But without any minutes restriction or schedule-related limitations (he's playing in back-to-backs!), Embiid has entered the "you're gonna need a bigger boat" stage of his career, where most teams have no satisfiable way to combat his all-around brilliance.
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A nondeterministic algorithm for determining whether a 2-satisfiability instance is not satisfiable, using only a logarithmic amount of writable memory, is easy to describe: simply choose (nondeterministically) a variable v and search (nondeterministically) for a chain of implications leading from v to its negation and then back to v. If such a chain is found, the instance cannot be satisfiable. By the Immerman–Szelepcsényi theorem, it is also possible in nondeterministic logspace to verify that a satisfiable 2-satisfiability instance is satisfiable. 2-satisfiability is NL-complete,.
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In a revisionist account, considers that Löwenheim's proof was complete. gave a (correct) proof using formulas in what would later be called Skolem normal form and relying on the axiom of choice: :Every countable theory which is satisfiable in a model M, is satisfiable in a countable substructure of M. also proved the following weaker version without the axiom of choice: : Every countable theory which is satisfiable in a model is also satisfiable in a countable model. simplified . Finally, Anatoly Ivanovich Maltsev (Анато́лий Ива́нович Ма́льцев, 1936) proved the Löwenheim–Skolem theorem in its full generality .
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An instance of this non-uniform problem can be then written as a set of literals of the form R_i(x_1,\ldots,x_n). Among these instances/sets of literals, some are satisfiable and some are not; whether a set of literals is satisfiable depends on the relations, which are specified by the non-uniform problem. In the other way around, a non-uniform problem tells which sets of literals represent satisfiable instances and which ones represent unsatisfiable instances. Once relations are named, a non- uniform problem expresses a set of sets of literals: those associated to satisfiable (or unsatisfiable) instances.
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The formula obtained by this transformation is satisfiable if and only if the original formula is.
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Let A be a countable admissible set. Let L be an A-finite relational language. Suppose \Gamma is a set of L_A-sentences, where \Gamma is a \Sigma_1 set with parameters from A, and every A-finite subset of \Gamma is satisfiable. Then \Gamma is satisfiable.
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See Algorithms for solving SAT below. SAT is trivial if the formulas are restricted to those in disjunctive normal form, that is, they are disjunction of conjunctions of literals. Such a formula is indeed satisfiable if and only if at least one of its conjunctions is satisfiable, and a conjunction is satisfiable if and only if it does not contain both x and NOT x for some variable x. This can be checked in linear time.
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The problem of deciding whether a MITL formula is satisfiable over a signal is in PSPACE-complete.
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We approach the proof of Theorem 2 by successively restricting the class of all formulas φ for which we need to prove "φ is either refutable or satisfiable". At the beginning we need to prove this for all possible formulas φ in our language. However, suppose that for every formula φ there is some formula ψ taken from a more restricted class of formulas C, such that "ψ is either refutable or satisfiable" → "φ is either refutable or satisfiable". Then, once this claim (expressed in the previous sentence) is proved, it will suffice to prove "φ is either refutable or satisfiable" only for φ's belonging to the class C. If φ is provably equivalent to ψ (i.e.
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The direction heuristic decides which variable assignment (true or false) to explore first. In satisfiable problem instances, choosing a satisfiable branch first is beneficial. The cutoff heuristic decides when to stop expanding a cube and instead forward it to a sequential conflict-driven solver. Preferably the cubes are similarly complex to solve.
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As a result, the formula F_2 = \forall x_1 \dots \forall x_n R(x_1,\dots,x_n,f(x_1,\dots,x_n)) is satisfiable, because it has the model obtained by adding the evaluation of f to M. This shows that F_1 is satisfiable only if F_2 is satisfiable as well. Conversely, if F_2 is satisfiable, then there exists a model M' that satisfies it; this model includes an evaluation for the function f such that, for every value of x_1,\dots,x_n, the formula R(x_1,\dots,x_n,f(x_1,\dots,x_n)) holds. As a result, F_1 is satisfied by the same model because one may choose, for every value of x_1,\ldots,x_n, the value y=f(x_1,\dots,x_n), where f is evaluated according to M'.
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If a quantifier-free sentence \theta is satisfiable then w(\theta) >0. The principle of super regularity (universal certainty), SReg. If a sentence \theta is satisfiable then w(\theta) >0. The constant irrelevance principle, IP. If sentences \theta, \phi have no constants in common then w(\theta \wedge \phi) = w(\theta) \cdot w(\phi).
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This generalization is commonly called satisfiability modulo theories. The question whether a sentence in propositional logic is satisfiable is a decidable problem. In general, the question whether sentences in first-order logic are satisfiable is not decidable. In universal algebra and equational theory, the methods of term rewriting, congruence closure and unification are used to attempt to decide satisfiability.
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A propositional logic formula, also called Boolean expression, is built from variables, operators AND (conjunction, also denoted by ∧), OR (disjunction, ∨), NOT (negation, ¬), and parentheses. A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to its variables. The Boolean satisfiability problem (SAT) is, given a formula, to check whether it is satisfiable.
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Therefore, each learnt clause can be inferred from the original clauses and other learnt clauses by a sequence of resolution steps. If cN is the new learnt clause, then ϕ is satisfiable if and only if ϕ ∪ {cN} is also satisfiable. Moreover, the modified backtracking step also does not affect soundness or completeness, since backtracking information is obtained from each new learnt clause.
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In the case of the positive propositional calculus, the satisfiability problem is trivial, as every formula is satisfiable, while the validity problem is co-NP complete.
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If Theorem 1 holds, and φ is not satisfiable in any structure, then ¬φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds. If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.
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In conjunction with the formula, each of the cubes forms a new formula. These formulas can be solved independently and concurrently by conflict-driven solvers. As the disjunction of these formulas is equivalent to the original formula, the problem is reported to be satisfiable, if one of the formulas is satisfiable. The look-ahead solver is favorable for small but hard problems, so it is used to gradually divide the problem into multiple sub-problems.
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Unit propagation, applied repeatedly as new unit clauses are generated, is a complete satisfiability algorithm for sets of propositional Horn clauses; it also generates a minimal model for the set if satisfiable: see Horn-satisfiability.
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One of these cliques is large if and only if it corresponds to a proof string that many proof checkers accept. If the original satisfiability instance is satisfiable, it will have a valid proof string, one that is accepted by all runs of the checker, and this string will correspond to a large clique in the graph. However, if the original instance is not satisfiable, then all proof strings are invalid, each proof string has only a small number of checker runs that mistakenly accept it, and all cliques are small. Therefore, if one could distinguish in polynomial time between graphs that have large cliques and graphs in which all cliques are small, or if one could accurately approximate the clique problem, then applying this approximation to the graphs generated from satisfiability instances would allow satisfiable instances to be distinguished from unsatisfiable instances.
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A small pre-calculated CNF expression that relates the inputs and outputs is appended (via the "and" operation) to the output expression. Note that inputs to these gates can be either the original literals or the introduced variables representing outputs of sub- gates. Though the output expression contains more variables than the input, it remains equisatisfiable, meaning that it is satisfiable if, and only if, the original input equation is satisfiable. When a satisfying assignment of variables is found, those assignments for the introduced variables can simply be discarded.
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For classical logics, it is generally possible to reexpress the question of the validity of a formula to one involving satisfiability, because of the relationships between the concepts expressed in the above square of opposition. In particular φ is valid if and only if ¬φ is unsatisfiable, which is to say it is not true that ¬φ is satisfiable. Put another way, φ is satisfiable if and only if ¬φ is invalid. For logics without negation, such as the positive propositional calculus, the questions of validity and satisfiability may be unrelated.
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In logic, two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice of variables. As a result, equisatisfiability is different from logical equivalence, as two equivalent formulae always have the same models. Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to be correct if the original and resulting formulae are equisatisfiable.
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Observe that the problem FSAT introduced above can be solved using only polynomially many calls to a subroutine which decides the SAT problem: An algorithm can first ask whether the formula \varphi is satisfiable. After that the algorithm can fix variable x_1 to TRUE and ask again. If the resulting formula is still satisfiable the algorithm keeps x_1 fixed to TRUE and continues to fix x_2, otherwise it decides that x_1 has to be FALSE and continues. Thus, FSAT is solvable in polynomial time using an oracle deciding SAT.
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In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation theorem-proving technique for sentences in propositional logic and first-order logic. In other words, iteratively applying the resolution rule in a suitable way allows for telling whether a propositional formula is satisfiable and for proving that a first-order formula is unsatisfiable. Attempting to prove a satisfiable first-order formula as unsatisfiable may result in a nonterminating computation; this problem doesn't occur in propositional logic. The resolution rule can be traced back to Davis and Putnam (1960); Here: p.
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This account is based mainly on . To understand the early history of model theory one must distinguish between syntactical consistency (no contradiction can be derived using the deduction rules for first-order logic) and satisfiability (there is a model). Somewhat surprisingly, even before the completeness theorem made the distinction unnecessary, the term consistent was used sometimes in one sense and sometimes in the other. The first significant result in what later became model theory was Löwenheim's theorem in Leopold Löwenheim's publication "Über Möglichkeiten im Relativkalkül" (1915): :For every countable signature σ, every σ-sentence which is satisfiable is satisfiable in a countable model.
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In computational complexity theory, (SAT, ε-UNSAT) is a language that is used in the proof of the PCP theorem, which relates the language NP to probabilistically checkable proof systems. For a given 3-CNF formula, Φ, and a constant, ε < 1, Φ is in (SAT, ε-UNSAT) if it is satisfiable and not in (SAT, ε-UNSAT) if the maximum number of satisfiable clauses (MAX-3SAT) is less than or equal to (1-ε) times the number of clauses in Φ. If neither of these conditions are true, the membership of Φ in (SAT, ε-UNSAT) is undefined.
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A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem.
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For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.
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One can form a 2-satisfiability instance at random, for a given number n of variables and m of clauses, by choosing each clause uniformly at random from the set of all possible two-variable clauses. When m is small relative to n, such an instance will likely be satisfiable, but larger values of m have smaller probabilities of being satisfiable. More precisely, if m/n is fixed as a constant α ≠ 1, the probability of satisfiability tends to a limit as n goes to infinity: if α < 1, the limit is one, while if α > 1, the limit is zero. Thus, the problem exhibits a phase transition at α = 1.
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In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers. Every first-order formula may be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization (sometimes spelled Skolemnization). The resulting formula is not necessarily equivalent to the original one, but is equisatisfiable with it: it is satisfiable if and only if the original one is satisfiable. Reduction to Skolem normal form is a method for removing existential quantifiers from formal logic statements, often performed as the first step in an automated theorem prover.
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In this case, the rule for expanding the formula has to be applied so that its conclusion(s) are appended to all of these branches that are still open, before one can conclude that the tableau cannot be further expanded and that the formula is therefore satisfiable.
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In mathematical logic, satisfiability and validity are elementary concepts of semantics. A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.See, for example, Boolos and Jeffrey, 1974, chapter 11. A formula is valid if all interpretations make the formula true.
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The basic backtracking algorithm runs by choosing a literal, assigning a truth value to it, simplifying the formula and then recursively checking if the simplified formula is satisfiable; if this is the case, the original formula is satisfiable; otherwise, the same recursive check is done assuming the opposite truth value. This is known as the splitting rule, as it splits the problem into two simpler sub-problems. The simplification step essentially removes all clauses that become true under the assignment from the formula, and all literals that become false from the remaining clauses. The DPLL algorithm enhances over the backtracking algorithm by the eager use of the following rules at each step: ; Unit propagation : If a clause is a unit clause, i.e.
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If satisfiability were also a semi- decidable problem, then the problem of the existence of counter-models would be too (a formula has counter-models iff its negation is satisfiable). So the problem of logical validity would be decidable, which contradicts the Church- Turing theorem, a result stating the negative answer for the Entscheidungsproblem.
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A portfolio is a set of different algorithms or different configurations of the same algorithm. All solvers in a parallel portfolio run on different processors to solve of the same problem. If one solver terminates, the portfolio solver reports the problem to be satisfiable or unsatisfiable according to this one solver. All other solvers are terminated.
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This addition makes the constraint store unsatisfiable. The interpreter then backtracks, removing the last addition from the constraint store. The evaluation of the second clause adds `X=1` and `Y>0` to the constraint store. Since the constraint store is satisfiable and no other literal is left to prove, the interpreter stops with the solution `X=1, Y=1`.
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A decision problem is in NP if it can be solved by a non- deterministic algorithm in polynomial time. An instance of the Boolean satisfiability problem is a Boolean expression that combines Boolean variables using Boolean operators. An expression is satisfiable if there is some assignment of truth values to the variables that makes the entire expression true.
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This follows immediately from the definition of a strongly compact cardinal as being a cardinal such that every collection of formulae of infinitary logic each of length smaller than , that is satisfiable for every subcollection of fewer than formulae, is globally satisfiable. See e.g. . The original De Bruijn–Erdős theorem is the case of this generalization, since a set is finite if and only if its cardinality is less than . However, some assumption such as the one of being a strongly compact cardinal is necessary: if the generalized continuum hypothesis is true, then for every infinite cardinal , there exists a graph of cardinality such that the chromatic number of is greater than , but such that every subgraph of whose vertex set has smaller power than has chromatic number at most .
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The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.
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Building upon previous work on the PCP theorem, Johan Håstad showed that, assuming P ≠ NP, no polynomial-time algorithm for MAX 3SAT can achieve a performance ratio exceeding 7/8, even when restricted to satisfiable instances of the problem in which each clause contains exactly three literals. Both the Karloff–Zwick algorithm and the above simple algorithm are therefore optimal in this sense..
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Every categorical theory is complete. However, the converse does not hold. Any theory T categorical in some infinite cardinal is very close to being complete. More precisely, the Łoś–Vaught test states that if a satisfiable theory has no finite models and is categorical in some infinite cardinal at least equal to the cardinality of its language, then the theory is complete.
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The basic model considers for simplicity needs that are 100% satisfiable and services that 100% satisfy the needs. This means that only the solutions that solve the problem are relevant. This logically means that the need defines the possible satisfying solutions – a set of solutions (many different products/services) that all can fulfill the user need. LUM is not limited to looking at one solution separately.
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Perhaps the simplest problem for alternating machines to solve is the quantified Boolean formula problem, which is a generalization of the Boolean satisfiability problem in which each variable can be bound by either an existential or a universal quantifier. The alternating machine branches existentially to try all possible values of an existentially quantified variable and universally to try all possible values of a universally quantified variable, in the left-to-right order in which they are bound. After deciding a value for all quantified variables, the machine accepts if the resulting Boolean formula evaluates to true, and rejects if it evaluates to false. Thus at an existentially quantified variable the machine is accepting if a value can be substituted for the variable which renders the remaining problem satisfiable, and at a universally quantified variable the machine is accepting if any value can be substituted and the remaining problem is satisfiable.
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By applying the sparsification lemma and then using new variables to split the clauses, one may then obtain a set of O(2εn) 3-CNF formulas, each with a linear number of variables, such that the original k-CNF formula is satisfiable if and only if at least one of these 3-CNF formulas is satisfiable. Therefore, if 3-SAT could be solved in subexponential time, one could use this reduction to solve k-SAT in subexponential time as well. Equivalently, if sk > 0 for any k > 3, then s3 > 0 as well, and the exponential time hypothesis would be true. The limiting value s∞ of the sequence of numbers sk is at most equal to sCNF, where sCNF is the infimum of the numbers δ such that satisfiability of conjunctive normal form formulas without clause length limits can be solved in time O(2δn).
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There are also 15 other ways of setting all the variables so that the formula becomes true. Therefore, the 2-satisfiability instance represented by this expression is satisfiable. Formulas in this form are known as 2-CNF formulas. The "2" in this name stands for the number of literals per clause, and "CNF" stands for conjunctive normal form, a type of Boolean expression in the form of a conjunction of disjunctions.
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The Berry paradox as formulated above arises because of systematic ambiguity in the word "definable". In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.
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This set contains the constraints the interpreter has assumed satisfiable in order to proceed in the evaluation. As a result, if this set is detected unsatisfiable, the interpreter backtracks. Equations of terms, as used in logic programming, are considered a particular form of constraints which can be simplified using unification. As a result, the constraint store can be considered an extension of the concept of substitution that is used in regular logic programming.
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They may find a solution of a problem, but they may fail even if the problem is satisfiable. They work by iteratively improving a complete assignment over the variables. At each step, a small number of variables are changed in value, with the overall aim of increasing the number of constraints satisfied by this assignment. The min- conflicts algorithm is a local search algorithm specific for CSPs and is based on that principle.
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For example, is a positive literal, is a negative literal, is a clause. The formula is in conjunctive normal form; its first and third clauses are Horn clauses, but its second clause is not. The formula is satisfiable, by choosing x1 = FALSE, x2 = FALSE, and x3 arbitrarily, since (FALSE ∨ ¬FALSE) ∧ (¬FALSE ∨ FALSE ∨ x3) ∧ ¬FALSE evaluates to (FALSE ∨ TRUE) ∧ (TRUE ∨ FALSE ∨ x3) ∧ TRUE, and in turn to TRUE ∧ TRUE ∧ TRUE (i.e. to TRUE).
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The model starts from the observation that there is a "user need", i.e. it is expected that there is a "clearly definiable, fully satisfiable want" that the user wants satisfied (it can also be said that the user has a problem that he/she wants solved). So there is a place for a solution, product, or service. The user need defines the set of possible solutions (products, services etc.) that fulfill the user need.
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This equivalence is useful because the definition of first-order satisfiability implicitly existentially quantifies over the evaluation of function symbols. In particular, a first-order formula \Phi is satisfiable if there exists a model M and an evaluation \mu of the free variables of the formula that evaluate the formula to true. The model contains the evaluation of all function symbols; therefore, Skolem functions are implicitly existentially quantified. In the example above, \forall x .
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In contrast to the generative-enumerative (proof- theoretic) approach to syntax assumed by transformational grammar, arc pair grammar takes a model-theoretic approach. In arc pair grammar, linguistic laws and language-specific rules of grammar are formalized as axiomatic logical statements. Sentences of a language, understood as structures of a certain type, follow the set of linguistic laws and language-specific statements. This reduces grammaticality to the logically satisfiable notion of model-theoretic satisfaction.
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In terms of Kripke semantics, S5 is characterized by models where the accessibility relation is an equivalence relation: it is reflexive, transitive, and symmetric. Determining the satisfiability of an S5 formula is an NP-complete problem. The hardness proof is trivial, as S5 includes the propositional logic. Membership is proved by showing that any satisfiable formula has a Kripke model where the number of worlds is at most linear in the size of the formula.
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Backjumping allows saving part of the search by backtracking "more than one variable" in some cases. Constraint learning infers and saves new constraints that can be later used to avoid part of the search. Look-ahead is also often used in backtracking to attempt to foresee the effects of choosing a variable or a value, thus sometimes determining in advance when a subproblem is satisfiable or unsatisfiable. Constraint propagation techniques are methods used to modify a constraint satisfaction problem.
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One sees here again a symmetry because of the disjunctive semantics on the right hand side. If the left side is empty, then one or more right-side propositions must be true. If the right side is empty, then one or more of the left-side propositions must be false. The doubly extreme case ' ⊢ ', where both the antecedent and consequent lists of formulas are empty is "not satisfiable".. In this case, the meaning of the sequent is effectively ' ⊤ ⊢ ⊥ '.
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Called "Requested Range Not Satisfiable" previously. ;417 Expectation Failed : The server cannot meet the requirements of the Expect request-header field. ;418 I'm a teapot (RFC 2324, RFC 7168) :This code was defined in 1998 as one of the traditional IETF April Fools' jokes, in RFC 2324, Hyper Text Coffee Pot Control Protocol, and is not expected to be implemented by actual HTTP servers. The RFC specifies this code should be returned by teapots requested to brew coffee.
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The Karloff–Zwick algorithm, in computational complexity theory, is a randomised approximation algorithm taking an instance of MAX-3SAT Boolean satisfiability problem as input. If the instance is satisfiable, then the expected weight of the assignment found is at least 7/8 of optimal. There is strong evidence (but not a mathematical proof) that the algorithm achieves 7/8 of optimal even on unsatisfiable MAX-3SAT instances. Howard Karloff and Uri Zwick presented the algorithm in 1997..
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R(x,f(x)) is satisfiable if and only if there exists a model M, which contains an evaluation for f, such that \forall x . R(x,f(x)) is true for some evaluation of its free variables (none in this case). This may be expressed in second order as \exists f \forall x . R(x,f(x)). By the above equivalence, this is the same as the satisfiability of \forall x \exists y . R(x,y).
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The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent.
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For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable. SAT is the first problem that was proven to be NP- complete; see Cook–Levin theorem. This means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT.
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An example of uninterpreted functions in SMT-LIB, an input standard for SMT Solvers: (declare-fun f (Int) Int) (assert (= (f 10) 1)) This is satisfiable: `f` is an uninterpreted function. All that is known about `f` is its signature, so it is possible that `f(10) = 1`. (declare-fun f (Int) Int) (assert (= (f 10) 1)) (assert (= (f 10) 42)) This is unsatisfiable: although `f` has no interpretation, it is impossible that it returns different values for the same input.
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Sun was awarded the Rollo Davidson Prize, given annually to a young probability theorist, in 2017. The award citation credited her research (with Jian Ding and Allan Sly) proving the existence of a threshold density such that random -satisfiability instances whose ratio of clauses to variables is below the threshold are almost always satisfiable, and instances whose ratio is above the threshold are almost always unsatisfiable. She was an invited plenary speaker at the 40th Stochastic Processes and their Applications conference.
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In this example propositional logic assertions are checked using functions to represent the propositions a and b. The following Z3 script checks to see if ¬(a ∧ b ) ≡ (¬ a ∨ ¬ b): (declare-fun a () Bool) (declare-fun b () Bool) (assert (not (= (not (and a b)) (or (not a)(not b))))) (check-sat) Result: unsat Note that the script asserts the negation of the proposition of interest. The unsat result means that the negated proposition is not satisfiable, thus proving the desired result (De Morgan's laws).
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In mathematical logic, given an unsatisfiable Boolean propositional formula in conjunctive normal form, a subset of clauses whose conjunction is still unsatisfiable is called an unsatisfiable core of the original formula. Many SAT solvers can produce a resolution graph which proves the unsatisfiability of the original problem. This can be analyzed to produce a smaller unsatisfiable core. An unsatisfiable core is called a minimal unsatisfiable core, if every proper subset (allowing removal of any arbitrary clause or clauses) of it is satisfiable.
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Two of these vertices are connected by an edge if they represent compatible variable assignments for different clauses. That is, there is an edge from to whenever and and are not each other's negations. If denotes the number of clauses in the CNF formula, then the -vertex cliques in this graph represent consistent ways of assigning truth values to some of its variables in order to satisfy the formula. Therefore, the formula is satisfiable if and only if a -vertex clique exists.
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Formally speaking, an SMT instance is a formula in first-order logic, where some function and predicate symbols have additional interpretations, and SMT is the problem of determining whether such a formula is satisfiable. In other words, imagine an instance of the Boolean satisfiability problem (SAT) in which some of the binary variables are replaced by predicates over a suitable set of non- binary variables. A predicate is a binary-valued function of non-binary variables. Example predicates include linear inequalities (e.g.
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If the range is valid, the server sends it to the client with a 206 Partial Content status code and a Content-Range header listing the range sent. If the range is invalid, the server responds with a 416 Requested Range Not Satisfiable status code. Clients which request byte- serving might do so in cases in which a large file has been only partially delivered and a limited portion of the file is needed in a particular range. Byte Serving is therefore a method of bandwidth optimization.
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Every tableau can be considered as a graphical representation of a formula, which is equivalent to the set the tableau is built from. This formula is as follows: each branch of the tableau represents the conjunction of its formulae; the tableau represents the disjunction of its branches. The expansion rules transforms a tableau into one having an equivalent represented formula. Since the tableau is initialized as a single branch containing the formulae of the input set, all subsequent tableaux obtained from it represent formulae which are equivalent to that set (in the variant where the initial tableau is the single node labeled true, the formulae represented by tableaux are consequences of the original set.) A tableau for the satisfiable set {a⋀c,¬a⋁b}: all rules have been applied to every formula on every branch, but the tableau is not closed (only the left branch is closed), as expected for satisfiable sets The method of tableaux works by starting with the initial set of formulae and then adding to the tableau simpler and simpler formulae until contradiction is shown in the simple form of opposite literals.
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For the related MAX-E3SAT problem, in which all clauses in the input 3SAT formula are guaranteed to have exactly three literals, the simple randomized approximation algorithm which assigns a truth value to each variable independently and uniformly at random satisfies 7/8 of all clauses in expectation, irrespective of whether the original formula is satisfiable. Further, this simple algorithm can also be easily derandomized using the method of conditional expectations. The Karloff–Zwick algorithm, however, does not require the restriction that the input formula should have three literals in every clause.
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The hidden transformation reformulates a constraint satisfaction problem in such a way all constraints have at most two variables. The new problem is satisfiable if and only if the original problem was, and solutions can be converted easily from one problem to the other. There are a number of algorithms for constraint satisfaction that work only on constraints that have at most two variables. If a problem has constraints with a larger arity (number of variables), conversion into a problem made of binary constraints allows for execution of these solving algorithms.
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In a probabilistically checkable proof system, a proof is represented as a sequence of bits. An instance of the satisfiability problem should have a valid proof if and only if it is satisfiable. The proof is checked by an algorithm that, after a polynomial- time computation on the input to the satisfiability problem, chooses to examine a small number of randomly chosen positions of the proof string. Depending on what values are found at that sample of bits, the checker will either accept or reject the proof, without looking at the rest of the bits.
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Music critic Jason D. Taylor, writing for the Allmusic, gave the release a mixed to positive review. He argued, "Sinch experiments with hidden samples and awkward vocal tricks to keep the listener unprepared for the next leap into the unknown, and they do a satisfiable job at holding one's attention throughout the remainder of the album." He added that, while seeming "unfocused" and somewhat weak in certain areas, their alternative rock influenced sound can rise beyond contemporaries such as Creed and Default, particularly praising the single "Something More" as a "goldmine".
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If a sentence φ evaluates to True under a given interpretation M, one says that M satisfies φ; this is denoted M \vDash \varphi. A sentence is satisfiable if there is some interpretation under which it is true. Satisfiability of formulas with free variables is more complicated, because an interpretation on its own does not determine the truth value of such a formula. The most common convention is that a formula with free variables is said to be satisfied by an interpretation if the formula remains true regardless which individuals from the domain of discourse are assigned to its free variables.
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It follows now that we need only prove Theorem 2 for formulas φ in normal form. Next, we eliminate all free variables from φ by quantifying them existentially: if, say, x1...xn are free in φ, we form \psi=\exists x_1 ... \exists x_n \phi. If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then eg \psi = \forall x_1 ... \forall x_n eg \phi is provable, and then so is ¬φ, thus φ is refutable. We see that we can restrict φ to be a sentence, that is, a formula with no free variables.
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Schaefer gives a construction allowing an easy polynomial-time reduction from 3-SAT to one-in- three 3-SAT. Let "(x or y or z)" be a clause in a 3CNF formula. Add six fresh boolean variables a, b, c, d, e, and f, to be used to simulate this clause and no other. Then the formula R(x,a,d) ∧ R(y,b,d) ∧ R(a,b,e) ∧ R(c,d,f) ∧ R(z,c,FALSE) is satisfiable by some setting of the fresh variables if and only if at least one of x, y, or z is TRUE, see picture (left).
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Skolem (1934) pioneered the construction of non-standard models of arithmetic and set theory. Skolem (1922) refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic. The resulting axiom is now part of the standard axioms of set theory. Skolem also pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets.
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The rules typically are expressed in terms of finite sets of formulae, although there are logics for which we must use more complicated data structures, such as multisets, lists, or even trees of formulas. Henceforth, "set" denotes any of {set, multiset, list, tree}. If there is such a rule for every logical connective then the procedure will eventually produce a set which consists only of atomic formulae and their negations, which cannot be broken down any further. Such a set is easily recognizable as satisfiable or unsatisfiable with respect to the semantics of the logic in question.
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This clause states one condition under which the statement `A(X,Y)` holds: `X+Y` is greater than zero and both `B(X)` and `C(Y)` are true. As in regular logic programming, programs are queried about the provability of a goal, which may contain constraints in addition to literals. A proof for a goal is composed of clauses whose bodies are satisfiable constraints and literals that can in turn be proved using other clauses. Execution is performed by an interpreter, which starts from the goal and recursively scans the clauses trying to prove the goal.
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The system E is satisfiable in S if there is a map f from X to S, which extends to a semigroup morphism f from X+ to S, such that for all (u,v) in E we have f(u) = f(v) in S. Such an f is a solution, or satisfying assignment, for the system E.Lothaire (2011) p. 444 Two systems of equations are equivalent if they have the same set of satisfying assignments. A system of equations if independent if it is not equivalent to a proper subset of itself. A semigroup is compact if every independent system of equations is finite.
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In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable.
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Satplan (better known as Planning as Satisfiability) is a method for automated planning. It converts the planning problem instance into an instance of the Boolean satisfiability problem, which is then solved using a method for establishing satisfiability such as the DPLL algorithm or WalkSAT. Given a problem instance in planning, with a given initial state, a given set of actions, a goal, and a horizon length, a formula is generated so that the formula is satisfiable if and only if there is a plan with the given horizon length. This is similar to simulation of Turing machines with the satisfiability problem in the proof of Cook's theorem.
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A 2-satisfiability instance in conjunctive normal form can be transformed into an implication graph by replacing each of its disjunctions by a pair of implications. For example, the statement (x_0\lor x_1) can be rewritten as the pair ( eg x_0 \rightarrow x_1), ( eg x_1 \rightarrow x_0). An instance is satisfiable if and only if no literal and its negation belong to the same strongly connected component of its implication graph; this characterization can be used to solve 2-satisfiability instances in linear time. In CDCL SAT-solvers, unit propagation can be naturally associated with an implication graph that captures all possible ways of deriving all implied literals from decision literals, which is then used for clause learning.
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In terms of the implicative form of a 2-CNF formula, this rule amounts to finding two implications \lnot a\Rightarrow b and b\Rightarrow \lnot c, and inferring by transitivity a third implication \lnot a\Rightarrow \lnot c. Krom writes that a formula is consistent if repeated application of this inference rule cannot generate both the clauses (x\lor x) and (\lnot x\lor\lnot x), for any variable x. As he proves, a 2-CNF formula is satisfiable if and only if it is consistent. For, if a formula is not consistent, it is not possible to satisfy both of the two clauses (x\lor x) and (\lnot x\lor\lnot x) simultaneously.
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However, his method leads to a polynomial time bound for solving 2-satisfiability problems. By grouping together all of the clauses that use the same variable, and applying the inference rule to each pair of clauses, it is possible to find all inferences that are possible from a given 2-CNF instance, and to test whether it is consistent, in total time , where is the number of variables in the instance. This formula comes from multiplying the number of variables by the number of pairs of clauses involving a given variable, to which the inference rule may be applied. Thus, it is possible to determine whether a given 2-CNF instance is satisfiable in time .
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In the literature, an approximation ratio for a maximization (minimization) problem of c - ϵ (min: c + ϵ) means that the algorithm has an approximation ratio of c ∓ ϵ for arbitrary ϵ > 0 but that the ratio has not (or cannot) be shown for ϵ = 0. An example of this is the optimal inapproximability — inexistence of approximation — ratio of 7 / 8 + ϵ for satisfiable MAX-3SAT instances due to Johan Håstad. As mentioned previously, when c = 1, the problem is said to have a polynomial-time approximation scheme. An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of size n goes to infinity as n does.
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Another important NL- complete problem is 2-satisfiability (Papadimitriou 1994 Thrm. 16.3), the problem of determining whether a boolean formula in conjunctive normal form with two variables per clause is satisfiable. The problem of unique decipherability of a given variable-length code was shown to be co-NL-complete by ; Rytter used a variant of the Sardinas–Patterson algorithm to show that the complementary problem, finding a string that has multiple ambiguous decodings, belongs to NL. Because of the Immerman–Szelepcsényi theorem, it follows that unique decipherability is also NL-complete. Additional NL- complete problems on Propositional Logic, Algebra, Linear System, Graph, Finite Automata, Context-free Grammar are listed in Jones (1976).
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In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic, this means that for every sentence the theory contains either the sentence or its negation but not both. According to this test, if a satisfiable theory is κ-categorical (there exists an infinite cardinal κ such that it has only one model up to isomorphism of cardinality κ, with κ at least equal to the cardinality of its language) and in addition it has no finite model, then it is complete. This theorem was proved independently by and , after whom it is named.
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The opposites of these concepts are unsatisfiability and invalidity, that is, a formula is unsatisfiable if none of the interpretations make the formula true, and invalid if some such interpretation makes the formula false. These four concepts are related to each other in a manner exactly analogous to Aristotle's square of opposition. The four concepts can be raised to apply to whole theories: a theory is satisfiable (valid) if one (all) of the interpretations make(s) each of the axioms of the theory true, and a theory is unsatisfiable (invalid) if all (one) of the interpretations make(s) each of the axioms of the theory false. It is also possible to consider only interpretations that make all of the axioms of a second theory true.
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In 2000, Richard Kaye published a proof that it is NP-complete to determine whether a given grid of uncovered, correctly flagged, and unknown squares, the labels of the foremost also given, has an arrangement of mines for which it is possible within the rules of the game. The argument is constructive, a method to quickly convert any Boolean circuit into such a grid that is possible if and only if the circuit is satisfiable; membership in NP is established by using the arrangement of mines as a certificate.Kaye (2000). If, however, a minesweeper board is already guaranteed to be consistent, solving it is not known to be NP-complete, but it has been proven to be co-NP-complete.
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Obtaining a tableau where all branches are closed is a way for proving the unsatisfiability of the original set. In the propositional case, one can also prove that satisfiability is proved by the impossibility of finding a closed tableau, provided that every expansion rule has been applied everywhere it could be applied. In particular, if a tableau contains some open (non-closed) branches and every formula that is not a literal has been used by a rule to generate a new node on every branch the formula is in, the set is satisfiable. This rule takes into account that a formula may occur in more than one branch (this is the case if there is at least a branching point "below" the node).
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Classical propositional logic usually has a connective to denote material implication. If we write this connective as ⇒, then the formula A ⇒ B stands for "if A then B". It is possible to give a tableau rule for breaking down A ⇒ B into its constituent formulae. Similarly, we can give one rule each for breaking down each of ¬(A ∧ B), ¬(A ∨ B), ¬(¬A), and ¬(A ⇒ B). Together these rules would give a terminating procedure for deciding whether a given set of formulae is simultaneously satisfiable in classical logic since each rule breaks down one formula into its constituents but no rule builds larger formulae out of smaller constituents. Thus we must eventually reach a node that contains only atoms and negations of atoms.
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In computer science, the Sharp Satisfiability Problem (sometimes called Sharp- SAT or #SAT) is the problem of counting the number of interpretations that satisfies a given Boolean formula, introduced by Valiant in 1979. In other words, it asks in how many ways the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. For example, the formula a\lor eg b is satisfiable by three distinct boolean value assignments of the variables, namely, for any of the assignments (a = TRUE, b = FALSE), (a = FALSE, b = FALSE), (a = TRUE, b = TRUE), we have a\lor eg b = TRUE. #SAT is different from Boolean satisfiability problem (SAT), which asks if there exists a solution of Boolean formula.
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In computational complexity theory, a branch of computer science, the Max/min CSP/Ones classification theorems state necessary and sufficient conditions that determine the complexity classes of problems about satisfying a subset S of boolean relations. They are similar to Schaefer's dichotomy theorem, which classifies the complexity of satisfying finite sets of relations; however, the Max/min CSP/Ones classification theorems give information about the complexity of approximating an optimal solution to a problem defined by S. Given a set S of clauses, the Max constraint satisfaction problem (CSP) is to find the maximum number (in the weighted case: the maximal sum of weights) of satisfiable clauses in S. Similarly, the Min CSP problem is to minimize the number of unsatisfied clauses. The Max Ones problem is to maximize the number of boolean variables in S that are set to 1 under the restriction that all clauses are satisfied, and the Min Ones problem is to minimize this number. When using the classifications below, the problem's complexity class is determined by the topmost classification that it satisfies.
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