63 Sentences With "unsatisfiable" | Random Sentence Generator

" And "it seems our desires are unsatisfiable, what can we do about it?

But it seems our desires are unsatisfiable, what can we do about it?

Charyn's Jerzy, too, is often in the role of unrequited—or perhaps unsatisfiable—lover.

Okay, well, I don't know how to answer ... Our desires are unsatisfiable ... I don't ... That's the human condition.

There are flashes of brilliantly weird and fanciful images — an electric toothbrush gone mad, the staggering progress of a walking palm — but for the most part these landscapes are full of food and drink and sex and shopping, banal and strangely similar, hinting that these are all the same hunger, unsatisfiable.

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.

Algorithms for Computing Minimal Unsatisfiable Subsets Notice the terminology: whereas minimal unsatisfiable core was a local problem with an easy solution, the minimum unsatisfiable core is a global problem with no known easy solution.

Thus, such a core is a local minimum, though not necessarily a global one. There are several practical methods of computing minimal unsatisfiable cores.N. Dershowitz, Z. Hanna, and A. Nadel, A Scalable Algorithm for Minimal Unsatisfiable Core ExtractionStefan Szeider, Minimal unsatisfiable formulas with bounded clause-variable difference are fixed- parameter tractable A minimum unsatisfiable core contains the smallest number of the original clauses required to still be unsatisfiable. No practical algorithms for computing the minimum core are known.

The procedure modifies the tableau in such a way that the formula represented by the resulting tableau is equivalent to the original one. One of these conjunctions may contain a pair of complementary literals, in which case that conjunction is proved to be unsatisfiable. If all conjunctions are proved unsatisfiable, the original set of formulae is unsatisfiable.

For example, to show that C \lor D is unsatisfiable requires showing that C and D are each unsatisfiable; this corresponds to a branching point in the tree with parent C \lor D and children C and D.

Runs of DPLL-based algorithms on unsatisfiable instances correspond to tree resolution refutation proofs.

Since the formula represented by a tableau is the disjunction of the formulae represented by its branches, contradiction is obtained when every branch contains a pair of opposite literals. Once a branch contains a literal and its negation, its corresponding formula is unsatisfiable. As a result, this branch can be now "closed", as there is no need to further expand it. If all branches of a tableau are closed, the formula represented by the tableau is unsatisfiable; therefore, the original set is unsatisfiable as well.

A sufficient condition of tractability is that a non-uniform problem is tractable if the set of its unsatisfiable instances can be expressed by a Boolean Datalog query. In other words, if the set of sets of literals that represent unsatisfiable instances of the non-uniform problem is also the set of sets of literals that satisfy a Boolean Datalog query, then the non-uniform problem is tractable.

Proof confluence is the property of a tableau calculus to obtain a proof for an arbitrary unsatisfiable set from an arbitrary tableau, assuming that this tableau has itself been obtained by applying the rules of the calculus. In other words, in a proof confluent tableau calculus, from an unsatisfiable set one can apply whatever set of rules and still obtain a tableau from which a closed one can be obtained by applying some other rules.

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.

The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses S then there is a Herbrand interpretation that satisfies them. Moreover, Herbrand's theorem states that if S is unsatisfiable then there is a finite unsatisfiable set of ground instances from the Herbrand universe defined by S. Since this set is finite, its unsatisfiability can be verified in finite time. However there may be an infinite number of such sets to check. It is named after Jacques Herbrand.

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.

If a tableau calculus is complete, every unsatisfiable set of formulae has an associated closed tableau. While this tableau can always be obtained by applying some of the rules of the calculus, the problem of which rules to apply for a given formula still remains. As a result, completeness does not automatically imply the existence of a feasible policy of application of rules that always leads to a closed tableau for every given unsatisfiable set of formulae. While a fair proof procedure is complete for ground tableau and tableau without unification, this is not the case for tableau with unification.

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.

At the same time, if one can show unsatisfiability of eg B, there is no need to check eg A. As a result, while there are two possible way to expand eg \Box A \wedge eg \Box B, one of these two ways is always sufficient to prove unsatisfiability if the formula is unsatisfiable. For example, one may expand the tableau by considering an arbitrary world where eg A holds. If this expansion leads to unsatisfiability, the original formula is unsatisfiable. However, it is also possible that unsatisfiability cannot be proved this way, and that the world where eg B holds should have been considered instead.

The tool finds abstract test cases by calculating a finite model for each leaf in a testing tree . Finite models are calculated by restricting the type of each VIS variable to a finite set and then by calculating the Cartesian product between these sets. Each leaf predicate is evaluated on each element of this Cartesian product until one satisfies the predicate (meaning that an abstract test case was found) or until it is exhausted (meaning that either the test class is unsatisfiable or the finite model is inadequate). In the last case, the user has the chance to assist the tool in finding the right finite model or to prune the test class because it is unsatisfiable.

She grew up in the Capt. Jack C. Hardy House in Brookhaven, Mississippi. With Ragsdale received her early education from her mother. At an early age, Ragsdale became an unsatisfiable reader, always seeking the weird, the unreal, the mystic; or else, the vivid, the passionate, the glowing in prose and poetry.

A Markov logic network (MLN) is a probabilistic logic which applies the ideas of a Markov network to first-order logic, enabling uncertain inference. Markov logic networks generalize first-order logic, in the sense that, in a certain limit, all unsatisfiable statements have a probability of zero, and all tautologies have probability one.

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.

Constraints encountered during this scan are placed in a set called constraint store. If this set is found out to be unsatisfiable, the interpreter backtracks, trying to use other clauses for proving the goal. In practice, satisfiability of the constraint store may be checked using an incomplete algorithm, which does not always detect inconsistency.

In contrast, the CNF formula a ∧ ¬a, consisting of two clauses of one literal, is unsatisfiable, since for a=TRUE or a=FALSE it evaluates to TRUE ∧ ¬TRUE (i.e., FALSE) or FALSE ∧ ¬FALSE (i.e., again FALSE), respectively. For some versions of the SAT problem, it is useful to define the notion of a generalized conjunctive normal form formula, viz.

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.

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`.

The resolution rule, as defined by Robinson, also incorporated factoring, which unifies two literals in the same clause, before or during the application of resolution as defined above. The resulting inference rule is refutation-complete, p. 350 (= p. 286 in the 1st edition of 1995) in that a set of clauses is unsatisfiable if and only if there exists a derivation of the empty clause using only resolution, enhanced by factoring.

A tableau calculus is a set of rules that allows building and modification of a tableau. Propositional tableau rules, tableau rules without unification, and tableau rules with unification, are all tableau calculi. Some important properties a tableau calculus may or may not possess are completeness, destructiveness, and proof confluence. A tableau calculus is called complete if it allows building a tableau proof for every given unsatisfiable set of formulae.

The first use of the labeling literal is to actual check satisfiability or partial satisfiability of the constraint store. When the interpreter adds a constraint to the constraint store, it only enforces a form of local consistency on it. This operation may not detect inconsistency even if the constraint store is unsatisfiable. A labeling literal over a set of variables enforces a satisfiability check of the constraints over these variables.

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.

The resolution rule is a single rule of inference that, together with unification, is sound and complete for first-order logic. As with the tableaux method, a formula is proved by showing that the negation of the formula is unsatisfiable. Resolution is commonly used in automated theorem proving. The resolution method works only with formulas that are disjunctions of atomic formulas; arbitrary formulas must first be converted to this form through Skolemization.

The SAT problem is self-reducible, that is, each algorithm which correctly answers if an instance of SAT is solvable can be used to find a satisfying assignment. First, the question is asked on the given formula Φ. If the answer is "no", the formula is unsatisfiable. Otherwise, the question is asked on the partly instantiated formula Φ{x1=TRUE}, i.e. Φ with the first variable x1 replaced by TRUE, and simplified accordingly.

Without proof confluence, the application of a 'wrong' rule may result in the impossibility of making the tableau complete by applying other rules. Propositional tableaux and tableaux without unification have strongly complete proof procedures. In particular, a complete proof procedure is that of applying the rules in a fair way. This is because the only way such calculi cannot generate a closed tableau from an unsatisfiable set is by not applying some applicable rules.

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.

A tableau calculus is simply a set of rules that tells how a tableau can be modified. A proof procedure is a method for actually finding a proof (if one exists). In other words, a tableau calculus is a set of rules, while a proof procedure is a policy of application of these rules. Even if a calculus is complete, not every possible choice of application of rules leads to a proof of an unsatisfiable set.

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..

Satisfiability can sometimes be established by enforcing a form of local consistency and then checking the existence of an empty domain or constraint relation. This is in general a correct but incomplete unsatisfiability algorithm: a problem may be unsatisfiable even if no empty domain or constraint relation is produced. For some forms of local consistency, this algorithm may also require exponential time. However, for some problems and for some kinds of local consistency, it is correct and polynomial-time.

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.

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.

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.

Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv., whether its negation is unsatisfiable). The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers—a feature absent from sentences of propositional logic.

In classical logic, particularly in propositional and first-order logic, a proposition \varphi is a contradiction if and only if \varphi\vdash\bot. Since for contradictory \varphi it is true that \vdash\varphi\rightarrow\psi for all \psi (because \bot\rightarrow\psi), one may prove any proposition from a set of axioms which contains contradictions. This is called the "principle of explosion", or "ex falso quodlibet" ("from falsity, anything follows"). In a complete logic, a formula is contradictory if and only if it is unsatisfiable.

If a variable appears in a real or finite domain expression, it can only take a value in the reals or the finite domain. Such a variable cannot take a term made of a functor applied to other terms as a value. The constraint store is unsatisfiable if a variable is bound to take both a value of the specific domain and a functor applied to terms. After a constraint is added to the constraint store, some operations are performed on the constraint store.

Such problems are usually solved via search, in particular a form of backtracking or local search. Constraint propagation are other methods used on such problems; most of them are incomplete in general, that is, they may solve the problem or prove it unsatisfiable, but not always. Constraint propagation methods are also used in conjunction with search to make a given problem simpler to solve. Other considered kinds of constraints are on real or rational numbers; solving problems on these constraints is done via variable elimination or the simplex algorithm.

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.

Propositional proof systems can be interpreted as nondeterministic algorithms for recognizing tautologies. Proving a superpolynomial lower bound on a proof system P thus rules out the existence of a polynomial-time algorithm for SAT based on P. For example, runs of DPLL algorithm on unsatisfiable instances correspond to tree-like Resolution refutations. Therefore, exponential lower bounds for tree-like Resolution (see below) rule out the existence of efficient DPLL algorithms for SAT. Similarly, exponential Resolution lower bounds imply that SAT solvers based on Resolution, such as CDCL algorithms cannot solve SAT efficiently (in worst-case).

In the case of weak connectedness, confluence holds provided that the clause used for expanding the root is relevant to unsatisfiability, that is, it is contained in a minimally unsatisfiable subset of the set of clauses. Unfortunately, the problem of checking whether a clause meets this condition is itself a hard problem. In spite of non-confluence, a closed tableau can be found using search, as presented in the "Searching for a closed tableau" section above. While search is made necessary, connectedness reduces the possible choices of expansion, thus making search more efficient.

If we assume that L1(Y) is proportional to Y, this amounts to a constant shift. He concludes that if the intercept with the IS curve is on the presumed horizontal section of the LM curve, then 'merely monetary means will not force down the rate of interest any further'. He regards this possibility as distinguishing Keynes's economic theories from those of the classics, and as characterising them as 'the economics of depression'. In later economic circumstances the risk of speculators having an unsatisfiable demand for money disappeared.

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.

A general solution for this problem is that of searching the space of tableaux until a closed one is found (if any exists, that is, the set is unsatisfiable). In this approach, one starts with an empty tableau and then recursively applies every possible applicable rule. This procedure visits a (implicit) tree whose nodes are labeled with tableaux, and such that the tableau in a node is obtained from the tableau in its parent by applying one of the valid rules. Since each branch can be infinite, this tree has to be visited breadth-first rather than depth-first.

In particular, this is the leaf containing the literal of the clause that unifies with the negation of a literal in the branch (or the negation of the literal in the parent, in case of strong connection). Both conditions of connectedness lead to a complete first-order calculus: if a set of clauses is unsatisfiable, it has a closed connected (strongly or weakly) tableau. Such a closed tableau can be found by searching in the space of tableaux as explained in the "Searching for a closed tableau" section. During this search, connectedness eliminates some possible choices of expansion, thus reducing search.

Unsatisfiability may then be proved from the subset of formulae referring to a single successor. This holds if a world may have more than one successor, which is true for most modal logic. If this is the case, a formula like eg \Box A \wedge eg \Box B is true if a successor where eg A holds exists and a successor where eg B holds exists. In the other way around, if one can show unsatisfiability of eg A in an arbitrary successor, the formula is proved unsatisfiable without checking for worlds where eg B holds.

Equality constraints on terms can be simplified, that is solved, via unification: A constraint `t1=t2` can be simplified if both terms are function symbols applied to other terms. If the two function symbols are the same and the number of subterms is also the same, this constraint can be replaced with the pairwise equality of subterms. If the terms are composed of different function symbols or the same functor but on different number of terms, the constraint is unsatisfiable. If one of the two terms is a variable, the only allowed value the variable can take is the other term.

Algorithms for finding strongly connected components may be used to solve 2-satisfiability problems (systems of Boolean variables with constraints on the values of pairs of variables): as showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable v such that v and its complement are both contained in the same strongly connected component of the implication graph of the instance.. Strongly connected components are also used to compute the Dulmage–Mendelsohn decomposition, a classification of the edges of a bipartite graph, according to whether or not they can be part of a perfect matching in the graph..

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.

Either the CNF formula Φ is found to comprise a consistent set of literals--that is, there is no `l` and `¬l` for any literal `l` in the formula. If this is the case, the variables can be trivially satisfied by setting them to the respective polarity of the encompassing literal in the valuation. Otherwise, when the formula contains an empty clause, the clause is vacuously false because a disjunction requires at least one member that is true for the overall set to be true. In this case, the existence of such a clause implies that the formula (evaluated as a conjunction of all clauses) cannot evaluate to true and must be unsatisfiable.

Equality between terms is a kind of constraint that is always present, as the interpreter generates equality of terms during execution. As an example, if the first literal of the current goal is `A(X+1)` and the interpreter has chosen a clause that is `A(Y-1):-Y=1` after rewriting is variables, the constraints added to the current goal are `X+1=Y-1` and Y=1. The rules of simplification used for function symbols are obviously not used: `X+1=Y-1` is not unsatisfiable just because the first expression is built using `+` and the second using `-`. Reals and function symbols can be combined, leading to terms that are expressions over reals and function symbols applied to other terms.

The modern history of the concept of the love addict – ignoring such precursors as Robert Burton's dictum that 'love extended is mere madness'Robert Burton, The Anatomy of Melancholy (New York 1951) p. 769 – extends to the early decades of the 20th century. Freud's study of the Wolf Man highlighted 'his liability to compulsive attacks of falling physically in love ... a compulsive falling in love that came on and passed off by sudden fits';Sigmund Freud, Case Studies II (PFL 9) p. 273 and p. 361 but it was Sandor Rado who in 1928 first popularized the term "love addict" – 'a person whose needs for more love, more succor, more support grow as rapidly as the frustrated people around her try to fill up what is, in effect, a terrible and unsatisfiable inner emptiness.

The results of look ahead is used to decide the next variable to evaluate and the order of values to give to this variable. In particular, for any unassigned variable and value, look-ahead estimates the effects of setting that variable to that value. The choice of the next variable and the choice of the next value to give it are complementary, in that the value is typically chosen in such a way a solution (if any) is found as quickly as possible, while the next variable is typically chosen in such a way unsatisfiability (if the current partial solution is unsatisfiable) is proven as quickly as possible. The choice of the next variable to evaluate is particularly important, as it may produce exponential differences in running time.

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.

To Guy, she pretends she is a frigid, sexually timid virgin: she tells him that her childhood in a dreadful orphanage and her friendship with a tragic girl called Enola Gay who is raped by a "pitiless Iraqi" and who produces a child called Little Boy, has left her unable to form a sexual relationship with any man, but that Guy has awakened the possibility in her. Feigning love for Guy, she teases him sexually at every opportunity, pretending she is too afraid and too unready to "go the whole way" with him, until his unsatisfiable and excruciating lust induces him to leave his wife and child and to give her a very large sum of money which he believes will help her bring the fictional Enola Gay and Little Boy to London. Nicola insists that Guy leave his wife and son to consummate their relationship, and Guy does so, destroying his family life. To Keith, Nicola styles herself as a rich, knowing woman of the world, a former one-night stand of the Shah of Iran, who recognises him for what he truly is – a darts prodigy and future darts and TV personality.