γ Γ A E Γ no elimination Γ E Γ Γ x A E Γ [t/x]a Γ [t/x]a I Γ x A Γ x A Γ Γ,:[a/x]A E a, We also have a new rle for hypotheses which was an implicit property of the hypothetical jdgments before Γ 1,:A, Γ 2 Other general assmptions abot hypotheses, namely that they may be sed arbitrarily often in a derivation and that their order does not matter, are indirectly raft of Agst 23, 1999

23 23 Localizing Hypotheses 19 reflected in these rles Note that if we erase the context Γ from the jdgments throghot a derivation, we obtain a derivation in the original notation When we discssed local redctions in order to establish local sondness, we sed the notation E for the reslt of sbstitting the derivation of for all ses of the hypothesis labelled in E We wold now like to reformlate the property with localized hypotheses In order to prove that the (now explicit) hypotheses behave as expected, we se the principle of strctral indction over derivations Simply pt, we prove a property for all derivations by showing that, whenever it holds for the premisses of an inference, it holds for the conclsion Note that we have to show the property otright when the rle nder consideration has no premisses, which amonts to the base cases forof the indction Theorem 21 (Strctral Properties of Hypotheses) The following properties hold for intitionistic natral dedction 1 (Exchange) If Γ 1, 1 :A, Γ 2, 2 :B, Γ 2 then Γ 1, 2 :B, Γ 2, 1 :A, Γ 2 2 (Weakening) If Γ 1, Γ 2 then Γ 1,:A, Γ 2 3 (Contraction) If Γ 1, 1 :A, Γ 2, 2 :A, Γ 2 then Γ 1,:A, Γ 2, Γ 3 4 (Sbstittion) If Γ 1,:A, Γ 2 and Γ 1 then Γ 1, Γ 2 Proof: The proof is in each case by straightforward indction over the strctre of the first given derivation In the case of exchange, we appeal to the indctive assmption on the derivations of the premisses and constrct a new derivation with the same inference rle Algorithmically, this means that we exchange the hypotheses labelled 1 and 2 in every jdgment in the derivation In the case of weakening and contraction, we proceed similarly, either adding the new hypothesis :A to every jdgment in the derivation (for weakening), or replacing ses of 1 and 2 by (for contraction) For sbstittion, we apply the indctive assmption to the premisses of the given derivation ntil we reach hypotheses If the hypothesis is different from we can simply erase :A (which is nsed) to obtain the desired derivation If the hypothesis is :A the derivation looks like = Γ 1,:A, Γ 2 so C = A in this case We are also given a derivation E of Γ 1 and have to constrct a derivation F of Γ 1, Γ 2 Bt we can jst repeatedly apply weakening to E to obtain F Algorithmically, this means that, as expected, we raft of Agst 23, 1999

24 20 Natral edction sbstitte the derivation E (possibly weakened) for ses of the hypotheses :A in Note that in or original notation, this weakening has no impact, since nsed hypotheses are not apparent in a derivation 2 It is also possible to localize the derivations themselves, sing proof terms As we will see in Section??, these proof terms form a λ-calcls closely related to fnctional programming When parameters, hypotheses, and proof terms are all localized or main jdgment becomes decidable In the terminology of Martin-Löf [ML94], the main jdgment is then analytic rather than synthetic We no longer need to go otside the jdgment itself in order to collect evidence for it: An analytic jdgment encapslates its own evidence 24 Exercises Exercise 21 Prove the following by natral dedction sing only intitionistic rles when possible We se the convention that,, and associate to the right, that is, A B C stands for A (B C) A B isa syntacticabbreviation for (A B) (B A) Also, we assme that and bind more tightly than, thatis,a B C stands for (A B) C The scope of a qantifier extends as far to the right as consistent with the present parentheses For example, ( x P (x) C) C wold be disambigated to ( x (P (x) C)) ( C) 1 B A 2 (B C) (A B) (A C) 3 (Peirce s Law) ((A B) A) A 4 (B C) (A B) (A C) 5 (A B) (A B) 6 (A x P (x)) x (A P (x)) 7 (( x P (x)) C) x (P (x) C) 8 x y (P (x) P (y)) Exercise 22 We write A if B follows from hypothesis A and A for A and B Which of the following eight parametric jdgments are derivable intitionistically? 1 ( x A) B x (A B) 2 A ( x B) x (A B) 3 ( x A) B x (A B) 4 A ( x B) x (A B) raft of Agst 23, 1999

25 24 Exercises 21 Provide natral dedctions for the valid jdgments Yo may assme that the bond variable x does not occr in B (items 1 and 3) or A (items 2 and 4) Exercise 23 Show that the three ways of extending the intitionistic proof system are eqivalent, that is, the same formlas are dedcible in all three systems Exercise 24 Assme we had omitted disjnction and existential qantification and their introdction and elimination rles from the list of logical primitives In the classical system, give a definition of disjnction and existential qantification (in terms of other logical constants) and show that the introdction and elimination rles now become admissible rles of inference A rle of inference is admissible if any dedction sing the rle can be transformed into one withot sing the rle Exercise 25 Assme we wold like to design a system of natral dedction for a simple temporal logic The main jdgment is now A is tre at time t written as t A 1 Explain how to modify the given rles for natral dedction to this more general jdgment and show the rles for implication and niversal qantification 2 Write ot introdction and elimination rles for the temporal operator A which shold be tre if A is tre at the next point in time enote the next time after t byt +1 3 Show the local redctions and expansions which show the local sondness and completness of yor rles 4 Write ot introdction and elimination rles for the temporal operator 2A which shold be tre if A is tre at all times 5 Show the local redctions and expansions Exercise 26 esign introdction and elimination rles for the connectives 1 A B, sally defined as (A B) (B A), 2 A B (exclsive or), sally defined as (A B) ( A B), withot recorse to other logical constants or operators Also show the corresponding local redctions and expansions raft of Agst 23, 1999

26 22 Natral edction raft of Agst 23, 1999

27 Bibliography [Cr30] [Gen35] [Her95] [Hil22] [Hil35] HB Crry Grndlagen der kombinatorischen Logik American Jornal of Mathematics, 52: , , 1930 Gerhard Gentzen Unterschngen über das logische Schließen Mathematische Zeitschrift, 39: , , 1935 Translated nder the title Investigations into Logical edctions in [Sza69] Hgo Herbelin Séqents q on calcle Ph thesis, Universite Paris 7, Janary 1995 avid Hilbert Nebegründng der Mathematik (erste Mitteilng) In Abhandlngen as dem mathematischen Seminar der Hambrgischen Universität, pages , 1922 Reprinted in [Hil35] avid Hilbert Gesammelte Abhandlngen, volme 3 Springer-Verlag, Berlin, 1935 [How69] W A Howard The formlae-as-types notion of constrction Unpblished manscript, 1969 Reprinted in To H B Crry: Essays on Combinatory Logic, Lambda Calcls and Formalism, 1980 [LS86] Joachim Lambek and Philip J Scott Introdction to Higher Order Categorical Logic Cambridge University Press, Cambridge, England, 1986 [ML85a] Per Martin-Löf On the meanings of the logical constants and the jstifications of the logical laws Technical Report 2, Scola di Specializzazione in Logica Matematica, ipartimento di Matematica, Università di Siena, 1985 [ML85b] Per Martin-Löf Trth of a proposition, evidence of a jdgement, validity of a proof Notes to a talk given at the workshop Theory of Meaning, Centro Fiorentino di Storia e Filosofia della Scienza, Jne 1985 [ML94] Per Martin-Löf Analytic and synthetic jdgements in type theory In Paolo Parrini, editor, Kant and Contemporary Epistemology, pages Klwer Academic Pblishers, 1994 raft of Agst 23, 1999

28 24 BIBLIOGRAPHY [Par92] Michel Parigot λµ-calcls: An algorithmic interpretation of classical natral dedction In A Voronkov, editor, Proceedings of the International Conference on Logic Programming and Atomated Reasoning, pages , St Petersbrg, Rssia, Jly 1992 Springer-Verlag LNCS 624 [Pfe95] Frank Pfenning Strctral ct elimination In Kozen, editor, Proceedings of the Tenth Annal Symposim on Logic in Compter Science, pages , San iego, California, Jne 1995 IEEE Compter Society Press [Pra65] ag Prawitz Natral edction Almqist & Wiksell, Stockholm, 1965 [Sza69] M E Szabo, editor The Collected Papers of Gerhard Gentzen North- Holland Pblishing Co, Amsterdam, 1969 raft of Agst 23, 1999