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Proofs, Computability, Undecidability, Complexity, And the Lambda Calculus An Introduction Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] c Jean Gallier Please, do not reproduce without permission of the author April 28, 2020 2 Preface The main goal of this book is to present a mix of material dealing with 1. Proof systems. 2. Computability and undecidability. 3. The Lambda Calculus. 4. Some aspects of complexity theory. Historically, the theory of computability and undecidability arose from Hilbert's efforts to completely formalize mathematics and from G¨odel'sfirst incompleteness theorem that showed that such a program was doomed to fail. People realized that to carry out both Hilbert's program and G¨odel'swork it was necessary to define precisely what is the notion of a computable function and the notion of a mechanically checkable proof. The first definition given around 1934 was that of the class of computable function in the sense of Herbrand{ G¨odel{Kleene. The second definition given by Church in 1935-1936 was the notion of a function definable in the λ-calculus. The equivalence of these two definitions was shown by Kleene in 1936. Shortly after in 1936, Turing introduced a third definition, that of a Turing- computable function. Turing proved the equivalence of his definition with the Herbrand{ G¨odel{Kleenedefinition in 1937 (his proofs are rather sketchy compared to Kleene's proofs). All these historical papers can be found in a fascinating book edited by Martin Davis [25]. Negative results pointing out limitations of the notion of computability started to appear: G¨odel'sfirst (and second) incompleteness result, but also Church's theorem on the undecid- ability of validity in first-order logic, and Turing's result on the undecidability of the halting problem for Turing machines. Although originally the main focus was on the notion of func- tion, these undecidability results triggered the study of computable and noncomputable sets of natural numbers. Other definitions of the computable functions were given later. From our point of view, the most important ones are 1. RAM programs and RAM-computable functions by Shepherdson and Sturgis (1963), and anticipated by Post (1944); see Machtey and Young [36]. 2. Diophantine-definable sets (Davis{Putnam{Robinson{Matiyasevich); see Davis [8, 9]. 3 4 We find the RAM-progam model quite attractive because it is a very simplified realistic model of the true architecture of a modern computer. Technically, we also find it more convenient to assign G¨odelnumbers to RAM programs than assigning G¨odelnumbers to Turing machines. Every RAM program can be converted to a Turing machine and vice- versa in polynomial time (going from a Turing machine to a RAM is quite horrific), so the two models are equivalent in a strong sense. So from our perspective Turing machines could be dispensed with, but there is a problem. The problem is that the Turing machine model seems more convenient to cope with time or space restrictions, that is, to define complexity classes. There is actually no difficulty in defining nondeterministic RAM programs and to impose a time restriction on the program counter or a space restriction on the size of registers, but nobody seems to follows this path. This seems unfortunate to us because it appears that it would be easier to justify the fact that certain reductions can be carried out in polynomial time (or space) by writing a RAM program rather than by constructing a Turing machine. Regarding this issue, we are not aware than anyone actually provides Turing machines computing these reductions, even for SAT. In any case, we will stick to the tradition of using Turing machines when discussing complexity classes. In addition to presenting the RAM-program model, the Turing machine model, the Herbrand–G¨odel{Kleenedefinition of the computable functions, and showing their equiv- alence, we provide an introduction to recursion theory (see Chapter 7). In particular, we discuss creative and productive sets (see Rogers [44]). This allows us to cover most of the main undecidability results. These include 1. The undecidability of the halting problem for RAM programs (and Turing machines). 2. Rice's theorem for the computable functions. 3. Rice's extended theorem for the listable sets. 4. A strong form of G¨odel'first incompleteness theorem (in terms of creative sets) follow- ing Rogers [44]. 5. The fact that the true first-order sentences of arithmetic are not even listable (a pro- ductive set) following Rogers [44]. 6. The undecidability of the Post correspondence problem (PCP) using a proof due to Dana Scott. 7. The undecidability of the validity in first-order logic (Church's theorem), using a proof due to Robert Floyd. 8. The undecidability of Hilbert's tenth problem (the DPRM theorem) following Davis [8]. 5 9. Another strong form of G¨odel' first incompleteness theorem, as a consequence of dio- phantine definability following Davis [8]. The following two topics are rarely covered in books on the theory of computation and undecidability. In Chapter 6 we introduce Church's λ-calculus and show how the computable functions and the partial computable functions are definable in the λ-calculus, using a method due to Barendregt [4]. We also give a glimpse of the second-order polymorphic λ-calculus of Girard. In Chapter 8 we discuss the definability of the listable sets in terms of Diophantine equations (zeros of polynomials with integer coefficients) and state the famous result about the undecidability of Hilbert's tenth problem (the DPRM theorem). We follow the masterly exposition of Davis [8, 9]. A possibly unsusual aspect of our book is that we begin with two chapters on mathemat- ical reasoning and logic. Given the origins of the theory of computation and undecidability, we feel that this is very appropriate. We present proof systems in natural deduction style (a la Prawitz), which makes it easy to discuss the special role of the proof{by{contradiction principle, and to introduce intuitionistic logic, which is the result of removing this rule from the set of inference rules. It is also quite natural to explain how proofs in intuitionistic propositional logic are represented by simply-typed λ-terms. Then it is easy to introduce the \Curry{Howard isomorphism." This is a prelude to the introduction of the \pure" (untyped) λ-calculus. Our treatment of complexity theory is limited to , , co- , , , (PSPACE) and (NPSPACE) and is fairly standard.P NP However,NP weEXP proveNEXP that SATPS is -completeNPS by first proving (following Lewis and Papadimitriou [35]) that a bounded tilingNP problem is -complete. NP In Chapter 12 we treat the result that primality testing is in in more details than most other sources, relying on an improved version of a theoremNP of Lucas as discussed in Crandall and Pomerance [6]. The only result that we omit is the existence of primitive roots in (Z=pZ)∗ when p is prime. In Chapter 13 we prove Savitch's theorem ( = ). We state the fact that the validity of quantified boolean formulae is -completePS NPS and provide parts of the proof. We conclude with the beautiful proof of StatmanPS [46] that provability in intuitionistic logic is -complete. We not give all the details but we prove the correctness of Statman's amazing translationPS of a valid QBF into an intuitionistically provable proposition. We feel strongly that one does not learn mathematics without reading (and struggling through) proofs, so we tried to provide as many proofs as possible. Among some of the omissions, we do not prove the DPRM nor do we show how to construct a G¨odelsentence in the proof of the first incompleteness theorem; Rogers [44] leaves this as an exercise! We also do not give a complete proof of Statman's result. However, whenever a proof is omitted, we provide a pointer to a source that contains such a proof. 6 Acknowledgement: We would like to thank Jo~aoSedoc and Marcelo Siqueira, for reporting typos and for helpful comments. I was initiated to the theory of computation and undecid- ability by my advisor Sheila Greibach who taught me how to do research. My most sincere thanks to Sheila for her teachings and the inspiration she provided. In writing this book we were inspired and sometimes borrowed heavily from the beautiful papers and books written by the following people who have my deepest gratitude: Henk Barendregt, Richard Crandall, Martin Davis, Herbert Enderton, Harvey Friedman, Jean-Yves Girard, John Hopcroft, Bill Howard, Harry Lewis, Zohar Manna, Christos Papadimitriou, Carl Pomerance, Dag Prawitz, Helmut Schwichtenberg, Dana Scott, Rick Statman, Jeff Ullman, Hartley Rogers, and Paul Young. Of course, we must acknowledge Alonzo Church, Gerhard Gentzen, Kurl G¨odel, Stephen Kleene and Alan Turing for their extraordinary seminal work. Contents Contents 7 1 Mathematical Reasoning And Basic Logic 11 1.1 Introduction . 11 1.2 Logical Connectives, Definitions . 12 1.3 Meaning of Implication and Proof Templates for Implication . 16 1.4 Proof Trees and Deduction Trees . 20 1.5 Proof Templates for .............................. 22 : 1.6 Proof Templates for ; ; ........................... 26 ^ _ ≡ 1.7 De Morgan Laws and Other Useful Rules of Logic . 34 1.8 Formal Versus Informal Proofs; Some Examples . 35 1.9 Truth Tables and Truth Value Semantics . 39 1.10 Proof Templates for the Quantifiers . 42 1.11 Sets and Set Operations . 50 1.12 Induction and Well{Ordering Principle . 55 1.13 Summary . 57 Problems . 59 2 Mathematical Reasoning And Logic, A Deeper View 63 2.1 Introduction . 63 2.2 Inference Rules, Deductions, Proof Systems ) and ) .......... 64 Nm NGm 2.3 Proof Rules, Deduction and Proof Trees for Implication .
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