Proof Complexity of Non-Classical Logics
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The Deduction Rule and Linear and Near-Linear Proof Simulations
The Deduction Rule and Linear and Near-linear Proof Simulations Maria Luisa Bonet¤ Department of Mathematics University of California, San Diego Samuel R. Buss¤ Department of Mathematics University of California, San Diego Abstract We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof. A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by \nearly linear" is meant the ratio of proof lengths is O(®(n)) where ® is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n ¢ ®(n)). Also we show ¤Supported in part by NSF Grant DMS-8902480. 1 that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n). -
Circuit Complexity, Proof Complexity and Polynomial Identity Testing
Electronic Colloquium on Computational Complexity, Report No. 52 (2014) Circuit Complexity, Proof Complexity and Polynomial Identity Testing Joshua A. Grochow and Toniann Pitassi April 12, 2014 Abstract We introduce a new and very natural algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial- size algebraic circuits (VNP 6= VP). As a corollary to the proof, we also show that super- polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complex- ity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. (The existence of efficient proofs for our PIT axioms appears to be somewhere in between the major conjecture that PIT2 P and the known result that PIT2 P=poly.) We use these PIT axioms to shed light on AC0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. -
The Gödel and the Splitting Translations
The Godel¨ and the Splitting Translations David Pearce1 Universidad Politecnica´ de Madrid, Spain, [email protected] Abstract. When the new research area of logic programming and non-monotonic reasoning emerged at the end of the 1980s, it focused notably on the study of mathematical relations between different non-monotonic formalisms, especially between the semantics of stable models and various non-monotonic modal logics. Given the many and varied embeddings of stable models into systems of modal logic, the modal interpretation of logic programming connectives and rules be- came the dominant view until well into the new century. Recently, modal inter- pretations are once again receiving attention in the context of hybrid theories that combine reasoning with non-monotonic rules and ontologies or external knowl- edge bases. In this talk I explain how familiar embeddings of stable models into modal log- ics can be seen as special cases of two translations that are very well-known in non-classical logic. They are, first, the translation used by Godel¨ in 1933 to em- bed Heyting’s intuitionistic logic H into a modal provability logic equivalent to Lewis’s S4; second, the splitting translation, known since the mid-1970s, that al- lows one to embed extensions of S4 into extensions of the non-reflexive logic, K4. By composing the two translations one can obtain (Goldblatt, 1978) an ade- quate provability interpretation of H within the Goedel-Loeb logic GL, the sys- tem shown by Solovay (1976) to capture precisely the provability predicate of Peano Arithmetic. These two translations and their composition not only apply to monotonic logics extending H and S4, they also apply in several relevant cases to non-monotonic logics built upon such extensions, including equilibrium logic, non-monotonic S4F and autoepistemic logic. -
Reflection Principles, Propositional Proof Systems, and Theories
Reflection principles, propositional proof systems, and theories In memory of Gaisi Takeuti Pavel Pudl´ak ∗ July 30, 2020 Abstract The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this paper we will revisit some results about the reflection principles for propositional proofs systems using a finer scale of reflection principles. We will use the result that proving lower bounds on Resolution proofs is hard in Resolution. This appeared first in the recent article of Atserias and M¨uller [2] as a key lemma and was generalized and simplified in some spin- off papers [11, 13, 12]. We will also survey some results about arithmetical theories and proof systems associated with them. We will show a connection between a conjecture about proof complexity of finite consistency statements and a statement about proof systems associated with a theory. 1 Introduction arXiv:2007.14835v1 [math.LO] 29 Jul 2020 This paper is essentially a survey of some well-known results in proof complexity supple- mented with some observations. In most cases we will also sketch or give an idea of the proofs. Our aim is to focus on some interesting results rather than giving a complete ac- count of known results. We presuppose knowledge of basic concepts and theorems in proof complexity. An excellent source is Kraj´ıˇcek’s last book [19], where you can find the necessary definitions and read more about the results mentioned here. -
Contents Part 1. a Tour of Propositional Proof Complexity. 418 1. Is There A
The Bulletin of Symbolic Logic Volume 13, Number 4, Dec. 2007 THE COMPLEXITY OF PROPOSITIONAL PROOFS NATHAN SEGERLIND Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents Part 1. A tour of propositional proof complexity. 418 1. Is there a way to prove every tautology with a short proof? 418 2. Satisfiability algorithms and theories of arithmetic 420 3. A menagerie of Frege-like proof systems 427 4. Reverse mathematics of propositional principles 434 5. Feasible interpolation 437 6. Further connections with satisfiability algorithms 438 7. Beyond the Frege systems 441 Part 2. Some lower bounds on refutation sizes. 444 8. The size-width trade-off for resolution 445 9. The small restriction switching lemma 450 10. Expansion clean-up and random 3-CNFs 459 11. Resolution pseudowidth and very weak pigeonhole principles 467 Part 3. Open problems, further reading, acknowledgments. 471 Appendix. Notation. 472 References. 473 Received June 27, 2007. This survey article is based upon the author’s Sacks Prize winning PhD dissertation, however, the emphasis here is on context. The vast majority of results discussed are not the author’s, and several results from the author’s dissertation are omitted. c 2007, Association for Symbolic Logic 1079-8986/07/1304-0001/$7.50 417 418 NATHAN SEGERLIND Part 1. -
Modal Logics That Bound the Circumference of Transitive Frames
Modal Logics that Bound the Circumference of Transitive Frames. Robert Goldblatt∗ Abstract For each natural number n we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than n and no strictly ascending chains. The case n =0 is the G¨odel-L¨ob provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When n = 1, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to in- tuitionistic propositional logic. A topological semantic analysis shows that the n-th member of the sequence of extensions of S4 is the logic of hereditarily n +1-irresolvable spaces when the modality ♦ is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of ♦ as the derived set (of limit points) operation. The variety of modal algebras validating the n-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by n. Moreovereach algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them. 1 Algebraic Logic and Logical Algebra The field of algebraic logic has been described as having two main aspects (see the introductions to Daigneault 1974 and Andr´eka, N´emeti, and Sain 2001). -
A Short History of Computational Complexity
The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns. -
On the Blok-Esakia Theorem
On the Blok-Esakia Theorem Frank Wolter and Michael Zakharyaschev Abstract We discuss the celebrated Blok-Esakia theorem on the isomorphism be- tween the lattices of extensions of intuitionistic propositional logic and the Grze- gorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the Blok-Esakia theorem to extensions of intuitionistic logic with modal operators and coimplication. In memory of Leo Esakia 1 Introduction The Blok-Esakia theorem, which was proved independently by the Dutch logician Wim Blok [6] and the Georgian logician Leo Esakia [13] in 1976, is a jewel of non- classical mathematical logic. It can be regarded as a culmination of a long sequence of results, which started in the 1920–30s with attempts to understand the logical as- pects of Brouwer’s intuitionism by means of classical modal logic and involved such big names in mathematics and logic as K. Godel,¨ A.N. Kolmogorov, P.S. Novikov and A. Tarski. Arguably, it was this direction of research that attracted mathemat- ical logicians to modal logic rather than the philosophical analysis of modalities by Lewis and Langford [43]. Moreover, it contributed to establishing close connec- tions between logic, algebra and topology. (It may be of interest to note that Blok and Esakia were rather an algebraist and, respectively, a topologist who applied their results in logic.) Frank Wolter Department of Computer Science, University of Liverpool, U.K. e-mail: wolter@liverpool. ac.uk Michael Zakharyaschev Department of Computer Science and Information Systems, Birkbeck, University of London, U.K. -
Substitution and Propositional Proof Complexity
Substitution and Propositional Proof Complexity Sam Buss Abstract We discuss substitution rules that allow the substitution of formulas for formula variables. A substitution rule was first introduced by Frege. More recently, substitution is studied in the setting of propositional logic. We state theorems of Urquhart’s giving lower bounds on the number of steps in the substitution Frege system for propositional logic. We give the first superlinear lower bounds on the number of symbols in substitution Frege and multi-substitution Frege proofs. “The length of a proof ought not to be measured by the yard. It is easy to make a proof look short on paper by skipping over many links in the chain of inference and merely indicating large parts of it. Generally people are satisfied if every step in the proof is evidently correct, and this is permissable if one merely wishes to be persuaded that the proposition to be proved is true. But if it is a matter of gaining an insight into the nature of this ‘being evident’, this procedure does not suffice; we must put down all the intermediate steps, that the full light of consciousness may fall upon them.” [G. Frege, Grundgesetze, 1893; translation by M. Furth] 1 Introduction The present article concentrates on the substitution rule allowing the substitution of formulas for formula variables.1 The substitution rule has long pedigree as it was a rule of inference in Frege’s Begriffsschrift [9] and Grundgesetze [10], which contained the first in-depth formalization of logical foundations for mathematics. Since then, substitution has become much less important for the foundations of Sam Buss Department of Mathematics, University of California, San Diego, e-mail: [email protected], Supported in part by Simons Foundation grant 578919. -
Esakia's Theorem in the Monadic Setting
Esakia’s theorem in the monadic setting Luca Carai joint work with Guram Bezhanishvili BLAST 2021 June 13, 2021 • Translation of intuitionistic logic into modal logic (G¨odel) • Algebraic semantics (Birkhoff, McKinsey-Tarski) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic logic (Glivenko) Important developements in 1930s and 1940s in the study of intuitionistic logic: • Axiomatization (Kolmogorov, Glivenko, Heyting) Intuitionism is an important direction in the mathematics of the 20th century. Intuitionistic logic is the logic of constructive mathematics. It has its origins in Brouwer’s criticism of the use of the principle of the excluded middle. • Translation of intuitionistic logic into modal logic (G¨odel) • Algebraic semantics (Birkhoff, McKinsey-Tarski) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic logic (Glivenko) Intuitionism is an important direction in the mathematics of the 20th century. Intuitionistic logic is the logic of constructive mathematics. It has its origins in Brouwer’s criticism of the use of the principle of the excluded middle. Important developements in 1930s and 1940s in the study of intuitionistic logic: • Axiomatization (Kolmogorov, Glivenko, Heyting) • Translation of intuitionistic logic into modal logic (G¨odel) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic -
Constructive Embeddings of Intermediate Logics
Constructive embeddings of intermediate logics Sara Negri University of Helsinki Symposium on Constructivity and Computability 9-10 June 2011, Uppsala The G¨odel-Tarski-McKinsey embedding of intuitionistic logic into S4 ∗ 1. G¨odel(1933): `Int A ! `S4 A soundness ∗ 2. McKinsey & Tarski (1948): `S4 A ! `Int A faithfulness ∗ 3. Dummett & Lemmon (1959): `Int+Ax A iff `S4+Ax∗ A A modal logic M is a modal companion of a superintuitionistic ∗ logic L if `L A iff `M A . So S4 is a modal companion of Int, S4+Ax∗ is a modal companion of Int+Ax. 2. and 3. are proved semantically. We look closer at McKinsey & Tarski (1948): Proof by McKinsey & Tarski (1948) uses: I 1. Completeness of intuitionistic logic wrt Heyting algebras (Brouwerian algebras) and of S4 wrt topological Boolean algebras (closure algebras) I 2. Representation of Heyting algebras as the opens of topological Boolean algebras. I 3. The proof is indirect because of 1. and non-constructive because of 2. (Uses Stone representation of distributive lattices, in particular Zorn's lemma) The result was generalized to intermediate logics by Dummett and Lemmon (1959). No syntactic proof of faithfulness in the literature except the complex proof of the embedding of Int into S4 in Troelstra & Schwichtenberg (1996). Goal: give a direct, constructive, syntactic, and uniform proof. Background on labelled sequent systems Method for formulating systems of contraction-free sequent calculus for basic modal logic and its extensions and for systems of non-classical logics in Negri (2005). General ideas for basic modal logic K and other systems of normal modal logics. -
Proof Complexity in Algebraic Systems and Bounded Depth Frege Systems with Modular Counting
PROOF COMPLEXITY IN ALGEBRAIC SYSTEMS AND BOUNDED DEPTH FREGE SYSTEMS WITH MODULAR COUNTING S. Buss, R. Impagliazzo, J. Kraj¶³cek,· P. Pudlak,¶ A. A. Razborov and J. Sgall Abstract. We prove a lower bound of the form N (1) on the degree of polynomials in a Nullstellensatz refutation of the Countq polynomials over Zm, where q is a prime not dividing m. In addition, we give an explicit construction of a degree N (1) design for the Countq principle over Zm. As a corollary, using Beame et al. (1994) we obtain (1) a lower bound of the form 2N for the number of formulas in a constant-depth N Frege proof of the modular counting principle Countq from instances of the counting M principle Countm . We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations. Further we show that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connective MODp follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987), Smolensky (1987) (in fact, these two proof systems are basically equivalent). Introduction A propositional proof system is intuitively a system for establishing the validity of propo- sitional tautologies in some ¯xed complete language. The formal de¯nition of propositional proof system is that it is a polynomial time function f which maps strings over an alphabet § onto the set of propositional tautologies (Cook & Reckhow 1979).