Propositional Tautologies Research Articles

Saturation properties for compositional truth with propositional correctness

It is an open question whether compositional truth with the principle of propositional soundness: “All arithmetical sentences which are propositional tautologies are true” is conservative over Peano Arithmetic. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the truth predicate, thus showing significant limitations to the possible conservativity proof.

Compositional truth with propositional tautologies and quantifier-free correctness

In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as Δ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _0$$\\end{document}-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.

NEW RELATIONS AND SEPARATIONS OF CONJECTURES ABOUT INCOMPLETENESS IN THE FINITE DOMAIN

AbstractIn [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

ℋ-Colouring Dichotomy in Proof Complexity

Abstract The $\mathcal {H}$-colouring problem for undirected simple graphs is a computational problem from a huge class of the constraint satisfaction problems (CSPs): an $\mathcal {H}$-colouring of a graph $\mathcal {G}$ is just a homomorphism from $\mathcal {G}$ to $\mathcal {H}$ and the problem is to decide for fixed $\mathcal {H}$, given $\mathcal {G}$, if a homomorphism exists or not. The dichotomy theorem for the $\mathcal {H}$-colouring problem was proved by Hell and Nešetřil (1990, J. Comb. Theory Ser. B, 48, 92–110) (an analogous theorem for all CSPs was recently proved by Zhuk (2020, J. ACM, 67, 1–78) and Bulatov (2017, FOCS, 58, 319–330)), and it says that for each $\mathcal {H}$, the problem is either $p$-time decidable or $NP$-complete. Since negations of unsatisfiable instances of CSP can be expressed as propositional tautologies, it seems to be natural to investigate the proof complexity of CSP. We show that the decision algorithm in the $p$-time case of the $\mathcal {H}$-colouring problem can be formalized in a relatively weak theory and that the tautologies expressing the negative instances for such $\mathcal {H}$ have polynomial proofs in propositional proof system $R^*(log)$, a mild extension of resolution. In fact, when the formulas are expressed as unsatisfiable sets of clauses, they have $p$-size resolution proofs. To establish this, we use a well-known connection between theories of bounded arithmetic and propositional proof systems. This upper bound follows also from a different construction in [1]. We complement this result by a lower bound result that holds for many weak proof systems for a special example of $NP$-complete case of the $\mathcal {H}$-colouring problem, using known results about the proof complexity of the pigeonhole principle. The main goal of our work is to start the development of some of the theories beyond the CSP dichotomy theorem in bounded arithmetic. We aim eventually—in a subsequent work—to formalize in such a theory the soundness of Zhuk’s algorithm, extending the upper bound proved here from undirected simple graphs to the general case of directed graphs in some logical calculi.

Trakhtenbrot Theorem and First-Order Axiomatic Extensions of MTL

In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies associated to finite models is not recursively enumerable. In 1999, P. Hajek generalized this result to the first-order versions of ?ukasiewicz, Godel and Product logics, w.r.t. their standard algebras. In this paper we extend the analysis to the first-order versions of axiomatic extensions of MTL. Our main result is the following. Let $${\mathbb{K}}$$K be a class of MTL-chains. Then the set of all first-order tautologies associated to the finite models over chains in $${\mathbb{K}}$$K, fTAUT$${_{\forall}^{\mathbb{K}}}$$?K, is $${\Pi_{1}^{0}}$$?10 -hard. Let TAUT$${_\mathbb{K}}$$K be the set of propositional tautologies of $${\mathbb{K}}$$K. If TAUT$${_{\mathbb{K}}}$$K is decidable, we have that fTAUT$${_{\forall}^{\mathbb{K}}}$$?K is in $${\Pi_{1}^{0}}$$?10. We have similar results also if we expand the language with the Δ operator.

Idealization, epistemic logic, and epistemology

Many criticisms of epistemic logic have centered around its use of devices such as idealized knowers with logical omniscience and perfect self-knowledge. One possible response to such criticisms is to say that these idealizations are normative devices, and that epistemic logic tells us how agents ought to behave. This paper will take a different approach, treating epistemic logic as descriptive, and drawing the analogy between its formal models and idealized scientific models on that basis. Treating it as descriptive matches the way in which some philosophers, including one of its founders, Jaako Hintikka, have thought about epistemic logic early in its history. Further, the analogy between the two fields will give us a way to defuse criticisms that see epistemic logic as unrealistic. For example, criticizing models of epistemic logic in which agents know all propositional tautologies as being unrealistic would be like criticizing frictionless planes in physics for being unrealistic. Each one would certainly be an unsuitable model for studying some kinds of phenomena, but is entirely appropriate for others. After outlining the analogy between epistemic and scientific models, we will discuss some ways in which idealizations are used by different research programs in epistemic logic.

Polynomial Simulation and Refutation of Complex Formulas of Resolution over Linear Equations in Propositional Proof System

In this paper, we present that the propositional proof system R(lin) (Resolution over Linear Equations) established by Ran Raz and Iddo Tzameret is not a super system, there exists a sequence of tautologies, which require proof complexity exponential in size of tautologies. We show that there are the sequence of unsatisfiable collections of disjuncts of linear equations, which require exponential lower bounds in R(lin) and have polynomially bounded refutations by incorporating renaming inference rule to R(lin) system. Some additional properties of R(lin) have been described that many of the “hard” provable in R outstanding examples of propositional tautologies (contradictions) have polynomially bounded proofs in R(lin).

Proving Propositional Tautologies in a Natural Dialogue

The paper proposes a dialogue system LND which brings together and unifies two traditions in studying dialogue as a game: the dialogical logic introduced by Lorenzen; and persuasion dialogue games as specified by Prakken. The first approach allows the representation of formal dialogues in which the validity of argument is the topic discussed. The second tradition has focused on natural dialogues examining, e.g., informal fallacies typical in real-life communication. Our goal is to unite these two approaches in order to allow communicating agents to benefit from the advantages of both, i.e., to equip them with the ability not only to persuade each other about facts, but also to prove that a formula used in an argument is a classical propositional tautology. To this end, we propose a new description of the dialogical logic which meets the requirements of Prakken's generic specification for natural dialogues, and we introduce rules allowing to embed a formal dialogue in a natural one. We also show the correspondence result between the original and the new version of the dialogical logic, i.e., we show that a winning strategy for a proponent in the original version of the dialogical logic means a winning strategy for a proponent in the new version, and conversely.

On Optimal Heuristic Randomized Semidecision Procedures, with Applications to Proof Complexity and Cryptography

The existence of an optimal propositional proof system is a major open question in proof complexity; many people conjecture that such systems do not exist. Krajíček and Pudlák (J. Symbol. Logic 54(3):1063, 1989) show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (Theor. Comput. Sci. 412(4–5):478, 2011) recently presented a conjecture implying that such an algorithm does not exist.We show that if one allows errors, then such optimal algorithms do exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow an algorithm, called a heuristic acceptor, to claim a small number of false “theorems” and err with bounded probability on other inputs. The amount of false “theorems” is measured according to a polynomial-time samplable distribution on non-tautologies. Our result remains valid for all recursively enumerable languages and can also be viewed as the existence of an optimal weakly automatizable heuristic proof system. The notion of a heuristic acceptor extends the notion of a classical acceptor; in particular, an optimal heuristic acceptor for any distribution simulates every classical acceptor for the same language.We also note that the existence of a co-NP-language L with a polynomial-time samplable distribution on \(\overline{L}\) that has no polynomial-time heuristic acceptors is equivalent to the existence of an infinitely-often one-way function.

ON THE PROOF COMPLEXITY OF THE NISAN–WIGDERSON GENERATOR BASED ON A HARD NP ∩ coNP FUNCTION

Let g be a map defined as the Nisan–Wigderson generator but based on an NP ∩ coNP -function f. Any string b outside the range of g determines a propositional tautology τ(g)b expressing this fact. Razborov [27] has conjectured that if f is hard on average for P/poly then these tautologies have no polynomial size proofs in the Extended Frege system EF. We consider a more general Statement (S) that the tautologies have no polynomial size proofs in any propositional proof system. This is equivalent to the statement that the complement of the range of g contains no infinite NP set. We prove that Statement (S) is consistent with Cook' s theory PV and, in fact, with the true universal theory T PV in the language of PV. If PV in this consistency statement could be extended to "a bit" stronger theory (properly included in Buss's theory [Formula: see text]) then Razborov's conjecture would follow, and if TPV could be added too then Statement (S) would follow. We discuss this problem in some detail, pointing out a certain form of reflection principle for propositional logic, and we introduce a related feasible disjunction property of proof systems.

Proof systems that take advice

One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [J. Symbolic Logic, 1979], where they defined propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajı´ček [J. Symbolic Logic, 2007] have recently started the investigation of proof systems which are computed by poly-time functions using advice.In this paper we concentrate on three fundamental questions regarding this new model. First, we investigate whether a given language L admits a polynomially bounded proof system with advice. Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that this question is tightly linked with the question whether L has small nondeterministic instance complexity.The second question concerns the existence of optimal proof systems with advice. For propositional proof systems, Cook and Krajı´ček gave a surprising positive answer which we extend to all languages.These results show that providing proof systems with advice yields a more powerful model, but this model is also less directly applicable in practice. Our third question therefore asks whether the usage of advice in propositional proof systems can be simplified or even eliminated. While in principle, the advice can be very complex, we show that propositional proof systems with logarithmic advice are also computable in poly-time with access to a sparse NP-oracle. Employing a recent technique of Buhrman and Hitchcock [CCC, 2008] we also manage to transfer the advice from the proof to the proven formula, which leads to a more practical computational model.

A new mapping between combinatorial proofs and sequent calculus proofs read out from logical flow graphs

Combinatorial proofs are abstract invariants for sequent calculus proofs, similarly to homotopy groups which are abstract invariants for topological spaces. Sequent calculus fails to be surjective onto combinatorial proofs, and here we extract a syntactically motivated closure of sequent calculus from which there is a surjection onto a complete set of combinatorial proofs. We characterize a class of canonical sequent calculus proofs for the full set of propositional tautologies and derive a new completeness theorem for combinatorial propositions. For this, we define a new mapping between combinatorial proofs and sequent calculus proofs, different from the one originally proposed, which explicitly links the logical flow graph of a proof to a skew fibration between graphs of formulas. The categorical properties relating the original and the new mappings are explicitly discussed.

Distinguishing standard SBL‐algebras with involutive negations by propositional formulas

AbstractPropositional fuzzy logics given by a combination of a continuous SBL t‐norm with finitely many idempotents and of an involutive negation are investigated. A characterization of continuous t‐norms which, in combination with different involutive negations, yield either isomorphic algebras or algebras with distinct and incomparable sets of propositional tautologies is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Complexity of Propositional Proofs

AbstractPropositional 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.

Efficiently checking propositional refutations in HOL theorem provers

This paper describes the integration of zChaff and MiniSat, currently two leading SAT solvers, with Higher Order Logic (HOL) theorem provers. Both SAT solvers generate resolution-style proofs for (instances of) propositional tautologies. These proofs are verified by the theorem provers. The presented approach significantly improves the provers' performance on propositional problems, and exhibits counterexamples for unprovable conjectures. It is also shown that LCF-style theorem provers can serve as viable proof checkers even for large SAT problems. An efficient representation of the propositional problem in the theorem prover turns out to be crucial; several possible solutions are discussed.

Polynomial-size Frege and resolution proofs of st-connectivity and Hex tautologies

A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st-connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere. We prove that the propositional tautologies which encode the st-connectivity principle have polynomial-size Frege proofs and polynomial-size TC 0 -Frege proofs. For bounded-width grid graphs, the st-connectivity tautologies have polynomial-size resolution proofs. A key part of the proof is to show that the group with two generators, both of order two, has word problem in alternating logtime (Alogtime) and even in TC 0 . Conversely, we show that constant depth Frege proofs of the st-connectivity tautologies require near-exponential size. The proof uses a reduction from the pigeonhole principle, via tautologies that express a “directed single source” principle SINK, which is related to Papadimitriou's search classes PPAD and PPADS (or, PSK). The st-connectivity principle is related to Urquhart's propositional Hex tautologies, and we establish the same upper and lower bounds on proof complexity for the Hex tautologies. In addition, the Hex tautology is shown to be equivalent to the SINK tautologies and to the one-to-one onto pigeonhole principle.

Integrating a SAT Solver with an LCF-style Theorem Prover

This paper describes the integration of a leading SAT solver with Isabelle/HOL, a popular interactive theorem prover. The SAT solver generates resolution-style proofs for (instances of) propositional tautologies. These proofs are verified by the theorem prover. The presented approach significantly improves Isabelle's performance on propositional problems, and furthermore exhibits counterexamples for unprovable conjectures.

Intuitionistic Completeness and Classical Logic

We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense.

Generating hard tautologies using predicate logic and the symmetric group

We introduce methods to generate uniform families of hard propositional tautologies. The tautologies are essentially generated from a single propositional formula by a natural action of the symmetric group Sn. The basic idea is that any Second Order Existential sentence Ψ can be systematically translated into a conjunction φ of a finite collection of clauses such that the models of size n of an appropriate Skolemization Ψ are in one-to-one correspondence with the satisfying assignments to φn: the Sn-closure of φ, under a natural action of the symmetric group Sn. Each φn is a CNF and thus has depth at most 2. The size of the φn’s is bounded by a polynomial in n. Under the assumption NEXPTIME 6= coNEXPTIME, for any such sequence φn for which the spectrum S := {n : φn satisfiable} is NEXPTIME-complete, the tautologies ¬φn 6∈S do not have polynomial length proofs in any propositional proof system. Our translation method shows that most sequences of tautologies being studied in propositional proof complexity can be systematically generated from Second Order Existential sentences The International PhD Research School at BRICS, Aarhus, Denmark; Email: smriis@daimi.aau.dk Part of this work was done while visiting the Fields Institute, Toronto, Canada CISE Department, University of Florida, Gainesville, FL 32611-6120; Email: sitharam@cise.ufl.edu Supported in part by NSF Grant CCR 94-09809.

Generating Hard Tautologies Using Predicate Logic and the Symmetric Group

We introduce methods to generate uniform families of hard propositional tautologies. The tautologies are essentially generated from a single propositional formula by a natural action of the symmetric group Sn.<br />The basic idea is that any Second Order Existential sentence Psi can be systematically translated into a conjunction phi of a finite collection of clauses such that the models of size n of an appropriate Skolemization Psi~ are in one-to-one correspondence with the satisfying assignments to phi_n: the Sn-closure of phi, under a natural action of the symmetric group Sn. Each phi_n is a CNF and thus has depth at most 2. The size of the phi_n's is bounded by a polynomial in n. Under the assumption NEXPTIME |= co- NEXPTIME, for any such sequence phi_n for which the spectrum S := {n : phi_n satisfiable} is NEXPTIME-complete, the tautologies not phi_(n not in S) do not have polynomial length proofs in any propositional proof system.<br /> Our translation method shows that most sequences of tautologies being studied in propositional proof complexity can be systematically generated from Second Order Existential sentences and moreover, many natural mathematical statements can be converted into sequences of propositional tautologies in this manner.<br /> We also discuss algebraic proof complexity issues for such sequences of tautologies. To this end, we show that any Second Order Existential sentence Psi can be systematically translated into a finite collection of polynomial equations Q = 0 such that the models of size n of an appropriate skolemization Psi~ are in one-to-one correspondence with the solutions to Qn = 0: the Sn-closure of Q = 0, under a natural action of the symmetric group Sn. The degree of Qn is the same as that of Q, and hence is independent of n, and the number of variables is no more than a polynomial in n. Furthermore, using results in [19] and [20], we briefly describe how, for the corresponding sequences of tautologies phi_n, the rich structure of the Sn closed, uniformly generated, algebraic systems Qn has profound consequences on the algebraic proof complexity of phi_n.